Symmetry of models versus models of symmetry

TL;DR

This paper distinguishes between models that are symmetric and models that represent a subject's belief in symmetry, introducing two notions of symmetry in belief models and exploring their implications.

Contribution

It introduces two distinct notions of symmetry in belief models—weak and strong invariance—and discusses their mathematical and philosophical significance.

Findings

Differentiate between symmetry of models and models of symmetry

Introduce weak and strong invariance in belief models

Show relevance to probabilistic modeling and exchangeability

Abstract

A model for a subject's beliefs about a phenomenon may exhibit symmetry, in the sense that it is invariant under certain transformations. On the other hand, such a belief model may be intended to represent that the subject believes or knows that the phenomenon under study exhibits symmetry. We defend the view that these are fundamentally different things, even though the difference cannot be captured by Bayesian belief models. In fact, the failure to distinguish between both situations leads to Laplace's so-called Principle of Insufficient Reason, which has been criticised extensively in the literature. We show that there are belief models (imprecise probability models, coherent lower previsions) that generalise and include the Bayesian belief models, but where this fundamental difference can be captured. This leads to two notions of symmetry for such belief models: weak invariance (representing symmetry of beliefs) and strong invariance (modelling beliefs of symmetry). We discuss various mathematical as well as more philosophical aspects of these notions. We also discuss a few examples to show the relevance of our findings both to probabilistic modelling and to statistical inference, and to the notion of exchangeability in particular.