We develop the theory and practice of an approach to modelling and probabilistic inference in causal networks that is suitable when application-specific or analysis-specific constraints should inform such inference or when little or no data for the learning of causal network structure or probability values at nodes are available. Constrained Bayesian Networks generalize a Bayesian Network such that probabilities can be symbolic, arithmetic expressions and where the meaning of the network is constrained by finitely many formulas from the theory of the reals. A formal semantics for constrained Bayesian Networks over first-order logic of the reals is given, which enables non-linear and non-convex optimisation algorithms that rely on decision procedures for this logic, and supports the composition of several constrained Bayesian Networks. A non-trivial case study in arms control, where few or no data are available to assess the effectiveness of an arms inspection process, evaluates our approach. An open-access prototype implementation of these foundations and their algorithms uses the SMT solver Z3 as decision procedure, leverages an open-source package for Bayesian inference to symbolic computation, and is evaluated experimentally.
-- see e.g. @cite_37 -- refer to the theory and practice of associating a convex set of probability measures with directed, acyclic graphs. Credal networks are also referred to as the Theory of Imprecise Probabilities @cite_1 or as the Quasi-Bayesian Theory @cite_29 .
@cite_37 @cite_29 @cite_1
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In this paper the theoretical and practical implications of dropping-from the basic Bayesian coherence principles- the assumption of comparability of every pair of acts is examined. The resulting theory is shown to be still perfectly coherent and has Bayesian theory as a particular case. In particular we question the need of weakening or ruling out some of the axioms that constitute the coherence principles; what are their practical implications; how this drive to the notion of partial information or partial uncertainty in a certain sense; how this partial information is combined with sample information and how this relates to Bayesian methods. We also point out the relation of this approach to rational behaviour with the more (and apparently unrelated) general notion of domination structures as applied to multicrieria decision making. 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.