The Relevance of Proofs of the Rationality of Probability Theory to Automated Reasoning and Cognitive Models

TL;DR

This paper critically examines the relevance of proofs that rationality implies probability theory, arguing they do not guarantee probabilistic models are suitable or practical for automated reasoning or cognitive modeling.

Contribution

It highlights that existing theorems only show the existence of some probabilistic models, not necessarily useful or appropriate ones for specific reasoning tasks.

Findings

Theorems do not ensure probabilistic models are computationally tractable.

Many probabilistic models can be more irrational than non-probabilistic ones.

Different probabilistic models can be suitable for similar induction tasks.

Abstract

A number of well-known theorems, such as Cox's theorem and de Finetti's theorem. prove that any model of reasoning with uncertain information that satisfies specified conditions of "rationality" must satisfy the axioms of probability theory. I argue here that these theorems do not in themselves demonstrate that probabilistic models are in fact suitable for any specific task in automated reasoning or plausible for cognitive models. First, the theorems only establish that there exists some probabilistic model; they do not establish that there exists a useful probabilistic model, i.e. one with a tractably small number of numerical parameters and a large number of independence assumptions. Second, there are in general many different probabilistic models for a given situation, many of which may be far more irrational, in the usual sense of the term, than a model that violates the axioms of probability theory. I illustrate this second point with an extended examples of two tasks of induction, of a similar structure, where the reasonable probabilistic models are very different.