Peter Walley

Towards a Unified Theory of Imprecise Probability

Abstract Coherent upper and lower probabilities, Choquet capacities of order 2, belief functions and possibility measures are amongst the most popular mathematical models for uncertainty and partial ignorance. Examples are given to show that these models are not sufficiently general to represent some common types of uncertainty. In particular, they are not sufficiently informative about expectations and conditional probabilities. Coherent lower previsions and sets of probability measures are considerably more general, but they may not be sufficiently informative for some purposes. Two other models for uncertainty, which involve partial preference orderings and sets of desirable gambles, are discussed. These are more informative and more general than the previous models, and they may provide a suitable mathematical foundation for a unified theory of imprecise probability.

A survey of concepts of independence for imprecise probabilities

Our aim in this paper is to clarify the notion of independence for imprecise probabilities. Suppose that two marginal experiments are each described by an imprecise probability model, i.e., by a convex set of probability distributions or an equivalent model such as upper and lower probabilities or previsions. Then there are several ways to define independence of the two experiments and to construct an imprecise probability model for the joint experiment. We survey and compare six definitions of independence. To clarify the meaning of the definitions and the relationships between them, we give simple examples which involve drawing balls from urns. For each concept of independence, we give a mathematical definition, an intuitive or behavioural interpretation, assumptions under which the definition is justified, and an example of an urn model to which the definition is applicable. Each of the independence concepts we study appears to be useful in some kinds of application. The concepts of strong independence and epistemic independence appear to be the most frequently applicable.

Graphoid properties of epistemic irrelevance and independence

This paper investigates Walleys concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can violate the symmetry, contraction and intersection axioms, that epistemic independence can violate contraction and intersection, and that this accords with informal notions of irrelevance and independence.

Upper probabilities based only on the likelihood function

In the problem of parametric statistical inference with a finite parameter space, we propose some simple rules for defining posterior upper and lower probabilities directly from the observed likelihood function, without using any prior information. The rules satisfy the likelihood principle and a basic consistency principle (‘avoiding sure loss’), they produce vacuous inferences when the likelihood function is constant, and they have other symmetry, monotonicity and continuity properties. One of the rules also satisfies fundamental frequentist principles. The rules can be used to eliminate nuisance parameters, and to interpret the likelihood function and to use it in making decisions. To compare the rules, they are applied to the problem of sampling from a finite population. Our results indicate that there are objective statistical methods which can reconcile three general approaches to statistical inference: likelihood inference, coherent inference and frequentist inference.

Reconciling frequentist properties with the likelihood principle

Abstract For the general problem of parametric statistical inference, several frequentist principles are formulated, including principles of hypothesis testing, set estimation, and conditional inference. These principles guarantee that, whatever the true parameter value, statistical procedures have little chance of producing misleading inferences. The frequentist principles are shown to be compatible with the likelihood principle and with principles of coherence. Two general methods are studied which satisfy both the likelihood and frequentist principles in finite samples. One method produces posterior upper and lower probabilities from a very large set of prior probability measures, which can be taken to be an e-contamination neighborhood with e slightly larger than 1 2 . The second method derives inferences from a normalized version of the observed likelihood function. Because inferences from the two methods encompass a wide range of frequentist, likelihood and Bayesian inferences, they are conservative and they have relatively low power. More powerful methods can be obtained by weakening the frequentist principles and making weak assumptions about the sampling rule. The results show that there are methods of statistical inference, based on particular types of imprecise probability model, which satisfy the likelihood principle, are coherent, and have good frequentist properties under a range of sampling models.

Statistical Reasoning with Imprecise Probabilities.

Part 1 Reasoning and behaviour: interpretations of probability beliefs and behaviour inference and decision reasoning and rationality assessment strategies survey of related work. Part 2 Coherent previsions: possibility probability currency upper and lower previsions avoiding sure loss coherence basic properties of coherent previsions coherent probabilities linear previsions and additive probabilities examples of coherent previsions interpretations of prevision and probability objections to behavioural theories of probability. Part 3 Extensions, envelopes and decisions: natural extension extension from a field lower envelopes of linear previsions linear extension invariant linear previsions compactness and extreme points of M(P) desirability and preference equivalent models for beliefs decision making. Part 4 Assessment and elicitation: a general elicitation procedure finitely-generated models and simplex representations steps in assessment process classificatory probability comparative probability other types of assessment. Part 5 The importance of imprecision: uncertainty, indeterminacy and imprecision sources from imprecision information from Bernoulli trials prior-data conflict Bayesian noninformative prioirs indecision axioms of precision practical reasons for precision Bayesian sensitivity analysis second-order probabilities fuzzy sets maximum entropy the Dempster-Shafer theory of belief functions. Part 6 Conditional previsions: updated and contingent previsions separate coherence coherence with unconditional previsions the generalized Bayes rule coherence axioms examples of conditional previsions extension of conditional and marginal previsions conglomerability countable additivity conditioning on events of probability zero updating beliefs. Part 7 Coherent statistical models: general concepts of coherence sampling models coherence of sampling model and posterior previsions inferences from improper priors confidence intervals and relevant subsets proper prior previsions standard Bayesian inference inferences from imprecise priors joint prior previsions. Part 8 Statistical reasoning: a general theroy of natural extension extension to prior previsions extension to predictive previsions extension to posterior previsions posteriors for imprecise sampling models the likelihood principle. Part 9 Structural judgements: independent events independent experiments constructing joint previsions from independent marginals permutability exchangeability Robust Bernoulli models structural judgements. Appendices: verifying coherence N-coherence win and place betting on horses topological structure and L and P separating hyperplane theorems desirability upper and lower variances operational measuremnet procedures the World Cup football experiment regular extension W-coherence.