Subjective inference

TL;DR

This paper characterizes how agents' subjective inferences about event corroboration can be represented mathematically, extending Bayesian models to include signed measures and revealing limitations in inference accuracy.

Contribution

It introduces a unique signed measure representation for subjective inferences that satisfy Villegas' axioms without monotonicity, and links these to Bayesian posterior-prior differences.

Findings

Unique signed measure representation for inferences

Bayesian interpretation with a unique posterior and equivalent guesses

Observation of priors and posteriors can reveal inference errors

Abstract

An agent observes a clue, and an analyst observes an inference: a ranking of events on the basis of how corroborated they are by the clue. We prove that if the inference satisfies the axioms of Villegas (1964) except for the classic qualitative probability axiom of monotonicity, then it has a unique normalized signed measure representation (Theorem 1). Moreover, if the inference also declares the largest event equivalent to the smallest event, then it can be represented as a difference between a posterior and a prior such that the former is the conditional probability of the latter with respect to an assessed event that is interpreted as a clue guess. Across these Bayesian representations, the posterior is unique, all guesses are in a suitable sense equivalent, and the prior is determined by the weight it assigns to each possible guess (Theorem 2). However, observation of a prior and posterior compatible with the inference could reveal that all of these guesses are wrong.