[Paper Review] `Plausibilities of plausibilities': an approach through circumstances

PierGianLuca Porta Mana, Anders Månsson+1 more|ArXiv.org|Jul 17, 2006

Mathematical and Theoretical Analysis75 references3 citations

TL;DR

This paper reinterprets probability-like parameters and priors in statistical models as plausibilities associated with abstract 'circumstances'—hypothetical or real pieces of knowledge that uniquely determine a probability distribution. By indexing these circumstances via the distributions they yield, the framework provides a logical, non-infinite-sequence-based interpretation of priors and 'plausibilities of plausibilities', offering a foundation for state assignment in physics without relying on exchangeability or propensity concepts.

ABSTRACT

Probability-like parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of `circumstance', a term which stands for any piece of knowledge that is useful in assigning a probability and that satisfies some additional logical properties. The idea, which can be traced to Laplace and Jaynes, is that the usual inferential reasonings about the probability-like parameters of a statistical model can be conceived as reasonings about equivalence classes of `circumstances' - viz., real or hypothetical pieces of knowledge, like e.g. physical hypotheses, that are useful in assigning a probability and satisfy some additional logical properties - that are uniquely indexed by the probability distributions they lead to.

Motivation & Objective

• To provide a logical reinterpretation of probability-like parameters and priors in statistical models, avoiding reliance on infinite exchangeability or physical propensity concepts.
• To formalize the notion of 'circumstances'—pieces of knowledge that uniquely determine a plausibility distribution—as the foundational object for assigning probabilities.
• To offer a framework for understanding 'plausibilities of plausibilities' as arising from hierarchical reasoning over circumstances, rather than recursive probability nesting.
• To lay the groundwork for state assignment in classical and quantum mechanics by grounding priors in context-specific, indexed circumstances.
• To provide an alternative to de Finetti’s theorem and the physical/subjective probability distinction, using only probability theory and logical consistency.

Proposed method

• Introduces 'circumstances' as abstract, logically coherent pieces of knowledge (e.g., physical hypotheses) that lead to unique plausibility distributions.
• Defines a coarse-graining process that maps a set of base circumstances {Cⱼ} into a new set {S_q} indexed by the plausibility distributions they yield.
• Uses the condition that moment integrals over parameter spaces must be equal across equivalent circumstances to enforce consistency: ∫Γ′ qᵏⁱ₁ᵢ₁ pS′(q|I) dq = ∫Γ′′ qᵏⁱ₁ᵢ₁ pS′′(q|I) dq.
• Applies the product rule and marginalization in probability calculus to derive constraints on how plausibilities of events depend on underlying circumstances.
• Establishes that the plausibility of a circumstance S_q is given by P(S_q|I), which can be interpreted as the prior f(q|I) in standard statistical models.
• Proposes iterative reasoning over 'circumstances of circumstances' to model higher-order plausibilities, avoiding reliance on infinite sequences or recursive probability definitions.

Experimental results

Research questions

• RQ1 How can probability-like parameters in statistical models be given a coherent, non-metaphysical interpretation without invoking infinite exchangeable sequences?
• RQ2 What is the logical role of prior distributions in statistical inference, and how can they be grounded in concrete epistemic contexts?
• RQ3 Can 'plausibilities of plausibilities' be meaningfully interpreted without resorting to recursive or hierarchical probability nesting?
• RQ4 How can the concept of 'circumstance' serve as a unifying framework for state assignment in classical and quantum mechanics?
• RQ5 In what way does the circumstance-based approach differ from and improve upon de Finetti’s theorem or propensity interpretations?

Key findings

• Plausibility-like parameters q in statistical models can be interpreted as indexing unique circumstances S_q that lead to specific probability distributions.
• The prior f(q|I) in a statistical model is equivalent to the plausibility P(S_q|I), providing a direct epistemic interpretation of priors as plausibilities of circumstances.
• The framework ensures invariance of the coarse-grained circumstance set {S_q} under changes in the plausibilities of the original circumstances {C_j}, preserving consistency.
• Moment-matching conditions across different sets of circumstances (e.g., ∫ qᵏⁱ₁ᵢ₁ pS′(q|I) dq = ∫ qᵏⁱ₁ᵢ₁ pS′′(q|I) dq) ensure that equivalent plausibility assignments are preserved across different context representations.
• The approach provides a foundation for quantum and classical state assignment methods by grounding priors in context-specific, logically coherent circumstances rather than abstract or infinite constructs.
• The framework naturally supports iterative reasoning over multiple levels of circumstances, enabling a hierarchy of plausibilities without requiring infinite sequences or recursive probability definitions.