Henry E. Kyburg

Set-based Bayesianism

Problems for strict and convex Bayesianism are discussed. A set-based Bayesianism generalizing convex Bayesianism and intervalism is proposed. This approach abandons not only the strict Bayesian requirement of a unique real-valued probability function in any decision-making context but also the requirement of convexity for a set-based representation of uncertainty. Levis E-admissibility decision criterion is retained and is shown to be applicable in the nonconvex case.

Higher order probabilities and intervals

Abstract Many researchers have felt uncomfortable with the precision of degrees of belief that seems to be demanded by the subjective Bayesian treatment of uncertainty. Various responses have been suggested. The most common one has been to incorporate higher order probabilities in systems that reason in beliefs. These probabilities concern statements of first-order probability. Thus a first-order probability (e.g., the probability of heads on the next toss of this coin is 1 2 ) is the subject of a second-order probability; for example, the probability is .9 that the probability of heads on the next toss of this coin is 1 2 . This approach is explored and is found to be epistemologically wanting, although there are important intuitions about beliefs that are captured by it. Furthermore, this approach may, in some circumstances, be computationally attractive. We also briefly explore a number of other approaches, including taking probabilities to be intervals and construing probability values as fuzzy sets.

Conditionals and consequences

Abstract We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, non-classical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework.

Probabilistic inference and non-monotonic inference

Publisher Summary This chapter discusses probabilistic inference and non-monotonic inference. It argues that many of the usual arguments against probabilistic inference are equally applicable to other forms of nonmonotonic reasoning and that once the single hurdle of inconsistency is overcome, probabilistic inference offers advantages (in some contexts) or at least no disadvantages (in most contexts) compared to other forms of nonmonotonic reasoning. It also argues that the ability to live comfortably with certain sorts of inconsistency is an important feature of probabilistic inference and that it allows to take as a basis for planning exactly those individual probable conclusions that are collectively impossible to credit. All this is not to say that the various forms of nonmonotonic reasoning that have been explored are not useful for special purposes. It suggests both that specific instances of nonmonotonic argument can be justified by reference to probabilities and that any sort of inference that was incompatible with probabilistic inference would have some strikes against it.

The Rule of Detachment in Inductive Logic

Publisher Summary This chapter focuses on the the rule of detachment in inductive logic. A rule of detachment in general may be regarded as a permissive rule allowing the detachment of a conclusion from a particular set of premises. The premises, in induction, are usually regarded as simple observable facts; observations themselves; or statements or propositions based directly upon those observations. A rule of detachment is construed as a rule that permits the acceptance of a statement h representing a factual inductive conclusion, given that certain criteria are satisfied. These criteria include (1) a body of statements regarded as evidence, and satisfying some principle of acceptability, and also a condition of total evidence; (2) the probability of h, relative to this body of statements; and (3) the information content of h the simplicity and fruitfulness of h other factors not generally regarded as relevant by philosophers of science, such as political or moral utility.

Comparison of Approaches

The variety of approaches to statistical inference and the problem of decision in the face of uncertainty is regarded by many statisticians as a Good Thing. On analogy with mathematics, statistics may be looked on as a body of theory and techniques which may be applicable to a wide variety of circumstances. Applications, however, are not the business of the theoretical statistician as such. On this view, the wider the range of available techniques, the richer and more various the body of statistical theory, the better. It is undemocratic, if not sinful, to regard one approach as wrong-headed as opposed to another.

Statistical inference as default reasoning

Classical statistical inference is nonmonotonic in nature. We show how it can be form~ized in the default logic framework. The structure of statistical inference is the same as that represented by default rules. In particular, the prerequisite corresponds to the sample statistics, the justifications require that we do not have any reason to believe that the sample is misleading, and the consequent corresponds to the conclusion sanctioned by the statistical test.

The Fiducial Argument

The fiducial argument has been of considerable interest for a long period of time, despite the fact that no one (except possibly R. A. Fisher himself) was altogether clear as to what a fiducial argument is. When it applies it has a great deal of appeal, and there are relatively uncontroversial circumstances under which it applies. In the present section, we shall investigate such an uncontroversial example within the framework provided by the preceding chapters.

LOGIC, SCIENCE, AND ENGINEERING

My commentators have taken issue with me on matters falling into each of three areas. The areas thus provide an appropriate framework for my response. While the line between science and engineering may be a little fuzzy, I 1.ake the line between logic and science to be quite sharp. Nevertheless, I seem to draw it in a slightly different place from others. My concerns are logical concerns. But what is logic? I take logic to have been traditionally concerned with standards for valid inference. But it is more than this. If we were only concerned to ensure that the consequences of our inferences should not be false when our premises are true, we could attain that goal by a judicious universal suspension of belief in the results of argument. In traditional logic we want not only principles that will ensure validity but also principles that will be useful. Early in this century, one (controversial) goal was to provide logical principles from which all matheinatics could be derived. Whether that goal was achieved or not depends on what you mean by logical principles. If you include the axioms that characterize the membership relation as part of logic, then it can be argued that mathematics can be derived from logic. If you don’t, then you must clearly distinguish another set of truths, in addition to logical truths, namely, mathematical truths. We must, as Kant argued, accommodate “‘5 + 7 = 12,” which clearly cannot (contrary to Mill) be falsified by experience. There are other forms of usefulness, and indeed it can plausibly be argued that the reason that the development of a logic adequate for mathematical argument was important was that arguments in science and engineering make use of quantities that may be represented as mathematical objects. To argue that if A is five feet long, and B is seven feet long, and C is the collinear juxtaposition of A and B , then C is twelve feet long, is just the sort of argument that interests us. It is the sort that interests us precisely because it is the sort of argument that we can often characterize as sound. That is, not only does the conclusion follow from the premises, but the premises are true. Note the difference between ii rule of inference being sound (it preserves truth) and an argument being sound (its premises are true). Given inductive acceptance, we can’t be certain that an argument is sound, but we can surely have good reason to think it is. It is here that the alternative approach touted by Bacchus et al. comes to a screeching halt as a “logical” approach, The authors agree that given a prior distribution the probability of C given BK: A E is deductively determined, but argue that “the choice of a prior distribution is not deductive!” Quite so. In the tradition of logic, the inferences that interest us are those that are sound: They are valid, and they have true premises. For a particular inference to interest us, we must have some reason to believe that the premises are true. But not only do I find it hard to believe that any of the prior distributions suggested by Bacchus et al. are true, I find it hard to imagine what it would mean to call one of them true. I understand that such inferences can be valid; I denigrated them as “merely deductive” in part because I see no way in which they can be both interesting and sound. A number of these issues are also raised by Charles Morgan. It appears that for him, as I suspect it is for Bacchus and Halpern, probabilities have to do with belief, and are psychological. The “principles must be true of all probability distributions Prob. This observation corresponds well with our intuitive notion that the logical principles of inference

Randomization and Uncertain Inference

Statistical conclusions must be supported on the basis of a finite amount of experimentaldata. It would be desirable if our intelligent systems could arrive at warrantedconclusions in the same way. One problem is that the validity of an uncertain conclusionis sometimes held to depend not just on the data, but on the provenance ofthe data. To be specific, in many experiments randomization is an important part of the protocol, yet precisely the same data could be produced by an experiment inwhich randomization played no part. From a Bayesian point of view, randomization plays at most a small role. From a classical point of view, randomization is centralto ensuring that the long run error rates are controlled as they are claimed to be. We examine this controversy from an AI perspective, and propose that an evidentialor epistemic approach to probability allows us to retain the frequency bounds on error, and at the same time allows the data to speak for themselves. Neverthelessrandomization can play an important role in the experimental protocol.