[Paper Review] On a generalised form of subjective probability

Russell J. Bowater|arXiv (Cornell University)|Oct 25, 2018

Bayesian Modeling and Causal Inference12 references4 citations

TL;DR

This paper proposes a generalized form of subjective probability that addresses limitations in prior subjective probability theories by incorporating both the distribution function and the internal/external strength of that function relative to reference events. It resolves issues like non-additivity, discrete spacing, reference set dependency, and lack of universality, and successfully explains Ellsberg’s paradox and justifies fiducial inference over Bayesian inference in specific contexts.

ABSTRACT

This paper is motivated by the questions of how to give the concept of probability an adequate real-world meaning, and how to explain a certain type of phenomenon that can be found, for instance, in Ellsberg's paradox. It attempts to answer these questions by constructing an alternative theory to one that was proposed in earlier papers on the basis of various important criticisms that were raised against this earlier theory. The conceptual principles of the corresponding definition of probability are laid out and explained in detail. In particular, what is required to fully specify a probability distribution under this definition is not just the distribution function of the variable concerned, but also an assessment of the internal and/or the external strength of this function relative to other distribution functions of interest. This way of defining probability is applied to various examples and problems including, perhaps most notably, to a long-running controversy concerning the distinction between Bayesian and fiducial inference. The characteristics of this definition of probability are carefully evaluated in terms of the issues that it sets out to address.

Motivation & Objective

• To address criticisms of earlier subjective probability theories, particularly regarding additivity, discrete spacing, reference set dependency, and lack of universality.
• To provide a real-world interpretation of probability that accommodates rational ambiguity aversion, as seen in Ellsberg’s paradox.
• To unify the treatment of discrete, continuous, and categorical distributions under a single framework using internal and external strength of distribution functions.
• To justify fiducial inference as a rational alternative to Bayesian inference in cases where subjective priors are problematic or incoherent.
• To develop a practical, elicitation-based method for assessing probabilities that reflects human judgment under uncertainty.

Proposed method

• Introduces a generalized probability definition based on the similarity between the likelihood of an event and events in a reference set of standard physical experiments, such as drawing balls from an urn with unknown composition.
• Defines internal strength as the degree of similarity between a distribution function and a reference distribution, assessed via comparison with events in a reference set R.
• Defines external strength as the relative reliability or credibility of a distribution function compared to other candidate functions, based on consistency with observed data or expert judgment.
• Applies the strength concepts to both continuous and discrete distributions, including Bernoulli distributions, to ensure universality across variable types.
• Uses a reference set R based on ambiguous standard experiments (e.g., urns with unknown ball counts) to improve elicitation accuracy and reduce cognitive load in probability assessment.
• Employs a two-stage elicitation process: first, assess similarity (internal strength), then evaluate reliability (external strength), enabling a more robust and interpretable probability assignment.

Experimental results

Research questions

• RQ1 How can subjective probability be defined in a way that satisfies the additivity axiom while remaining grounded in real-world judgment?
• RQ2 Why do people exhibit ambiguity aversion in Ellsberg-type scenarios, and can this behavior be rationally justified within a coherent probability framework?
• RQ3 How can a unified theory of subjective probability be constructed that applies equally to discrete, continuous, and categorical distributions?
• RQ4 In what circumstances is fiducial inference more rational than Bayesian inference, and how can this be formalized using strength assessments?
• RQ5 Can the strength of a distribution function be objectively assessed in a way that supports reliable probability elicitation in practical decision-making?

Key findings

• The proposed theory resolves the additivity issue by embedding probability assessments within a strength-based framework that preserves coherence across events.
• The theory eliminates the need for discrete spacing by allowing continuous probability assignments through strength-based comparison with reference events.
• By introducing internal and external strength, the theory removes dependency on a fixed reference set, enhancing flexibility and reducing cognitive bias in elicitation.
• The theory provides a rational explanation for Ellsberg’s paradox by showing that ambiguity aversion arises from low internal strength in ambiguous distributions, not irrationality.
• The framework justifies fiducial inference in specific cases by demonstrating that the distribution function’s external strength can exceed that of subjective priors, making it a more defensible basis for inference.
• The theory achieves universality by allowing the same strength concepts to be applied to Bernoulli, continuous, and categorical distributions, removing the need for separate definitions.