Rafael B. Stern

In Defense of Randomization: a Subjectivist Bayesian Approach

In research situations usually approached by Decision Theory, it is only considered one researcher who collects a sample and makes a decision based on it. It can be shown that randomization of the sample does not improve the utility of the obtained results. Nevertheless, we present situations in which this approach is not satisfactory. First, we present a case in which randomization can be an important tool in order to achieve agreement between people with different opinions. Next, we present another situation in which there are two agents: the researcher—a person who collects the sample; and the decision‐maker—a person who makes decisions based on the sample collected. We show that problems emerge when the decision‐maker allows the researcher to arbitrarily choose a sample. We also show that the decision‐maker maximizes his expected utility requiring that the sample is collected randomly.

Learning with many experts: Model selection and sparsity

Experts classifying data are often imprecise. Recently, several models have been proposed to train classifiers using the noisy labels generated by these experts. How to choose between these models? In such situations, the true labels are unavailable. Thus, one cannot perform model selection using the standard versions of methods such as empirical risk minimization and cross validation. In order to allow model selection, we present a surrogate loss and provide theoretical guarantees that assure its consistency. Next, we discuss how this loss can be used to tune a penalization which introduces sparsity in the parameters of a traditional class of models. Sparsity provides more parsimonious models and can avoid overfitting. Nevertheless, it has seldom been discussed in the context of noisy labels due to the difficulty in model selection and, therefore, in choosing tuning parameters. We apply these techniques to several sets of simulated and real data.

The Gambler’s Fallacy: A Bayesian Approach

We study the problem of prediction in sequences of binary random variables. The models are then considered vis‐a‐vis the Gambler’s Fallacy. Another model in which the Gambler’s Fallacy need not be a fallacy is presented. The results may contribute for the judgment of how reasonable the assumption of infinite exchangeability is relative to typical human perception.

Sleeping Beauty’s Credences

The Sleeping Beauty problem has spawned a debate between “thirders” and “halfers” who draw conflicting conclusions about Sleeping Beautys credence that a coin lands heads. Our analysis is based on a probability model for what Sleeping Beauty knows at each time during the experiment. We show that conflicting conclusions result from different modeling assumptions that each group makes. Our analysis uses a standard “Bayesian” account of rational belief with conditioning. No special handling is used for self-locating beliefs or centered propositions. We also explore what fair prices Sleeping Beauty computes for gambles that she might be offered during the experiment.

The Logical Consistency of Simultaneous Agnostic Hypothesis Tests

Simultaneous hypothesis tests can fail to provide results that meet logical requirements. For example, if A and B are two statements such that A implies B, there exist tests that, based on the same data, reject B but not A. Such outcomes are generally inconvenient to statisticians (who want to communicate the results to practitioners in a simple fashion) and non-statisticians (confused by conflicting pieces of information). Based on this inconvenience, one might want to use tests that satisfy logical requirements. However, Izbicki and Esteves shows that the only tests that are in accordance with three logical requirements (monotonicity, invertibility and consonance) are trivial tests based on point estimation, which generally lack statistical optimality. As a possible solution to this dilemma, this paper adapts the above logical requirements to agnostic tests, in which one can accept, reject or remain agnostic with respect to a given hypothesis. Each of the logical requirements is characterized in terms of a Bayesian decision theoretic perspective. Contrary to the results obtained for regular hypothesis tests, there exist agnostic tests that satisfy all logical requirements and also perform well statistically. In particular, agnostic tests that fulfill all logical requirements are characterized as region estimator-based tests. Examples of such tests are provided.

Exchangeability and the law of maturity

The law of maturity is the belief that less-observed events are becoming mature and, therefore, more likely to occur in the future. Previous studies have shown that the assumption of infinite exchangeability contradicts the law of maturity. In particular, it has been shown that infinite exchangeability contradicts probabilistic descriptions of the law of maturity such as the gambler’s belief and the belief in maturity. We show that the weaker assumption of finite exchangeability is compatible with both the gambler’s belief and belief in maturity. We provide sufficient conditions under which these beliefs hold under finite exchangeability. These conditions are illustrated with commonly used parametric models.

Statistical Information: A Bayesian Perspective

We explore the meaning of information about quantities of interest. Our approach is divided in two scenarios: the analysis of observations and the planning of an experiment. First, we review the Sufficiency, Conditionality and Likelihood principles and how they relate to trivial experiments. Next, we review Blackwell Sufficiency and show that sampling without replacement is Blackwell Sufficient for sampling with replacement. Finally, we unify the two scenarios presenting an extension of the relationship between Blackwell Equivalence and the Likelihood Principle.

Goalkeeper Game: A new assessment instrument in neurology showed higher predictive power than MoCA for gait performance in people with parkinson\'s disease

Objective To investigate the use of the Goalkeeper Game (GG) to assess gait automaticity decline under dual task conditions in people with Parkinson’s disease (PPD) and compare its predictive power with the one of the MoCA test. Materials and Methods 74 PPD (H&Y stages: 23 in stage 1; 31 in stage 2; 20 in stage 3), without dementia (MoCA cut-off 23), tested in ON period with dopaminergic medication were submitted to single individual cognitive/motor evaluation sessions. The tests applied were: MoCA, GG, dynamic gait index (DGI) task and timed up and go test (TUG) under single and dual-task (DT) conditions. GG test resulted in 9 measures extracted via a statistical model. The predictive power of the GG measures and the MoCA score with respect to gait performance, as assessed by DGI and DT-TUG, were compared. Results The predictive models based on GG measures and MoCA score obtained, respectively, sensitivities of 65% and 56% for DGI scores and 59% and 57% for DT-TUG cost at a 50% specificity. GG application proved to be feasible and aroused more motivation in PPDs than MoCa. Conclusion GG, a friendly and ludic game, was able to reach a good power of gait performance prediction in people at initial and intermediate stages of PD evolution.

Teaching decision theory proof strategies using a crowdsourcing problem

ABSTRACT Teaching how to derive minimax decision rules can be challenging because of the lack of examples that are simple enough to be used in the classroom. Motivated by this challenge, we provide a new example that illustrates the use of standard techniques in the derivation of optimal decision rules under the Bayes and minimax approaches. We discuss how to predict the value of an unknown quantity, θ ∈ {0, 1}, given the opinions of n experts. An important example of such crowdsourcing problem occurs in modern cosmology, where θ indicates whether a given galaxy is merging or not, and Y1, …, Yn are the opinions from n astronomers regarding θ. We use the obtained prediction rules to discuss advantages and disadvantages of the Bayes and minimax approaches to decision theory. The material presented here is intended to be taught to first-year graduate students.

Prior Shift Using the Ratio Estimator

Several machine learning applications use classifiers as a way of quantifying the prevalence of positive class labels in a target dataset, a task named quantification. For instance, a naive a way of determining what proportion of people like a given product with no labeled reviews is to (i) train a classifier based on the Google Shopping reviews to predict whether a user likes a product given its review, and then (ii) apply this classifier to Facebook/Google+ posts about that product. It is well known that such a two-step approach, named Classify and Count, fails because of dataset shift, and thus, several improvements have been recently proposed under an assumption named prior shift. Unfortunately, these methods only explore the relationship between the covariates and the response via classifiers. Moreover, the literature lacks in the theoretical foundation to improve these techniques. We propose a new family of estimators named Ratio Estimator which is able to explore the relationship between the cov ariates and the response using any function \( g: \mathscr {X} \rightarrow \mathbb {R}\) and not only classifiers. We show that for some choices of g, our estimator matches standard estimators used in the literature. We also explore alternative ways of constructing functions g that lead to estimators with good performance, and compare them using real datasets. Finally, we provide a theoretical analysis of the method.