[Paper Review] From "plausibilities of plausibilities" to state-assignment methods. I. "Plausibilities of plausibilities": an approach through circumstances

PierGianLuca Porta Mana, A. Maansson+1 more|arXiv (Cornell University)|Jul 17, 2006

Philosophy and History of ScienceArts and Humanities2 citations

TL;DR

This paper reinterprets 'probabilities of probabilities' using the concept of 'circumstance' to unify classical and quantum state-assignment methods. By applying a probability-theoretic theorem, it shows that propositions can be uniquely parametrized by probability distributions in a way invariant to changes in their own probabilities, offering a foundational framework for state assignment.

ABSTRACT

This is the first part of a three-note study which starts from an analysis of "probabilities of probabilities" to arrive at old and new state-assignment methods in classical and quantum mechanics. In this note, probability-like parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of 'circumstance'. The idea is basically Laplace's and Jaynes', and rests on a theorem from probability theory which shows that a set of propositions can be uniquely parametrised by probability distributions. This parametrisation is invariant with respect to changes in the probabilities of the propositions themselves.

Motivation & Objective

• To resolve conceptual ambiguities in 'probabilities of probabilities' by grounding them in the notion of 'circumstance'.
• To unify classical and quantum state-assignment methods through a common probabilistic foundation.
• To provide a mathematically rigorous reinterpretation of prior distributions in statistical models using invariant parametrization.
• To extend Laplace's and Jaynes' principles by formalizing their application via a probability-theoretic theorem.
• To establish a framework where state assignments are invariant under changes in the probabilities of underlying propositions.

Proposed method

• Reinterprets probability-like parameters in statistical models as functions of 'circumstances'—contextual conditions that define the state of knowledge.
• Applies a foundational theorem from probability theory showing that a set of propositions admits a unique parametrization by probability distributions.
• Uses this parametrization to define prior distributions that remain invariant under reassignment of the probabilities of the propositions.
• Establishes that the parametrization is independent of the actual probabilities of the propositions, relying only on the structure of the propositions and their logical relationships.
• Introduces a formalism where 'circumstance' acts as a meta-level variable that encodes the state of information, enabling consistent state assignment.
• Derives the invariance of the parametrization under transformations of the probabilities of the propositions, ensuring consistency in state assignments.

Experimental results

Research questions

• RQ1 How can 'probabilities of probabilities' be consistently reinterpreted to avoid circularity in statistical modeling?
• RQ2 What role does the concept of 'circumstance' play in unifying classical and quantum state-assignment methods?
• RQ3 In what way does the parametrization of propositions by probability distributions remain invariant under changes in the probabilities of those propositions?
• RQ4 How does this framework extend Laplace's and Jaynes' principles to a more general and formal setting?
• RQ5 Can a unified foundation for state assignment be constructed using invariant parametrization of propositions?

Key findings

• The paper establishes that a set of propositions can be uniquely parametrized by probability distributions, independent of the actual probabilities assigned to the propositions.
• This parametrization is invariant under changes in the probabilities of the propositions, ensuring consistency in state assignment across different probability values.
• The framework provides a formal basis for interpreting prior distributions as functions of 'circumstance', resolving ambiguities in higher-order probability models.
• The approach unifies classical and quantum state-assignment methods by grounding them in a common probabilistic structure.
• The use of 'circumstance' as a meta-level variable enables a coherent treatment of prior information in statistical inference.
• The theorem underpinning the parametrization ensures that the state assignment process is logically consistent and independent of arbitrary probability assignments to propositions.