# A Noninformative Prior on a Space of Distribution Functions

## TL;DR

This paper extends the concept of noninformative priors in Bayesian analysis to infinite-dimensional spaces of distribution functions, providing a framework for unique posterior determination based on invariance principles.

## Contribution

It generalizes Jaynes' invariance-based prior construction to infinite-dimensional settings, introducing families of approximately invariant priors for distribution functions.

## Key findings

- Families of approximately invariant priors are constructed for distribution functions.
- Conditions are identified under which a Dirichlet Process posterior is uniquely determined by invariance.
- The approach links invariance principles to the derivation of Bayesian posteriors in complex models.

## Abstract

In a given problem, the Bayesian statistical paradigm requires the specification of a prior distribution that quantifies relevant information about the unknowns of main interest external to the data. In cases where little such information is available, the problem under study may possess an invariance under a transformation group that encodes a lack of information, leading to a unique prior---this idea was explored at length by E.T. Jaynes. Previous successful examples have included location-scale invariance under linear transformation, multiplicative invariance of the rate at which events in a counting process are observed, and the derivation of the Haldane prior for a Bernoulli success probability. In this paper we show that this method can be extended, by generalizing Jaynes, in two ways: (1) to yield families of approximately invariant priors, and (2) to the infinite-dimensional setting, yielding families of priors on spaces of distribution functions. Our results can be used to describe conditions under which a particular Dirichlet Process posterior arises from an optimal Bayesian analysis, in the sense that invariances in the prior and likelihood lead to one and only one posterior distribution.