Beyond subjective and objective in statistics
aa r X i v : . [ s t a t . O T ] A ug Beyond subjective and objective in statistics ∗ Andrew Gelman † Christian Hennig ‡ Abstract
We argue that the words “objectivity” and “subjectivity” in statistics discourse are usedin a mostly unhelpful way, and we propose to replace each of them with broader collections ofattributes, with objectivity replaced by transparency , consensus , impartiality , and correspon-dence to observable reality , and subjectivity replaced by awareness of multiple perspectives and context dependence . The advantage of these reformulations is that the replacement terms donot oppose each other. Instead of debating over whether a given statistical method is subjec-tive or objective (or normatively debating the relative merits of subjectivity and objectivity instatistical practice), we can recognize desirable attributes such as transparency and acknowledg-ment of multiple perspectives as complementary goals. We demonstrate the implications of ourproposal with recent applied examples from pharmacology, election polling, and socioeconomicstratification.
1. Introduction
We can’t do statistics without data, and as statisticians much of our efforts revolve around modelingthe links between data and substantive constructs of interest. We might analyze national surveydata on purchasing decisions as a way of estimating consumers’ responses to economic conditions; orgather blood samples over time on a sample of patients with the goal of estimating the metabolismof a drug, with the ultimate goal of coming up with a more effective dosing schedule; or we mightbe performing a more exploratory analysis, seeking clusters in a multivariate dataset with the aimof discovering patterns not apparent in simple averages of raw data.As applied researchers we are continually reminded of the value of integrating new data intoan analysis, and the balance between data quality and quantity. In some settings it is possible toanswer questions of interest using a single clean dataset, but more and more we are finding thatthis simple textbook approach does not work.External information can come in many forms, including (a) recommendations on what vari-ables to adjust for non-representativeness of a survey or imbalance in an experiment or observationalstudy; (b) the extent to which outliers should be treated as regular, erroneous, or as indicatingsomething that is meaningful but essentially different from the main body of observations; (c)substantial information on the role of variables, including potential issues with measurement, con-founding, and substantially meaningful effect sizes; (d) population distributions that are used inpoststratification, age adjustment, and other procedures that attempt to align inferences to a com-mon population of interest; (e) restrictions such as smoothness or sparsity that serve to regularizeestimates in high-dimensional settings; (f) the choice of functional form in a regression model (whichin economics might be chosen to work with a particular utility function, or in public health might ∗ We thank Sebastian Weber, Jay Kadane, Arthur Dempster, Michael Betancourt, Michael Zyphur, E. J. Wagen-makers, Deborah Mayo, James Berger, Prasanta Bandyopadhyay, Laurie Paul, Jan-Willem Romeijn, Gianluca Baio,Keith O’Rourke, and Laurie Davies for helpful comments. † Department of Statistics and Department of Political Science, Columbia University, New York. ‡ Department of Statistical Science, University College London. e motivated based on success in similar studies in the literature); and (g) numerical informationabout particular parameters in a model. Of all these, only the final item is traditionally given thename “prior information” in a statistical analysis, but all can be useful in serious applied work.Other relevant information concerns not the data generating process but rather how the data andresults of an analysis are to be used or interpreted.We were motivated to write the present paper because we felt that our applied work, and that ofothers, was impeded because of the conventional framing of certain statistical analyses as subjective.It seemed to us that, rather than being in opposition, subjectivity and objectivity both had virtuesthat were relevant in making decisions about statistical analyses. We have earlier noted (Gelmanand O’Rourke, 2015) that statisticians typically choose their procedures based on non-statisticalcriteria, and philosophical traditions and even the labels attached to particular concepts can affectreal-world practice.In this paper we reassess objectivity and subjectivity, exploding each into several sub-concepts,and we demonstrate the relevance of these ideas for three of our active applied research projects: ahierarchical population model in pharmacology, a procedure for adjustment of opt-in surveys, anda cluster analysis of data on socioeconomic stratification. We hope that readers will likewise seethe relevance of these ideas in their own applied work, where decisions must be made about howto combine information of varying quality from different sources.
The continuing interest in and discussion of objectivity and subjectivity in statistics is, we believe,a necessary product of a fundamental tension in science: On one hand, scientific claims should beimpersonal in the sense that a scientific argument should be understandable by anyone with thenecessary training, not just by the person promulgating it, and it should be possible for scientificclaims to be evaluated and tested by outsiders. On the other hand, the process of scientific inferenceand discovery involves individual choices; indeed, scientists and the general public celebrate thebrilliance and inspiration of greats such as Einstein, Darwin, and the like, recognizing the rolesof their personalities and individual experiences in shaping their theories and discoveries, andphilosophers of science have studied the interplay between personal attitudes and scientific theories(Kuhn, 1962). Thus it is clear that objective and subjective elements arise in the practice of science,and similar considerations hold in statistics.Within statistics, though, discourse on objectivity and subjectivity is at an impasse. Ideallythese concepts would be part of a consideration of the role of different sorts of information andassumptions in statistical analysis, but instead they often seemed to be used in restrictive andmisleading ways.One problem is that the terms “objective” and “subjective” are loaded with so many associationsand are often used in a mixed descriptive/normative way. Scientists whose methods are branded assubjective have the awkward choice of either saying, No, we are really objective, or else embracingthe subjective label and turning it into a principle. From the other direction, scientists who usemethods labeled as objective often seem so intent on eliminating subjectivity from their analyses,that they end up censoring themselves. This happens, for example, when researchers rely on p -values but refuse to recognize when their choice of analysis is contingent on data and that the theorybehind the p -values is therefore invalidated (as discussed by Simmons, Nelson, and Simonsohn,2011, and Gelman and Loken, 2014): significance testing is often used as a tool for a misguidedideology that leads researchers to hide, even from themselves, the iterative searching process bywhich a scientific theory is mapped into a statistical model or choice of data analysis (Box, 1983).2ore generally, misguided concerns about subjectivity can lead researchers to avoid incorporatingrelevant and available information into their analyses and adapting the analyses appropriately totheir research questions and potential uses of their results.Many users of the terms “objective” and “subjective” in discussions concerning statistics do notacknowledge that these terms are quite controversial in the philosophy of science, and that theyare used with a variety of different meanings and are therefore prone to misunderstandings.
2. Our proposal
We propose when talking about statistics to replace, where ever possible, the words “objectiv-ity” and “subjectivity” with broader collections of attributes, with objectivity replaced by trans-parency , consensus , impartiality , and correspondence to observable reality , and subjectivity replacedby awareness of multiple perspectives and context dependence .The advantage of this reformulation is that the replacement terms do not oppose each other.Instead of debating over whether a given statistical method is subjective or objective (or normativelydebating the relative merits of subjectivity and objectivity in statistical practice), we can recognizeattributes such as transparency and acknowledgment of multiple perspectives as complementarygoals. Merriam-Webster defines “objective” as “based on facts rather than feelings or opinions: not in-fluenced by feelings” and “existing outside of the mind: existing in the real world” (actually theconcept is quite controversial, see Section 4.3). Science is practiced by human beings, who only haveaccess to the real world through interpretation of their perceptions. Taking objectivity seriouslyas an ideal, scientists need to make the sharing of their perceptions and interpretations possible.When applied to statistics, the implication is that the choices in the data analysis (including theprior distribution, if any, but also the model for the data, methodology, and the choice of whatinformation to include in the first place) should be motivated based on factual, externally verifiableinformation and transparent criteria. This is similar to the idea of the concept of “institutionaldecision analysis” (Section 9.5 of Gelman, Carlin, et al., 2013), under which the mathematics offormal decision theory can be used to ensure that decisions can be justified based on clearly-statedcriteria. Different stakeholders will disagree on decision criteria, and different scientists will differon statistical modeling decisions, so, in general, there is no unique “objective” analysis, but wecan aim at communicating and justifying analyses in ways that support scrutiny and eventuallyconsensus. Similar thoughts have motivated the slogan “transparency is the new objectivity” injournalism (Weinberger, 2009).In the context of statistical analysis, a key aspect of objectivity is therefore a process of trans-parency , in which the choices involved are justified based on external, potentially verifiable sourcesor at least transparent considerations (ideally accompanied by sensitivity analyses if such consid-erations leave alternative options open), a sort of “paper trail” leading from external information,through modeling assumptions and decisions about statistical analysis, all the way to inferencesand decision recommendations. But transparency is not enough. We hold that science aims at consensus in potentially free exchange (see Section 4.4 for elaboration), which is one reason thatthe current crisis of non-replication is taken so seriously in psychology (Yong, 2012). Transparencycontributes to this building of consensus by allowing scholars to trace the sources and information3sed in statistical reasoning (Gelman and Basbøll, 2013). Furthermore, scientific consensus, as faras it deserves to be called “objective,” requires rationales, clear arguments and motivation, andelucidation how this relates to already existing knowledge. Following generally accepted rules andprocedures counters the dependence of results on the personalities of the individual researchers,although there is always a danger that such generally accepted rules and procedures are inappro-priate or suboptimal for the specific situation at hand. In any case, consensus can only be achievedif researchers attempt to be impartial by taking into account competing perspectives, avoiding tofavor pre-chosen hypotheses, and being open to criticism.The world outside the observer’s mind plays a key role in usual concepts of objectivity. Findingout about the real world is seen by many as the major objective of science, and this suggestscorrespondence to reality as the ultimate source of scientific consensus. This idea is not withoutits problems and meets some philosophical opposition; see Section 4.3. We acknowledge that the“real world” is only accessible to human beings through observation, and that scientific observationand measurement cannot be independent of human preconceptions and theories. As statisticianswe are concerned with making general statements based on systematized observations, and thismakes correspondence to observed reality a core concern regarding objectivity. This is not meantto imply that empirical statements about observations are the only meaningful ones that can bemade about reality; we think that scientific theories that cannot be verified (but potentially befalsified) by observations are meaningful thought constructs, particularly because observations aretruly independent of thought constructs.Formal statistical methods contribute to objectivity as far as they contribute to the fulfillmentof these desiderata, particularly by making procedures and their implied rationales transparent andunambiguous.For example, Bayesian statistics is commonly characterized as “subjective” by Bayesians andnon-Bayesians alike. But depending on how exactly prior distributions are interpreted and used(see Sections 5.3–5.5), they fulfill or aid some or all of the virtues listed above. Priors make theresearchers’ prior point of view transparent; different approaches of interpreting them provide dif-ferent rationales for consensus; “objective Bayesians” (see Section 5.4) try to make them impartial;and if suitably interpreted (see Section 5.5) they can be properly grounded in observations.
Merriam-Webster defines “subjective” as “relating to the way a person experiences things in hisor her own mind” and “based on feelings or opinions rather than facts.” Science is normallyseen as striving for objectivity, and therefore acknowledging subjectivity is not popular in science.But as noted above already, reality and the facts are only accessible through individual personalexperiences. Different people bring different information and different viewpoints to the table,and they will use scientific results in different ways. In order to enable clear communication andconsensus, differing perspectives need to be acknowledged, which contributes to transparency andthus to objectivity. Therefore, subjectivity is important to the scientific process. Subjectivity isvaluable in statistics in that it represents a way to incorporate the information coming from differingperspectives.We propose to replace the concept of “subjectivity” with awareness of multiple perspectives and context dependence . To the extent that subjectivity in statistics is a good thing, it is becauseinformation truly is dispersed, and, for any particular problem, different stakeholders have differentgoals. A counterproductive implication of the idea that science should be “objective” is that thereis a tendency in the communication of statistical analyses to either avoid or hide decisions that4annot be made in an automatic, seemingly “objective” fashion by the available data. Given thatall observations of reality depend on the perspective of an observer, interpreting science as strivingfor a unique (“objective”) perspective is illusory. Multiple perspectives are a reality to be reckonedwith and should not be hidden. Furthermore, by avoiding personal decisions, researchers oftenwaste opportunities to adapt their analyses appropriately to the context, the specific backgroundand their specific research aims, and to communicate their perspective more clearly. Therefore wesee the acknowledgment of multiple perspectives and context dependence as virtues, making clearerin which sense subjectivity can be productive and helpful.The term “subjective” is often used to characterize aspects of certain statistical proceduresthat cannot be derived in an automatic manner from the data to be analyzed, such as Bayesianprior distributions and tuning parameters (for example, the proportion of trimmed observationsin trimmed means, or the threshold in wavelet smoothing). Such decisions are entry points formultiple perspectives and context dependence. The first decisions of this kind are typically thechoice of data to be analyzed and the family of statistical models to be fit.To connect with the other half of our proposal, the recognition of different perspectives shouldbe done in a transparent way. We should not say we set a tuning parameter to 2.5 (say) justbecause that is our belief. Rather, we should justify the choice explaining clearly how it supportsthe research aims. This could be by embedding the choice in a statistical model that can ultimatelybe linked back to observable reality and empirical data, or by reference to desirable characteristics(or avoidance of undesirable artifacts) of the methodology given the use of the chosen parameter;actually, many tuning parameters are related to such characteristics and aims of the analysis ratherthan to some assumed underlying “belief” (see Section 3.3). In some cases, such a justificationmay be imprecise, for example because background knowledge may be only qualitative and notquantitative or not precise enough to tell possible alternative choices apart, but often it can beargued that even then conscious tuning or specification of a prior distribution comes with benefitscompared to using default methods of which the main attraction often is that seemingly “subjective”decisions can be avoided.To consider an important example, regularization requires such decisions. Default priors onregression coefficients are used to express the belief that coefficients are typically close to zero,and from a non-Bayesian perspective, lasso shrinkage can be interpreted as encoding an externalassumption of sparsity. Sparsity assumptions can be connected to an implicit or explicit model inwhich problems are in some sense being sampled from some distribution or probability measure ofpossible situations; see Section 5.5. This general perspective (which can be seen as Bayesian withan implicit prior on states of nature, or classical with an implicit reference set for the evaluation ofstatistical procedures) provides a potential basis to connect choices to experience; at least it makestransparent what kind of view of reality is encoded in the choices.Tibshirani (2014) writes that enforcing sparsity is not primarily motivated by beliefs aboutthe world, but rather by benefits such as computability and interpretability, hinting at the factthat considerations other than being “close to the real world” often play an important role instatistics and more generally in science. Even in areas such as social science where no underlyingtruly sparse structure exists, imposing sparsity can have advantages such as supporting stability(Gelman, 2013).In a wider sense, if one is performing a linear or logistic regression, for example, and consideringoptions of maximum likelihood, lasso, or hierarchical Bayes with a particular structure of priors,all of these choices are “subjective” in the sense of encoding aims regarding possible outputs andassumptions, and all are “objective” as far as these aims and assumptions are made transparent andthe assumptions can be justified based on past data and ultimately be checked given enough future5ata. So the conventional labeling of Bayesian analyses or regularized estimates as “subjective”misses the point.Alternatively to basing it on past data, the choice of tuning parameter can be based on knowl-edge of the impact of the choice on results and a clear explanation why a certain impact is desiredor not. In robust statistics, for example, the breakdown point of some methods can be tuned andmay be chosen lower than the optimal 50%, because if there is a too large percentage of datadeviating strongly from the majority, one may rather want the method to deliver a compromisebetween all observations, but if the percentage of outliers is quite low, one may rather want themto be disregarded, with borderline percentages depending on the application (particularly on towhat extent outliers are interpreted as erroneous observations rather than as somewhat special butstill relevant cases).
To summarize the above discussion, virtues that are often referred to as “objective” include:1. Transparency:(a) Clear and unambiguous definitions of concepts,(b) Open planning and following agreed protocols,(c) Full communication of reasoning, procedures, and potential limitations;2. Consensus:(a) Accounting for relevant knowledge and existing related work,(b) Following generally accepted rules where possible and reasonable,(c) Provision of rationales for consensus and unification;3. Impartiality:(a) Thorough consideration of relevant and potentially competing theories and points ofview,(b) Thorough consideration and if possible removal of potential biases: factors that mayjeopardize consensus and the intended interpretation of results,(c) Openness to criticism and exchange;4. Correspondence to observable reality:(a) Clear connection of concepts and models to observables,(b) Clear conditions for reproduction, testing, and falsification.This last bit is a challenge in statistics, as reproduction, testing, and falsification can only beassessed probabilistically in any real, finite-sample setting.What about subjectivity? The term “subjective” is often used as opposite to “objective” andas such often meant to be opposed to scientific virtues, or to be something that cannot fully beavoided and that therefore has to be only grudgingly accepted.But subjective perspectives are the building blocks for scientific consensus, and therefore thereare also scientific virtues associated with subjectivity:6. Awareness of multiple perspectives,2. Awareness of context dependence:(a) Recognition of dependence on specific contexts and aims,(b) Honest acknowledgment of the researcher’s position, goals, experiences, and subjectivepoint of view.In the subsequent discussion we shall label the items in the above lists as O1a–O4b or O1–O4 for groups of items (“O” for “connected to objectivity”), and S1, S2 (S2a, S2b) for the itemsconnected to subjectivity. Our intention is to sketch a system of virtues that allows a more preciseand detailed discussion where issues of objectivity and subjectivity are at stake.We are aware that in some situations some of these virtues may oppose each other, for example“consensus” can contradict “awareness of multiple perspectives,” and indeed dissent is essential toscientific progress. This tension between impersonal consensus and creative debate is an unavoidableaspect of science. Sometimes the consensus can only be that there are different legitimate points ofview. Furthermore, the listed virtues are not all fully autonomous; clear reference to observationsmay be both a main rationale for consensus and a key contribution to transparency; and the threesubjective virtues contribute to both transparency and openness to criticism and exchange.Not all items on the list apply to all situations. For example, in the following section we willapply the list to the foundations of statistics, but the items O1c and S2b rather apply to specificstudies.
3. Applied examples
In conventional statistics, assumptions are commonly minimized. Classical statistics and econo-metrics is often framed in terms of robustness, with the goal being methods that work with minimalassumptions. But the decisions about what information to include and how to frame the model—these are typically buried, not stated formally as assumptions but just baldly stated: “Here is theanalysis we did . . . ,” sometimes with the statement or implication that these have a theoreticalbasis but typically with little clear connection between subject-matter theory and details of mea-surements. From the other perspective, Bayesian analyses are often boldly assumption-based butwith the implication that these assumptions, being subjective, need no justification and cannot bechecked from data.We would like statistical practice, Bayesian and otherwise, to move toward more transparencyregarding the steps linking theory and data to models, and recognition of multiple perspectives inthe information that is included in this paper trail and this model. In this section we show howwe are trying to move in this direction in some of our recent research projects. We present theseexamples not as any sort of ideals but rather to demonstrate how we are grappling with these ideasand, in particular, the ways in which active awareness of the concepts of transparency, consensus,impartiality, correspondence to observable reality, multiple perspectives and context dependence ischanging our applied work.
Statistical inference in pharmacokinetics/pharmacodynamics involves many challenges: data areindirect and often noisy; the mathematical models are nonlinear and computationally expensive,requiring the solution of differential equations; and parameters vary by person but often with only7 small amount of data on each experimental subject. Hierarchical models and Bayesian inferenceare often used to get a handle on the many levels of variation and uncertainty (see, for example,Sheiner, 1984, and Gelman, Bois, and Jiang, 1996).One of us is currently working on a project in drug development involving a Bayesian modelthat was difficult to fit, even when using advanced statistical algorithms and software. Followingthe so-called folk theorem of statistical computing (Gelman, 2008), we suspected that the problemswith computing could be attributed to a problem with our statistical model. In this case, the issuedid not seem to be lack of fit, or a missing interaction, or unmodeled measurement error—problemswe had seen in other settings of this sort. Rather, the fit appeared to be insufficiently constrained,with the Bayesian fitting algorithm being stuck going through remote regions of parameter spacethat corresponded to implausible or unphysical parameter values.In short, the model as written was only weakly identified, and the given data and priors wereconsistent with all sorts of parameter values that did not make scientific sense. Our iterativeBayesian computation had poor convergence—that is, the algorithm was having difficulty approx-imating the posterior distribution—and the simulations were going through zones of parameterspace that were not consistent with the scientific understanding of our pharmacology colleagues.To put it another way, our research team had access to prior information that had not beenincluded in the model. So we took the time to specify a more informative prior. The initial modelthus played the role of a placeholder or default which could be elaborated as needed, following theiterative prescription of falsificationist Bayesianism (Box, 1980, Gelman et al., 2013, Section 5.5).In our experience, informative priors are not so common in applied Bayesian inference, andwhen they are used, they often seem to be presented without clear justification. In this instance,though, we decided to follow the principle of transparency and write a note explaining the genesisof each prior distribution. To give a sense of what we’re talking about, we present a subset of thesenotes here: • γ : mean of population distribution of log(BVA latent j / /
50) = 0 .
18 to indicate that we’re pretty sure the mean is between 40 and 60. • γ : mean of pop dist of log( k in j /k out j ), centered at 3.7 because we started with − . k in and − . k out , specified from the literature about the disease. We use a sd of 0.5 to represent a certain amountof ignorance: we’re saying that our prior guess for the population mean of k in /k out could easily be offby a factor of exp(0 .
5) = 1 . • γ : mean of pop dist of log k out j , centered at − . • γ : log E , centered at 0 with sd 2.0 because that’s what we were given earlier. We see this sort of painfully honest justification as a template for future Bayesian data analyses.The above snippet certainly does not represent an exemplar of best practices, but we see it as a“good enough” effort that presents our modeling decisions in the context in which they were made.To label this prior specification as “objective” or “subjective” would miss the point. Rather, wesee it as having some of the virtues of objectivity and subjectivity—notably, transparency (O1) andsome aspects of consensus (O2) and awareness of multiple perspectives (S1)—while recognizing itsclear imperfections and incompleteness. Other desirable features would derive from other aspectsof the statistical analysis—for example, we use external validation to approach correspondence toobservable reality (O4), and our awareness of context dependence (S2) comes from the placementof our analysis within the larger goal, which is to model dosing options for a particular drug.8ne concern about our analysis which we have not yet thoroughly addressed is sensitivity tomodel assumptions. We have established that the prior distribution makes a difference but it ispossible that different reasonable priors yield posteriors with greatly differing real-world implica-tions, which would raise concern about consensus (O2) and impartiality (O3). Our response tosuch concerns, if this sensitivity is indeed a problem, would be to more carefully document ourchoice of prior, thus doubling down on the principle of transparency (O1) and to compare to otherpossible prior distributions supported by other information, thus supporting impartiality (O3) andawareness of multiple perspectives (S1).As with “institutional decision analysis” (Gelman et al., 2003, section 22.5), the point is notthat our particular choices of prior distributions are “correct” (whatever that means), or optimal,or even good, but rather that they are transparent, and in a transparent way connected to knowl-edge. Subsequent researchers—whether supportive, critical, or neutral regarding our methods andsubstantive findings—should be able to interpret our priors (and, by implication, our posteriorinferences) as the result of some systematic process, a process open enough that it can be criticizedand improved as appropriate.
Wang et al. (2014) describe another of our recent applied Bayesian research projects, in this case astatistical analysis that allows highly stable estimates of public opinion by adjustment of data fromnon-random samples. The particular example used was an analysis of data from an opt-in surveyconducted on the Microsoft Xbox video game platform, a technique that allowed the research teamto, effectively, interview respondents in their living rooms, without ever needing to call or entertheir houses.The Xbox survey was performed during the two months before the 2012 U.S. presidentialelection. In addition to offering the potential practical benefits of performing a national surveyusing inexpensive data, this particular project made use of its large sample size and panel structure(repeated responses on many thousands of Americans) to learn something new about U.S. politics:we found that certain swings in the polls, which had been generally interpreted as representing largeswings in public opinion, actually could be attributed to differential nonresponse, with Democratsand Republicans in turn being more or less likely to respond during periods where there was goodor bad news about their candidate. This finding was consistent with some of the literature inpolitical science (see Erikson, Panagopoulos, and Wlezien, 2004), but the Xbox study representedan important empirical confirmation.Having established the potential importance of the work, we next consider its controversialaspects. For many decades, the gold standard in public opinion research has been probabilitysampling, in which the people being surveyed are selected at random from a list or lists (forexample, selecting households at random from a list of addresses or telephone numbers and thenselecting a person within each sampled household from a list of the adults who live there). From thisstandpoint, opt-in sampling of the sort employed in the Xbox survey lacks a theoretical foundation,and the estimates and standard errors thus obtained (and which we reported in our research papers)do not have a clear statistical interpretation.This criticism—that inferences from opt-in surveys lack a theoretical foundation–is interestingto us here because it is not framed in terms of objectivity or subjectivity. We do use Bayesianmethods for our survey adjustment but the criticism from certain survey practitioners is not aboutadjustment but rather about the data collection: they take the position that no good adjustmentis possible for data collected from a non-probability sample.9s a practical matter, our response to this criticism is that nonresponse rates in nationalrandom-digit-dialed telephone polls are currently in the range of 90%, which implies that real-world surveys of this sort are essentially opt-in samples in any case: If there is no theoreticaljustification for non-random samples then we are all dead, which leaves us all with the choice toeither abandon statistical inference entirely when dealing with survey data, or to accept that ourinferences are model-based and do our best (Gelman, 2014c).We shall now express this discussion using the criteria from Section 2.3. Probability samplinghas the clear advantage of transparency (O1) in that the population and sampling mechanism canbe clearly defined and accessible to outsiders, in a way that an opt-in survey such as the Xboxis not. In addition, the probability sampling has the benefits of consensus (O2), at least in theUnited States, where such surveys have a long history and are accepted in marketing and opinionresearch. Impartiality (O3) and correspondence to observable reality (O4) are less clearly presentbecause of the concern with nonresponse, just noted. We would argue that the large sample sizeand repeated measurements of the Xbox data, coupled with our sophisticated hierarchical Bayesianadjustment scheme, put us well on the road to impartiality (through the use of multiple sourcesof information, including past election outcomes, used to correct for biases in the form of knowndifferences between sample and observation) and correspondence to observable reality (in that themethod can be used to estimate population quantities that could be validated from other sources).Regarding the virtues associated with subjectivity, the various adjustment schemes representawareness of context dependence (S2) in that the choice of variables to match in the populationdepend on the context of political polling, both in the sense of which aspects of the population areparticularly relevant for this purpose, and in respecting the awareness of survey practitioners ofwhat variables are predictive of nonresponse. The researcher’s subjective point of view is involvedin the choice of exactly what information to include in weighting adjustments and exactly whatstatistical model to fit in regression-based adjustment. Users of probability sampling on grounds of“objectivity” may shrink from using such judgments, and may therefore ignore valuable informationfrom the context.
Cluster analysis aims at grouping together similar objects and separating dissimilar ones, and assuch is based, explicitly or implicitly, on some measure of dissimilarity measure. Defining such ameasure, for example using some set of variables characterizing the objects to be clustered, caninvolve many decisions. Here we consider an example of Hennig and Liao (2013), where we clustereddata from the 2007 U.S. Consumer Finances Survey, comprising variables on income, savings,housing, education, occupation, number of checking and savings accounts, and life insurance withthe aim of data-based exploration of socioeconomic stratification. The choice of variables and thedecisions of how they are selected, transformed, standardized, and weighted has a strong impact onthe results of the cluster analysis. This impact depends to some extent on the clustering techniquethat is afterward applied to the resulting dissimilarities, but will typically be considerable, even forcluster analysis techniques that are not directly based on dissimilarities. One of the various issuesdiscussed by Hennig and Liao (2013) was the transformation of the variables treated as continuous(namely income and savings amount), with the view of basing a cluster analysis on a Euclideandistance after transformation, standardization, and weighting of variables.There is some literature on choosing transformations, but the usual aims of transformation,namely achieving approximate additivity, linearity, equal variances, or normality, are often notrelevant for cluster analysis, where such assumptions only apply to model-based clustering, and10nly within the clusters, which are not known before transformation.The rationale for transformation when setting up a dissimilarity measure for clustering is ofa different kind. The measure needs to formalize appropriately which objects are to be treatedas “similar” or “dissimilar” by the clustering methods, and should therefore be put into the sameor different clusters, respectively. In other words, the formal dissimilarity between objects shouldmatch what could be called the “interpretative dissimilarity” between objects. This is an issueinvolving subject-matter knowledge that cannot be decided by the data alone.Hennig and Liao (2013) argue that the interpretative dissimilarity between different savingsamounts is governed rather by ratios than by differences, so that $2 million of savings is seen asabout as dissimilar from $1 million, as $2,000 is dissimilar from $1,000. This implies a logarithmictransformation. We do not argue that there is a precise argument that privileges the log trans-formation over other transformations that achieve something similar, and one might argue fromintuition that even taking logs may not be strong enough. We therefore recognize that any choiceof transformation is a provisional device and only an approximation to an ideal “interpretativedissimilarity,” even if such an ideal exists.In the dataset, there are no negative savings values as there is no information on debts, butthere are many people who report zero savings, and it is conventional to kluge the logarithmictransformation to become x log( x + c ) with some c >
0. Hennig and Liao then point out that,in this example, the choice of c has a considerable impact on clustering. The number of peoplewith very small but nonzero savings in the dataset is rather small. Setting c = 1, for example, thetransformation creates a substantial gap between the zero savings group and people with fairly low(but not very small) amounts of savings, and of course this choice is also sensitive to scaling (forexample, savings might be coded in dollars, or in thousands of dollars). The subsequent clusteranalysis (done by “partitioning around medoids”; Kaufman and Rousseeuw, 1990) would thereforeseparate the zero savings group strictly; no person with zero savings would appear together in acluster with a person with nonzero savings. For larger values for c , the dissimilarity between thezero savings group and people with a low savings amount becomes effectively small enough thatpeople with zero savings could appear in clusters together with other people, as long as values onother variables are similar enough.We do not believe that there is a true value of c . Rather, clusterings arising from differentchoices of c are legitimate but imply different interpretations. The clustering for c = 1 is based ontreating the zero savings group as very special, whereas the clustering for c = 200, say, implies thata difference in savings between 0 and $100 is taken as not such a big deal (although it is a biggerdeal in any case than the difference between $100 and $200). Similar considerations hold for issuessuch as selecting and weighting variables and coding ordinal variables.It can be frustrating to the novice in cluster analysis that such decisions for which there do notseem to be an objective basis can make such a difference, and there is apparently a very strongtemptation to ignore the issue and to just choose c = 1, which may look “natural” in the sense thatit maps zero onto zero, or even to avoid transformation at all in order to avoid the discussion, so thatno obvious lack of objectivity strikes the reader. Having the aim of socioeconomic stratification inmind, though, it is easy to argue that clusterings that result from ignoring the issue are less desirableand useful than a clustering obtained from making a however imprecisely grounded decision choosinga c >
1, therefore avoiding either separation of the zero savings group as a clustering artifact oran undue domination of the clustering by people with large savings in case of not applying anytransformation at all.We believe that this kind of tuning problem that cannot be interpreted as estimating an unknowntrue constant (and does therefore not lend itself naturally to an approach through a Bayesian prior)11s not exclusive to cluster analysis, and is often hidden in presentations of data analyses.In Hennig and Liao (2013), we pointed out the issue and did some sensitivity analysis aboutthe strength of the impact of the choice of c (O1, transparency). The way we picked the c in thatpaper made clear reference to the context dependence, while being honest that the subject-matterknowledge in this case provided only weak guidelines for making this decision (S2). We were alsoclear that alternative choices would amount to alternative perspectives rather than being just wrong(S1, O3).The issue how to foster consensus and to make a connection to observable reality (O2, O4) isof interest, but not treated here.It is, however, problematic to establish rationales for consensus that are based on ignoring thecontext and potentially multiple perspectives. There is a tendency in the cluster analysis literatureto seek formal arguments for making such decisions automatically (see, for example, Everitt et al.,2011, Section 3.7, on variable weighting; it is hard to find anything systematic in the clusteringliterature on transformations), for example trying to optimize “clusterability” of the dataset, orto prefer methods that are less sensitive to such decisions, because this amounts to making thedecisions implicitly without giving the researchers access to them. In other words, the data aregiven the authority to determine not only which objects are similar (which is what we want themto do), but also what similarity should mean. The latter should be left to the researcher, althoughwe acknowledge that the data can have a certain impact: for example the idea that dissimilarity ofsavings amounts is governed by ratios rather than differences is connected to (but not determinedby) the fact that the distribution of savings amounts is skewed, with large savings amounts sparselydistributed. Another feature of the cluster analysis in Hennig and Liao (2013) was a parametric bootstrap testfor homogeneity against clustering, see also Hennig and Lin (2015) for a more general elaboration.Clusterings can be computed regardless of whether the data are clustered in a sense that is relevantfor the application of interest. In this example, the test involved the construction of a null modelthat captured the features of the dataset such as the dependence between variables and marginaldistributions of the categorical variables as well as possible, without involving anything that couldbe interpreted as clustering structure. As opposed to the categorical variables, the marginal dis-tributions of the “continuous” variables such as the transformed savings amount were treated aspotentially indicating clustering, and therefore the null model used unimodal distributions for them.As test statistic we used a cluster validity statistic of the clustering computed on the data, witha parametric bootstrap used to compute a clustering in the same manner on data generated fromthe null model.We used a classical significance test rather than a Bayesian approach here because we werenot interested in posterior probabilities for either the null model to be true or prediction of futureobservations. Rather the question of interest was whether the observed clustering structure in thedata (as measured by the validity index) could be explained by a model without any feature thatwould be interpreted as “real clustering,” regardless of whether or to what extent we believe thismodel or not. However, we deviated from classical significance test logic in some ways, particularlynot using a test statistic that was optimal test against any specific alternative, instead choosinga statistic pointing in a rough direction (namely “clustering”) from the null model. Furthermore,setting up the null model required decisions on which potential characteristics of the dataset wouldbe interpretable as “clustering,” on could therefore not be incorporated in the null model that was12o be interpreted as “non-clustering.” A non-significant outcome of the test can then clearly beinterpreted as no evidence in the data for real clustering, whereas the interpretation of a significantoutcome depends on whether we can argue convincingly that the null model is as good as it getsat trying to model the data without clustering structure. Setting up a straw man null model forhomogeneity and rejecting it would have been easy and not informative.There is no point in arguing that our significance test was more objective than for examplea Bayesian analysis would have been, and actually our approach involved decisions such as thedistinction between data characteristics interpreted as “clustering” or “non-clustering” and thechoice of a test statistic that were made by by considerations other than seemingly objectivemathematical optimality or estimation from the data. Still the ultimate aim was to see whetherthe idea of a real clustering would be supported by the data (O4), in an impartial and transparentmanner (O1, O3), trying hard to give the null model a fair chance to fit the data, but involvingcontext dependent judgment (S2) and the transparent choice of a specific perspective (the chosenvalidity index) among a potential variety (S1), because we were after more qualitative statementsthan degrees of belief in certain models.
4. Objectivity and subjectivity in statistics and science
In discussions of the foundations of statistics, objectivity and subjectivity are seen as opposites.Objectivity is typically seen as a good thing; many see it as a major requirement for good science.Bayesian statistics is often presented as being subjective because of the choice of a prior distribution.Some Bayesians (notably Jaynes, 2003, and Berger, 2006) have advocated an objective approach,whereas others (notably de Finetti, 1974) have embraced subjectivity. It has been argued thatthe subjective/objective distinction is meaningless because all statistical methods, Bayesian orotherwise, require subjective choices, but the choice of prior distribution is sometimes held to beparticularly subjective because, unlike the data model, it cannot be determined for sure even inthe asymptotic limit. In practice, subjective prior distributions often have well known empiricalproblems such as overconfidence (Alpert and Raiffa, 1984, Erev, Wallsten, and Budescu, 1994),which motivates efforts to check and calibrate Bayesian models (Rubin, 1984, Little, 2012) and tosituate Bayesian inference within an error-statistical philosophy (Mayo, 1996, Gelman and Shalizi,2013).De Finetti can be credited with acknowledging honestly that subjective decisions cannot beavoided in statistics, but it is misleading to think that the required subjectivity always takesthe form of prior belief. The confusion arises from two directions: first, prior distributions arenot necessarily any more subjective than other aspects of a statistical model; indeed, in manyapplications priors can and are estimated from data frequencies (see Chapter 1 of Gelman, Carlin,et al., 2013, for several examples). Second, somewhat arbitrary choices come into many aspects ofstatistical models, Bayesian and otherwise, and therefore we think it is a mistake to consider theprior distribution as the exclusive gate at which subjectivity enters a statistical procedure.The objectivity vs. subjectivity issue also arises with statistical methods that require tuningparameters; decision boundaries such as the significance level of tests; and decisions regardinginclusion, exclusion, and transformation of data in preparation for analysis.On one hand, statistics is sometimes said to be the science of defaults: most applications ofstatistics are performed by non-statisticians who adapt existing general methods to their particularproblems, and much of the research within the field of statistics involves devising, evaluating,13nd improving such generally applicable procedures (Gelman, 2014b). It is then seen as desirablethat any required data-analytic decisions or tuning are performed in an objective manner, eitherdetermined somehow from the data or justified by some kind of optimality argument.On the other hand, practitioners must apply their subjective judgment in the choice of whatmethod to use, what assumptions to invoke, and what data to include in their analyses. Even using“no need for tuning” as a criterion for method selection or prioritizing bias, for example, or meansquared error, is a subjective decision. Settings that appear completely mechanical involve choice:for example, if a researcher has a checklist saying to apply linear regression for continuous data,logistic regression for binary data, and Poisson regression for count data, he or she still has theoption to code a response as continuous or to use a threshold to define a binary classification. Andsuch choices can be far from trivial; for example, when modeling elections or sports outcomes, onecan simply predict the winner or instead predict the numerical point differential or vote margin.Modeling the binary outcome can be simpler to explain but in general will throw away information,and subjective judgment arises in deciding what to do in this sort of problem (Gelman, 2013a).And in both classical and Bayesian statistics, subjective choices arise in defining the sample spaceand considering what information to condition on.
Scholars in humanistic studies such as history and literary criticism have considered the waysin which differently-situated observers can give different interpretations to what Luc Sante callsthe “factory of facts.” In political arguments, controversies often arise over “cherry picking” orselective use of data, a concern we can map directly to the statistical principle of random orrepresentative sampling, and the more general idea that information used in data collection beincluded in any statistical analysis (Rubin, 1978). In a different way, the concepts of transferenceand counter-transference, central to psychoanalysis, live at the boundary of personal impressionsand measurable facts, all subject to the constraint that, as Philip K. Dick put it, “Reality is thatwhich, when you stop believing in it, doesn’t go away.”The social sciences have seen endless arguments over the relative importance of objective condi-tions and what Keynes (1936) called “animal spirits.” In macroeconomics, for example, the debatehas been between the monetarists who tend to characterize recessions as necessary consequences ofunderlying economic conditions (as measured, for example, by current account balances, businessinvestment, and productivity), and the Keynesians who focus on more subjective factors such asstock market bubbles and firms’ investment decisions. These disagreements also turn methodologi-cal, with much dispute, for example, over the virtues and defects of various attempts to objectivelymeasure the supply and velocity of money, or consumer confidence, or various other inputs to eco-nomic models. The interplay between objective and subjective effects also arises in political science,for example in the question of whether to attribute the political successes of a Ronald Reagan ora Bill Clinton to their charisma and appealing personalities, to their political negotiating skills, orsimply to periods of economic prosperity that would have made a success out of just about anypolitical leader. Again, these disputes link to controversies regarding research methods: a focus onobjective, measurable factors can be narrow, but with a more subjective analysis it can be difficultto attain a scientific consensus. In fields such as social work it has been argued that one must workwith subjective realities in order to make objective progress (Saari, 2005), but this view is relevantto science more generally.In the social and physical sciences alike (as well as in hybrid fields such as psychophysics), thetwentieth century saw an intertwining of objectivity and subjectivity. From one direction, Heisen-14erg’s uncertainty principle told us that, at the quantum level, measurement depends fundamentallyon the observation process, an insight that is implicit in modern statistics and econometrics withlikelihood functions, measurement-error models, and sampling and missing-data mechanisms beingmanifestations of observation models. So in that sense there is no pure objectivity. From the otherdirection, psychologists have continued their effort to scientifically measure personality traits andsubjective states. For example, Kahneman (1999) defines “objective happiness” as “the average ofutility over a period of time.” Whether or not this definition makes much sense, it illustrates amovement in the social and behavioral sciences to measure, in supposedly objective manners, whatmight previously have been considered unmeasurable.Much of these discussions are relevant to statistics because of the role of quantification. Thereis an ideology widespread in many areas of science that sees quantification and numbers and theirstatistical analysis as key tools for objectivity. An important function of quantitative scientificmeasurement is the production of observations that are thought of as independent of individualpoints of view. Apart from the generally difficult issue of measurement validity, the focus on whatcan be quantified, however, may narrow down what can be observed, and may not necessarilydo the measured entities justice, see the examples from political science and psychology above.Another example is the use of quantitative indicators for human rights in different countries; al-though it has been argued that it is of major importance that such indicators should be objectiveto have appropriate impact on political decision making (Candler et al., 2011), many aspects oftheir definition and methodology are subject to controversy and reflect specific political interestsand views (Merry, 2011), and we think that it will help the debate to communicate such indicatorstransparently together with their limitations and the involved decisions rather than to sell them asobjective and unquestionable. In many places the present paper may read as if we treat the ob-servations to be analyzed by the statisticians as given, but we acknowledge the central importanceof measurement and the benefits and drawbacks of quantification. See Porter (1996), Desrosieres(2002), Douglas (2009) for more discussion of the connection between quantification and objectivity.As with choices in statistical modeling and analysis, we believe that when considering measure-ment the objective/subjective antagonism is less helpful than a more detailed discussion of whatquantification can achieve and what its limitations are.
Discussions involving objectivity and subjectivity often suffer from objectivity having multiplemeanings, in statistics and elsewhere (much of the following discussion will focus on the term“objectivity”; subjectivity is often considered as the opposite of objectivity and as such implicitlydefined). Ambiguity in these terms is often ignored. We believe that such discussions can becomeclearer by referring to the meanings that are relevant in any specific situation instead of using theambiguous terms “objectivity” and “subjectivity” without further explanation.Lorraine Daston has explored the ways in which objectivity has been used as a way to generalizescientific inquiry and make it more persuasive. As Daston (1992) puts it, scientific objectivity “isconceptually and historically distinct from the ontological aspect of objectivity that pursues the ul-timate structure of reality, and from the mechanical aspect of objectivity that forbids interpretationin reporting and picturing scientific results.” The core of the current use of the term “objectivity” isthe idea of impersonality of scientific statements and procedures. According to Daston and Galison(2007), the term has only been used in this way in science from the mid-nineteenth century; beforethen, “objective” and “subjective” were used with meanings almost opposite from the current onesand did not play a strong role in discussions about science. Daston (1994) specifically addresses15hanging concepts of subjectivity and objectivity of probabilities, and Zabell (2011) traces thehistorical development of these concepts.The idea of independence of the individual subject can be applied in various ways. Megill (1994)listed four basic senses of objectivity: “absolute objectivity” in the sense of “representing the thingsas they really are” (independently of an observer), “disciplinary objectivity” referring to a consensusamong experts within a discipline and highlighting the role of communication and negotiation, “pro-cedural objectivity” in the sense of following rules that are independent of the individual researcher,and “dialectical objectivity.” The latter somewhat surprisingly involves subjective contributions,because it refers to active human “objectification” required to make phenomena communicable andmeasurable so that they can then be treated in an objective way so that different subjects can un-derstand them in the same way. Statistics for example relies on the construction of well delimitedpopulations and categories within which averages and probabilities can be defined; see Desrosieres(2002).Daston and Galison (2007) call the ideal of scientific images that attempt to capture realityin an unmanipulated way “mechanical objectivity” as opposed to “structural objectivity,” whichemerged from the insight of scientists and philosophers such as Helmholtz and Poincare that obser-vation of reality cannot exclude the observer and will never be as reliable and pure as “mechanicalobjectivists” would hope. Instead, “structural objectivity” refers to mathematical and logical struc-tures. Porter (1996) lists the ideal of impartiality of observers as another sense of objectivity, andhighlights the important role of quantitative and formal reasoning for concepts of objectivity be-cause of their potential for removing ambiguities. In broad agreement with interpretations alreadylisted (and covered by our virtues), Reiss and Sprenger (2014) group key aspects of objectivity intothe categories “faithfulness to facts,” “absence of normative commitments and value-freedom,” and“absence of personal bias.” Fuchs (1997) notes that various modern meanings of objectivity ratherrefer to the absence of subjectivity and all kinds of biasing factors than to something positive.To us, the most problematic aspect of the term “objectivity” is that it incorporates normativeand descriptive aspects, and that these are often not clearly delimited. For example, a statisticalmethod that does not require the specification of any tuning parameters is objective in a descriptivesense (it does not require decisions by the individual scientist). Often this is presented as anadvantage of the method without further discussion, implying objectivity as a norm, but dependingon the specific situation the lack of flexibility caused by the impossibility of tuning may actually bea disadvantage (and indeed can lead to subjectivity at a different point in the analysis, when theanalyst must make the decision of whether to use an auto-tuned approach in a setting where itsinferences do not appear to make sense). The frequentist interpretation of probability is objective inthe sense that it locates probabilities in an objective world that exists independently of the observer,but the definition of these probabilities requires a subjective definition of a reference set. Althoughsome proponents of frequentism consider its objectivity (in the sense of impersonality, conditionalon the definition of the reference set) as a virtue, this property is ultimately only descriptive; itdoes not imply on its own that such probabilities indeed exist in the objective world, nor that theyare a worthwhile target for scientific inquiry.The interpretation of objectivity as a scientific virtue is connected to what are seen to be theaims and values of science. Scientific realists hold that finding out the truth about the observer-independent reality is the major aim of science. This makes “absolute objectivity” as discussedabove a core scientific ideal, as which it is still popular. But observer-independent reality is onlyaccessible through human observations, and the realist ideal of objectivity has been branded asmetaphysical, meaningless, and illusory by positivists including Karl Pearson (1911), and morecontemporarily by empiricists such as van Fraassen (1980). In the latter groups, objectivity is seen16s a virtue as well, although for them it does not refer to observer-independent reality but rather toa standardized, disciplined, and impartial application of scientific methodology enabling academicconsensus about observations.Reference to observations is an element that the empiricist, positivist, and realist ideas ofobjectivity have in common; Mayo and Spanos (2010) see checking theories against experience bymeans of what they call “error statistics” as a central tool to ensure objectivity, which according tothem is concerned with finding out about reality in an unbiased manner . In contrast, van Fraassen(1980) takes observability and the ability of theory to account for observed facts as objective from an anti -realist perspective. His construal of observability depends on the context, theory, and meansof observation, and his concept of objectivity is conditional on these conditions of observation,assuming that at least acceptance of observations and observability given these conditions shouldnot depend on the subject.Daston and Galison (2007) portray the rise of “mechanical objectivity” as a scientific virtue inreaction to shortcomings of the earlier scientific ideal of “truth-to-nature,” which refers to the ideathat science should discover and present an underlying ideal and universal (Platonic) truth belowthe observed phenomena. The move towards mechanical objectivity, inspired by the development ofphotographic techniques, implied a shift of perspective; instead of producing pure and ideal “true”types the focus moved to capturing nature “as it is,” with all irregularities and variations that hadbeen suppressed by a science devoted to “truth-to-nature.” Increasing insight in the shortcomingsand the theory-dependence of supposedly objective observational techniques led to the virtue of“trained judgment” as a response to mechanical objectivity. According to Daston and Galison(2007), the later virtues did not simply replace the older ones, but rather supplemented them, sothat nowadays all three still exist in science.Another typology of objectivity was set up by Douglas (2004), who distinguishes three modes ofobjectivity, namely human interaction with the world (connected to our “correspondence to observ-able reality”), individual thought processes (connected to our “impartiality”) and processes to reachan agreement (connected to our “consensus” and “transparency”). These modes are subdividedinto different “senses.” Regarding human interaction with the world, Douglas distinguishes objec-tivity connected to human manipulation and intervention and objectivity connected to stability ofresults when taking multiple approaches of observation. Regarding individual thought processes,a interesting distinction is made between prohibition against using values in place of evidence andagainst using any values at all. Douglas suspects that the latter is hard to achieve and will ratherencourage sweeping issues under the carpet. She writes that “hiding the decisions that scientistsmake, and the important role values should play in those decisions, does not exclude values.” Athird sense is the conscious attempt to be value-neutral. The three proposed senses of objectivityregarding processes to reach an agreement are the use of generally agreed procedures, explorationof whether and to what extent consensus exists, and an interactive discursive attempt to achieveconsensus.Further distinctions regarding objectivity appear in the philosophical literature. Reiss andSprenger (2014) distinguish the objectivity of a process, such as inference or procedure, fromthe objectivity of an outcome. Some of our aspects of objectivity, such as impartiality, concernthe former; while others, such as correspondence to observable reality, concern the latter; butthe connection is not always clear. Following Reichenbach (1938), there is much discussion inthe philosophy of science concerning the distinction between the “context of discovery” and the Mayo emphasizes that her approach does not require being a realist; according to our reading, she is in any caseconcerned with observer-independent reality, as opposed to the positivists, without subscribing to naive and all toooptimistic ideas about what we can know about it.
The attitude taken in the present paper is based on Hennig (2010), which was in turn inspired byconstructivist philosophy (Maturana, 1988, von Glasersfeld, 1995) and distinguishes personal real-18ty, social reality, and observer-independent reality. According to this perspective, human inquirystarts from observations that are made by personal observers (personal reality). Through commu-nication, people share observations and generate social realities that go beyond a personal point ofview. These shared realities include for example measurement procedures that standardize obser-vations, and mathematical models that connect observations to an abstract formal system that ismeant to create a thought system cleaned from individually different point of views. Nevertheless,human beings only have access to observer-independent reality through personal observations andhow these are brought together in social reality.According to Hennig (2010), science aims at arriving at a view of reality that is stable andreliable and can be agreed freely by general observers and is therefore as observer-independent aspossible. In this sense we see objectivity as a scientific ideal. But at the same time we acknowledgewhat gave rise to the criticism of objectivity: the existence of different individual perspectives andalso of perspectives that differ between social systems, and therefore the ultimate inaccessibility ofa reality that is truly independent of observers, is a basic human condition. Objectivity can onlybe attributed by observers, and if observers disagree about what is objective, there is no privilegedposition from which this can be decided. Ideal objectivity can never be achieved.This does not imply, however, that scientific disputes can never be resolved by scientific means.Yes, there is an element of “politics” involved in the adjudication of scholarly disagreements, but,as we shall discuss, the norm of transparency and other norms associated with both objectivityand subjectivity can advance such discussions. In general no particular observer has a privilegedposition but this does not mean that all positions are equal. We recognize subjectivity not to throwup our hands and give up on the possibility of scientific consensus but as a first step to exploringand, ideally, reconciling, the multiple perspectives that are inevitable in nearly any human inquiry.Denying the existence of different legitimate subjective perspectives and of their potential tocontribute to scientific inquiry cannot make sense in the name of objectivity. Heterogeneous pointsof view cannot be dealt with by imposing authority. Our attitude values the attempt to reachscientific agreement between different perspectives, but ideally such an agreement is reached by freeexchange between the different points of view. In practice, however, agreement will not normallybe universal, and in order to progress, science has to aim at a more restricted agreement betweenexperts who have enough background knowledge to either make sure that the agreement aboutsomething new is in line with what was already established earlier, or to know that and howit requires a revision of existing knowledge. But the resulting agreement is still intended to bepotentially open for everyone to join or to challenge. Therefore, in science there is always a tensionbetween the ideal of general agreement and the reality of heterogeneous perspectives.Furthermore our attitude to science is based on the idea that consensus is possible regardingstable and reliable statements about the observed reality (which may require elaborate measurementprocedures), and that science aims at nontrivial knowledge in the sense that it makes statementsabout observable reality that can and should be checked and potentially falsified by observation.Although there is no objective access to observer-independent reality, we acknowledge thatthere is an almost universal human experience of a reality perceived as located outside the observerand as not controllable by the observer. This reality is a target of science, although it cannot betaken for granted that it is indeed independent of the observer. We are therefore “active scientificrealists” in the sense of Chang (2012), who writes: “I take reality as whatever is not subject toone’s will, and knowledge as an ability to act without being frustrated by resistance from reality.This perspective allows an optimistic rendition of the pessimistic induction, which celebrates thefact that we can be successful in science without even knowing the truth. The standard realistargument from success to truth is shown to be ill-defined and flawed.” This form of realism is not19n contradiction to the criticism of realism by van Fraassen or the arguments against the desirabilityof certain forms of objectivity by constructivists or feminists as outlined above. Active scientificrealism implies that finding out the truth about objective reality is not the ultimate aim of science,but that science rather aims at supporting human actions. This means that scientific methodologyhas to be assessed relative to the specific aims and actions connected to its use. Another irreduciblesubjective element in science, apart from multiple perspectives on reality, is therefore the aim ofscientific inquiry, which cannot be standardized in an objective way. A typical statistical instanceof this is how much prediction accuracy in a restricted setting is valued compared with parsimonyand interpretability.Because science aims at agreement, communication is central to science, as are transparency andtechniques for supporting the clarity of communication. Among these techniques are formal andmathematical language, standardized measurement procedures, and scientific models. Objectivityas we see it is therefore a scientific ideal that can never fully be achieved. As much as scienceaims for objectivity, it has to acknowledge that it can only be built from a variety of subjectiveperspectives through communication.
5. Decomposing subjectivity and objectivity in the foundations of statistics
In this section, we use the above list of virtues to revisit aspects of the discussion on fundamentalapproaches to statistics, for which the terms “subjective” and “objective” typically play a dominantrole. We discuss what we perceive to be the major streams of the foundations of statistics, but withineach of these streams there exist several different approaches, which we cannot cover completely insuch a paper; rather we sketch the streams somewhat roughly and refer to only a single or a fewleading authors for details where needed.Here, we distinguish between interpretations of probability, and approaches for statistical infer-ence. Thus, we take frequentism to be an interpretation of probability, which does not necessarilyimply that Fisherian or Neyman-Pearson tests are preferred to Bayesian methods, despite the factthat frequentism is more often associated with the former than with the latter.We shall go through several philosophies of statistical inference, for each laying out the connec-tions we see to the virtues of objectivity and subjectivity outlined in Section 2.3.Exercising awareness of multiple perspectives, we emphasize that we do not believe that oneof these philosophies is the correct or best one, nor do we claim that reducing the different ap-proaches to a single one would be desirable. What is lacking here is not unification, but rather,often, transparency about which interpretation of probabilistic outcomes is intended when applyingstatistical modeling to specific problems. Particularly, we think that, depending on the situation,both “aleatory” or “epistemic” approaches to modeling uncertainty are legitimate and worthwhile,referring to data generating processes in observer-independent reality on one hand and rationaldegrees of belief on the other.
We label “frequentism” as the identification of the probability of an event in a certain experimentwith a limiting relative frequency of occurrences if the experiment were to be carried out infinitelyoften in some kind of independent manner. Frequentist statistics is based on evaluating proceduresbased on a long-term average over a “reference set” of hypothetical replicated data sets. In the widersense, we call probabilities “frequentist” when they formalize observer-independent tendencies orpropensities of experiments to yield certain outcomes (see, for example, Gillies, 2000), which are20hought of as replicable and yielding a behavior under infinite replication as suggested by what isassumed to be the “true” probability model.The frequentist mindset locates probabilities in the observer-independent world, so they arein this sense objective. This objectivity, however, is model-based, as an infinite amount of actualreplicates cannot exist, and most researchers, in most settings, would be skeptical about trulyidentical replicates and true independence or, when it comes to observational studies, about whetherobservations can be interpreted as drawn in a purely random manner from an appropriate referenceset.The decision to adopt the frequentist interpretation of probability regarding a certain phe-nomenon therefore requires idealization. It cannot be justified in a fully objective way, which heremeans, referring to our list of virtues, that it can neither be enforced by observation, nor is theregeneral enough consensus that this interpretation applies to any specific setup, although it is welldiscussed and supported in some physical settings such as radioactive decay (O2, O4). Once a fre-quentist model is adopted, however, it makes predictions about observations that can be checked,so the reference to the observable reality (O4) is clear.There is some disagreement about whether the frequentist definition of probability is clear andunambiguous (O1a). On one hand, the idea of a tendency of an experiment to produce certainoutcomes as manifested in observed and expected relative frequencies seems clear enough, given thatthe circumstances of the experiment are well defined and regardless of whether frequencies indeedbehave in the implied way. On the other hand, von Mises (1957) was not completely successful inhis attempt to avoid involving stochastic independence and identity in the definition of frequentistprobabilities through the concepts of the collective and the axiom of invariance under place selectionrules (Fine, 1973), and the issue has never been completely resolved.Frequentism implies that, in the observer-independent reality, true probabilities are unique,but there is considerable room for multiple perspectives (S1) regarding the definition of replicableexperiments, collectives, or reference sets. The idea of replication is often constructed in a rathercreative way. For example, frequentist time series models are used for time series data, implying anunderlying true distribution for every single time point, but there is no way to repeat observationsindependently at the same time point. This actually means that the effective sample size for timeseries data would be 1, if replication was not implicitly constructed in the statistical model, forexample by assuming independent innovations in ARMA-type models. Such models, or, moreprecisely, certain aspects of such models, can be checked against the data, but even if such a checkdoes not fail, it is still clear that there is no such thing in observable reality, even approximately,as a marginal “true” frequentist distribution of the value of the time series x t at fixed t , as impliedby the model, because x t is strictly not replicable.The issue that useful statistical models require a construction of replication (or exchangeability)on some level by the statistician, is, as we discuss below, not confined to frequentist models. Inorder to provide a rationale for the essential statistical task of pooling information from manyobservations to make inference relevant for future observations, all these observations need to beassumed to somehow represent the same process.The appropriateness of such assumptions in a specific situation can often only be tested in aquite limited way by observations. All kinds of informal arguments can apply about why it is agood or bad idea to consider a certain set of observations (or unobservable implied entities such aserror terms and latent variables) as independent and identically distributed frequentist replicates.Unfortunately, although such an openness to multiple perspectives and potential context-de-pendence (S2a) can be seen as positive from our perspective, these issues involved in the choicesof a frequentist reference set are often not clearly communicated and discussed. The existence of21 true model with implied reference set is typically taken for granted by frequentists, motivated atleast in part by the desire for objectivity.From the perspective taken here and in Hennig (2010), the frequentist interpretation of prob-ability can be adopted as an idealized model, a thought construction, without having to believethat frequentist probabilities really exist in the observer-independent world (many criticisms offrequentism such as most of the issues raised in Hajek, 2009, refer to a belief in the “existence” oflimits of hypothetical sequences that are impossible in the real world). This can be justified, on acase-by-case basis, if it is seen as useful for the scientific aims in the given situation, for examplebecause a specific frequentist model communicates (more or less) clearly the scientist’s view of acertain phenomenon (O1a), and implies the means for testing this against observations (O4). The term “error statistics” was coined by the philosopher Deborah Mayo (1996). We use it here torefer to an approach to statistical inference that is based on a frequentist interpretation of prob-ability and methods that can be characterized and evaluated by error probabilities. Traditionallythese would be the Type I and Type II errors of Neyman-Pearson hypothesis testing, but the error-statistical perspective could also apply to other constructs such as errors of sign and magnitude(“Type S” and “Type M” errors; Gelman and Carlin, 2014). Mayo (1996) introduced another keyconcept for error statistics, “severity,” which is connected with, but not identical to, the power oftests. It serves to quantify the extent to which a test result can corroborate a hypothesis (keepingin mind that testing specific statistical hypothesis can only ever shed light on specific aspects ofa scientific theory of interest; and that a specific test can only corroborate a specific aspect of ahypothesized statistical model). The severity principle states that a test result can only be evidenceof the absence of a certain discrepancy from a (null) hypothesis, if the probability is high, giventhat such a discrepancy indeed existed, that the test result would have been less in line with thehypothesis than what was observed .According to Mayo and Spanos (2010), objectivity is a core concern of error statistics, whichis specifically driven by providing methodology for reproduction, testing, and falsification (O4b).Mayo (2014) defined objective scientific measurement as being “relevant,” “reliably capable,” and“able to learn from error,” which can be interpreted as the error-statistical rationale for consensus(O2c). Error statistical methodology is portrayed as “reliably capable” as far as its potential toproduce inferential errors can be analyzed, and as far as the resulting error probabilities are low.The “ability to learn from error” refers to erroneous hypotheses, rejected by an error statisticalprocedure that optimally can pinpoint the reason for rejection and thus lead to an improvementof the hypothesis, rather than errors of the inferential method. The underlying idea, with whichwe agree, is that learning from error is a main driving force in science, a lifetime contract betweenthe mode of statistical investigation and its object. This corresponds to Chang’s active scientificrealism mentioned above, and it implies that for Mayo the reference to observations is central forobjectivity.Mayo’s “relevance” concerns the problem of inquiry of interest and is therefore related to virtueS2a, which we classified as related to subjectivity. As Mayo attempts to defend the objectivityof the error statistical approach against charges of subjectivity, she may not be happy about thisclassification, but we agree with her that this is an important virtue nonetheless, which, however,is not specifically connected to error statistics. Mayo applies the term “severity” also more generally, not confined to statistics. .Davies suggests that it is misleading to hypothesize models or parameters to be “true,” andthat one should instead take into account all models that are “adequate” for approximating thedata in the sense that they are not rejected by tests based on features of the data the statisticianis interested in, which does not require reference to unobservable true frequentist probabilities, buttakes into account error probabilities as well. Such an approach is tied to the observations in a moredirect way without making metaphysical assumptions about unobservable features of observer-independent reality (O1a, O4). However, it is possible that such a metaphysical assumption isimplicitly still needed if the researcher wants to use “data approximating models” to learn aboutobserver-independent reality, and that the class of all adequate models is too rich for meaningfulinference (as in more standard frequentist treatments, Davies focuses on models with independentand identically distributed random variables or error terms). Earlier work on robust statistics (seeHuber and Ronchetti, 2009) already introduced the idea of sets of models that neighbor a nominalmodel, from which the models in the neighborhood could not be reliably distinguished based onthe data.Even further flexibility in error statistical analyses comes from the fact that the assumption ofa single true underlying distribution does not determine the parametric or nonparametric familyof distributions, within which the true distribution is embedded. Although Neyman and Pearsonderived optimal tests considering specific alternatives to the null hypothesis, many kinds of alter-natives and test statistics could be of potential interest. Davies (2014) explicitly mentions thedependence of the choice of statistics for checking the adequacy of models on the context and theresearcher’s aims (S2a) instead of relying on Neyman-Pearson type optimality results.Overall, there is no shortage of entry points for multiple perspectives (S1) in the error statisticalapproach. This could be seen as something positive, but it runs counter to some extent to the waythe approach is advertised as objective by some of its proponents. Many frequentist and errorstatistical analyses could in our opinion benefit from acknowledging honestly their flexibility andthe researcher’s choices made, many of which cannot be determined by data alone. We call “subjectivist epistemic” the interpretation of probabilities as quantifications of strengthsof belief of an individual, where probabilities can be interpreted as derived from, or implementablethrough, bets that are coherent in that no opponent can cause sure losses by setting up some Davies (2014) uses the example for a wider discussion of modeling issues including regularization and defects ofthe likelihood.
Given the way objectivity is often advertised as a key scientific virtue (often without specifyingwhat exactly it means), it is not surprising that de Finetti’s emphasis on subjectivity is not sharedby all Bayesians, and that there have been many attempts to specify prior distributions in amore objective way. Currently the approach of E. T. Jaynes (2003) seems to be among the mostpopular. As with many of his predecessors such as Jeffreys and Carnap, Jaynes saw probability as ageneralization of binary logic to uncertain propositions. Cox (1961) proved that given a certain listof supposedly common-sense desiderata for a “plausibility” measurement, all such measurementsare equivalent, after suitable scaling, to probability measures. This theorem is the basis of Jaynes’objectivist Bayesianism, and the claim to objectivity comes from postulating that, given the sameinformation, everybody should come to the same conclusions regarding plausibilities: prior andposterior probabilities (O2c), a statement with which subjectivist Bayesians disagree.In practice, this objectivist ideal seems to be hard to achieve, and Jaynes (2003) admits thatsetting up objective priors including all information is an unsolved problem. One may wonderwhether his ideal is achievable at all. For example, in chapter 21, he gives a full Bayesian “solution”to the problem of dealing with and identifying outliers, which assumes that prior models have tobe specified for both “good” and “bad” data (between which therefore there has to be a properdistinction), including parameter priors for both models, as well as a prior probability for anynumber of observations to be “bad.” It is hard to see, and no information about this is provided byJaynes himself, how it can be possible to translate the unspecific information of knowing of someoutliers in many kinds of situations, some of which are more or less related, but none identical (say)to the problem at hand, into precise quantitative specifications as needed for Jaynes’ approach in25n objective way, all before seeing the data.Setting aside the difficulties or working with informally specified prior information, even themore elementary key issue of specifying an objective prior distribution formalizing the absenceof information is riddled with difficulties, and there are various principles for doing this whichdisagree in many cases (Kass and Wasserman, 1996). Objectivity seems to be an ambition ratherthan a description of what indeed can be achieved by setting up objectivist Bayesian priors. Moremodestly, therefore, Bernardo (1979) spoke of “reference priors,” avoiding the term “objective,”and emphasizing that it would be desirable to have a convention for such cases (O2b), but admittingthat it may not be possible to prove any general approach for arriving at such a convention uniquelycorrect or optimal in any rational sense.Apart from the issue of the objectivity of the specification of the prior, by and large the ob-jectivist Bayesian approach has similar advantages and disadvantages regarding our list of virtuesas the subjectivist Bayesian approach. Particularly it comes with the same difficulties regardingthe issue of falsifiability from observations. Prior probabilities are connected to logical analysisof the situation rather than to betting rates for future observations as in de Finetti’s subjectivistapproach, which makes the connection of objectivist Bayesian prior probabilities to observationseven weaker than in the subjectivist Bayesian approach (but probabilistic logic has applicationsother than statistical data analysis, for which this may not be a problem).The merit of objectivist Bayesianism is that the approach comes with a much stronger driveto justify prior distributions in a transparent way using principles that are as clear and general aspossible. This drive, together with some subjectivist honesty about the fact that despite tryinghard in the vast majority of applications the resulting prior will not deserve the “objectivity” stampand will still be subject to potential disagreement, can potentially combine the best of both of thesetraditional Bayesian worlds.
For both subjectivist and objectivist Bayesians, following de Finetti (1974) and Jaynes (2003),probability models including both parameter priors and sampling models do not model the datagenerating process, but rather represent plausibility or belief from a certain point of view. Plausi-bility and belief models can be modified by data in ways that are specified a priori, but they cannotbe falsified by data.In much applied Bayesian work, on the other hand, the sampling model is interpreted, explicitlyor implicitly, as representing the data-generating process in a frequentist or similar way, and pa-rameter priors and posteriors are interpreted as giving information about what is known about the“true” parameter values. It has been argued that such work does not directly run counter to thesubjectivist or objectivist philosophy, because the “true parameter values” can often be interpretedas expected large sample functions given the prior model (Bernardo and Smith, 1994), but the wayin which classical subjectivist or objectivist statistical data analysis is determined by the untestableprior assignments is seen as unsatisfactory by many statisticians. The suggestion of testing aspectsof the prior distribution by observations using error statistical techniques has been around for sometime (Box, 1980). Gelman and Shalizi (2013) incorporate this in an outline of what we refer tohere as “falsificationist Bayesianism,” a philosophy that openly deviates from both objectivist andsubjectivist Bayesianism, integrating Bayesian methodology with an interpretation of probabilitythat can be seen as frequentist in a wide sense and with an error statistical approach to testingassumptions in a bid to improve Bayesian statistics regarding virtue O4b.Falsificationist Bayesianism follows the frequentist interpretation of the probabilities formalized26y the sampling model given a true parameter, so that these models can be tested using errorstatistical techniques (with the limitations that such techniques have, as discussed in Section 5.2).Gelman and Shalizi argue, as some frequentists do, that such models are idealizations and shouldnot be believed to be literally true, but that the scientific process proceeds from simplified modelsthrough test and potential falsification by improving the models where they are found to be deficient.This reflects certain attitudes of Jaynes (2003), with the difference that Jaynes generally consideredprobability models as derivable from constraints of a physical system, whereas Gelman and Shalizifocus on examples in social or network science which are not governed by simple physical lawsand thus where one cannot in general derive probability distributions from first principles, so that“priors” (in the sense that we are using the term in this paper, encompassing both the data modeland the parameter model) are more clearly subjective.A central issue for falsificationist Bayesianism is the meaning and use of the parameter prior,which can have various interpretations, which gives falsificationist Bayesianism a lot of flexibility fortaking into account multiple perspectives, contexts, and aims (S1, S2a) but may be seen as a prob-lem regarding clarity and unification (O1a, O2c). Frequentists may wonder whether a parameterprior is needed at all. Here are some potential benefits of incorporating a parameter prior: • The parameter prior may formalize relevant prior information. • The parameter prior may be a useful device for regularization. • The parameter prior may formalize deliberately extreme points of view to explore sensitivityof the inference. • The parameter prior may make transparent a point of view involved in an analysis. • The parameter prior may facilitate a certain kind of behavior of the results that is connectedto the aims of analysis (such as penalizing complexity or models on which it is difficult to actby giving them low prior weight). • The Bayesian procedure involving a certain parameter prior may have better error statisticalproperties (such as the mean squared error of point estimates derived from the posterior)than a straightforward frequentist method, if such a method even exists. • Often finding a Bayesian parameter prior which emulates a frequentist/error statistical methodhelps understanding the implications of the method.Here are some ways to interpret the parameter prior: • The parameter prior may be interpreted in a frequentist way, as formalizing a more or lessidealized data generating process generating parameter values. The “generated” parametervalues may not be directly observable, but in some applications the idea of having, at leastindirectly, a sample of several parameter values from the parameter prior makes sense (“em-pirical Bayes”). In many other applications the idea is that only a single parameter fromthe parameter prior is actually realized, which then gives rise to all the observed data. Evenin these applications one could in principle postulate a data generating process behind theparameter, of which only one realization is observable, and only indirectly. This is a ratherbold idealization, but frequentists are no strangers to such idealizations either; see Section5.1. A similarly bold idealization would be to view “all kinds of potential studies with the27statistically) same parameter” as the relevant population, even if the studies are about dif-ferent topics with different variables, in which case more realizations exist, but it is hard toview a specific study of interest as a “random draw” from such a population.If parameter priors are interpreted in this sense, they can actually be tested and falsifiedusing error statistical methods; see Gelman, Meng and Stern (1996). In situations with onlyone parameter realization, the power of such tests is low, though, and any kind of severecorroboration will be hard to achieve. Also, if there is only a single realization of an idealizedparameter distribution, the information in the parameter posterior seems to rely strongly onidealization. • If the quality of the inference is to be assessed by error statistical measures, the parameterprior may be seen as a purely technical device. In this case, however, the posterior distri-bution does not have a proper interpretation, and only well defined statistics with knownerror statistical properties such as the mean or mode of the parameter posterior should beinterpreted. • Assuming that frequentist probabilities from sampling models should be equal to the sub-jectivist or objectivist epistemic probabilities if it is known that the sampling model is true(which Lewis, 1980, called “the principal principle”), the parameter prior can still be inter-preted as giving epistemic probabilities such as subjectivist betting rates, conditionally on thesampling model to hold, even if the sampling model is interpreted in a frequentist way. Thepossibility of rejecting the sampling model based on the data will invalidate both coherenceand Cox’s axioms, so that the foundation for the resulting epistemic probabilities becomesrather shaky. This does not necessarily have to stop an individual from interpreting and usingthem as betting rates, though.Given such a variety of uses and meanings, it is crucial for applications of falsificationist Bayesianismthat the choice of the parameter prior is clearly explained and motivated, so transparency is centralhere as well as for the other varieties of Bayesian statistics.Overall, falsificationist Bayesianism combines the virtue of error statistical falsifiability withthe virtues listed above as “subjective,” doing so via a flexibility that may be seen by some asproblematic regarding clarity and unification.
6. Other philosophies
There are important perspectives on statistics that lie outside the traditional frequentist-Bayesiandivide.In machine learning , the focus is on prediction rather than parameter estimation, thus theemphasis is on correspondence to observable reality (O4). Computer scientists are also interested intransparency; disclosure of data, and methods with full reproducibility (O1) but are sometimes lessattuned to multiple perspectives and context dependence (S1, S2). Such attributes are necessary inpractice (users have many “knobs” to tune in external validation, including the objective functionbeing optimized, the division into training and test sets, and the choice of corpus to use in theevaluation)—but are typically pushed to the background.In robust statistics , the point is to assess stability of inferences when assumptions are violated,or to make minimal assumptions. This connects to impartiality (O3). There is literature onclassical and Bayesian robustness; in any case consideration of model violations requires awarenessof multiple perspectives (S1). Striving for robustness (against disturbances of systems, observations,28ssumptions) can itself be seen as a scientific virtue, although it is not normally associated witheither objectivity or subjectivity.
Alternative models of uncertainty such as belief functions, imprecise probabilities or fuzzy logicaim to get around some of the limitations of probability theory (most notoriously, the difficulty ofdistinguishing between “known unknowns” and “unknown unknowns,” or risk and uncertainty inthe terminology of Knight, 1921). These approaches are typically framed not as subjective or ob-jective but rather as a way to incorporate radically uncertain information into a statistical analysis.One could say that these generalizations of probability theory aim at virtues O1c (communication ofpotential limitations), O2a (accounting for relevant knowledge, here regarding distinctions that arenot represented in classical probability modeling) and O4a (connection of models to observables).
Exploratory data analysis (EDA; Tukey, 1977) is all about data operations rather than models.In that sense, EDA resembles classical statistics in its positivist focus, but with the differencethat the goal is exploration rather than hypothesis testing or rigorous inference. EDA is sensitiveto multiple perspectives (S1) and context dependence (S2) in that discovery of the unexpected isalways relative to what was previously expected by the researcher. Regarding transparency (O1), itcould be argued that the refusal to use probability models with all their problems and particularlyreferences to what cannot be observed (see above) contributes to clarity. However, it can also beargued that some techniques of EDA can be usefully explained in terms of probability models, e.g.,as predictive checking (Gelman, 2003), but in traditional EDA such models are left incomplete orimplicit, and methods that come with implicit assumptions are portrayed as assumptionless, whichworks against transparency.
7. Discussion
At the level of discourse, we would like to move beyond a subjective vs. objective shouting match.But our goals are larger than this. Gelman and Shalizi (2013) on the philosophy of Bayesianstatistics sought not just to clear the air but also to provide philosophical and rhetorical spacefor Bayesians to feel free to check their models and for applied statisticians who were concernedabout model fit to feel comfortable with a Bayesian approach. In the present paper, our goalsare for scientists and statisticians to achieve more of the specific positive qualities into which wedecompose objectivity and subjectivity in Section 2.3. At the present time, we feel that concernsabout objectivity are getting in the way of researchers trying out different ideas and consideringdifferent sources of inputs to their model, while an ideology of subjectivity is limiting the degreeto which researchers are justifying and understanding their model.There is a tendency for hardcore believers in objectivity to needlessly avoid the use of valuableexternal information in their analyses, and for subjectivists, but also for statisticians who want tomake their results seem strong and uncontroversial, to leave their assumptions unexamined. Wehope that our new framing of transparency, consensus, avoidance of bias, reference to observablereality, multiple perspectives, dependence on context and aims, and honesty about the researcher’sposition and decisions will give researchers of all stripes the impetus and, indeed, permission, tointegrate different sources of information into their analyses, to state their assumptions more clearly,and to trace these assumptions backward to past data that justify them and forward to future datathat can be used to validate them.Also, we believe that the pressure to appear objective has led to confusion and even dishonestyregarding data coding and analysis decisions which cannot be motivated in supposedly objective29ays; see van Loo and Romeijn (2015) for a discussion of this point in the context of psychiatricdiagnosis. We prefer to encourage a culture in which it is acceptable to be open about the reasonsfor which decisions are made, which may at times be mathematical convenience, or the aim of thestudy, rather than strong theory or hard data. It should be recognized openly that the aim ofstatistical modeling is not always to make the model as close as possible to observer-independentreality (which always requires idealization anyway), and that some decisions are made, for example,in order to make outcomes more easily interpretable for specific target audiences.Our key points: (1) multiple perspectives correspond to multiple lines of reasoning, not merelyto mindless and unjustified guesses; and (2) what is needed is not just a prior distribution or a tuningparameter, but a statistical approach in which these choices can be grounded, either empirically orby connecting them in a transparent way to the context and aim of the analysis.For these reasons, we do not think it at all accurate to limit Bayesian inference to “the analysisof subjective beliefs.”
Yes, Bayesian analysis can be expressed in terms of subjective beliefs, butit can also be applied to other settings that have nothing to do with beliefs (except to the extentthat all scientific inquiries are ultimately about what is believed about the world).Similarly, we would not limit classical statistical inference to “the analysis of simple randomsamples.”
Classical methods of hypothesis testing, estimation, and data reduction can be appliedto all sorts of problems that do not involve random sampling. There is no need to limit theapplications of these methods to a narrow set of sampling or randomization problems; rather, it isimportant to clarify the foundation for using the mathematical models for a larger class of problems.
The list in Section 2.3 is the core of the paper. The list may not be complete, and such a listmay also be systematized in different ways. Particularly, we developed the list having particularlyapplied statistics in mind, and we may have missed aspects of objectivity and subjectivity that arenot connected in some sense to statistics. In any case, we believe that the given list can be helpfulin practice for researchers, for justifying and explaining their choices, and for recipients of researchwork, for checking to what extent the listed virtues are practiced in scientific work. A key issuehere is transparency, which is required for checking all the other virtues. Another key issue is thatsubjectivity in science is not something to be avoided at any cost, but that multiple perspectivesand context dependence are actually basic conditions of scientific inquiry, which should be explicitlyacknowledged and taken into account by researchers. We think that this is much more constructivethan the simple objective/subjective duality.We do not think this advice represents empty truisms of the “mom and apple pie” variety. Infact, we repeatedly encounter publications in top scientific journals that fall foul of these virtues,which indicates to us that the underlying principles are subtle and motivates this paper. We hopethat a change in names will clarify what can be done to improve statistical analyses in these twodimensions.Instead of pointing at specific bad examples, here is a list of some issues that can regularlybe encountered in scientific publications (see, for example, our discussions in Gelman, 2015, andGelman and Zelizer, 2015), and where we believe that exercising one or more of our listed virtueswould improve matters: • Presenting analyses that are contingent on data without explaining the exploration and se-lection process and without even acknowledging that it took place, • Justifying decisions by reference to specific literature without acknowledging that what was30ited may be controversial, not applicable in the given situation, or without proper justifica-tion in the cited literature as well (or not justifying the decisions at all), • Failure to reflect on whether model assumptions are reasonable in the given situation, whatimpact it would have if they were violated, or whether alternative models and approachescould be reasonable as well, • Choosing methods for the main reason that they do not require tuning or make decisionsautomatically and therefore seem “objective” without discussing whether the chosen methodscan handle the data more appropriately in the given situation than alternative methods withtuning, • Choosing methods for the main reason that they “do not require assumptions” without re-alizing that every method is based on implicit assumptions about how to treat the dataappropriately, regardless of whether these are stated in terms of statistical models, • Choosing Bayesian priors without justification or explanation of what they mean and imply, • Using nonstandard methodology without justifying the deviation from standard approaches(where they exist), • Using standard approaches without discussion of whether they are appropriate in the specificcontext.Most of these have to do with the unwillingness to admit to having made decisions, to justify them,and to take into account alternative possible views that may be equally reasonable. In some senseperhaps this can be justified based on a sociological model of the scientific process in which eachpaper presents just one view, and then the different perspectives battle it out. But we think thatthis idea ignores the importance of communication and facilitating consensus for science. Scientistsnormally believe that each analysis aims at the truth, and if different analyses give different results,this is not because there are different conflicting truths but rather because different analysts havedifferent aims, perspectives and access to different information. Letting the issue aside of whetherit makes sense to talk of the existence of different truths or not, we see aiming at general agreementin free exchange as essential to science, and the more perspectives are taken into account, the morethe scientific process is supported.We see the listed virtues as ideals which in practice cannot generally be fully achieved in anyreal project. For example, tracing all assumptions to observations and making them checkable byobservable data is impossible because one can always ask whether and why results from the specificobservations used should generalize to other times and other situations. As mentioned in Section5.1, ultimately a rationale for treating different situations as “identical and independent” or “ex-changeable” needs to be constructed by human thought (people may appeal to historical successesfor justifying such idealizations, but this does not help much regarding specific applications). Atsome point—but, we hope, not too early—researchers have to resort to somewhat arbitrary choicesthat can be justified only by logic or convention, if that.And it is likewise unrealistic to suppose that we can capture all the relevant perspectives on anyscientific problem. Nonetheless, we believe it is useful to set these as goals which, in contrast tothe inherently opposed concepts of “objectivity” and “subjectivity,” can be approached together.31 eferences
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