Statistical Reasoning with Imprecise Probabilities.

Abstract

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.