On Bayesian learning from Bernoulli observations

TL;DR

This paper justifies Bayesian updating for Bernoulli observations using loss functions and asymptotic analysis, supporting Bayesian consistency under i.i.d. assumptions.

Contribution

It provides a new theoretical justification for Bayesian updating in Bernoulli models based on loss functions and asymptotics, clarifying assumptions behind Bayesian consistency.

Findings

Bayesian updating is justified for Bernoulli observations under certain conditions.

The approach relies on loss functions and asymptotic behavior.

Supports the validity of Bayesian methods in Bernoulli models.

Abstract

We provide a reason for Bayesian updating, in the Bernoulli case, even when it is assumed that observations are independent and identically distributed with a fixed but unknown parameter $\theta_0$. The motivation relies on the use of loss functions and asymptotics. Such a justification is important due to the recent interest and focus on Bayesian consistency which indeed assumes that the observations are independent and identically distributed rather than being conditionally independent with joint distribution depending on the choice of prior.