# Finite-time optimality of Bayesian predictors

## TL;DR

This paper proves that in the most general setting of sequential probability forecasting, Bayesian predictors can achieve near-optimal cumulative loss within a logarithmic factor, regardless of model assumptions.

## Contribution

It establishes the first non-asymptotic bound showing Bayesian predictors' finite-time optimality in extremely general, assumption-free settings.

## Key findings

- Bayesian predictor's loss matches any predictor's loss up to log n additive term.
- The bound applies uniformly over all models in the set C and all time steps.
- A lower bound shows the unavoidable growth of loss difference over time.

## Abstract

The problem of sequential probability forecasting is considered in the most general setting: a model set C is given, and it is required to predict as well as possible if any of the measures (environments) in C is chosen to generate the data. No assumptions whatsoever are made on the model class C, in particular, no independence or mixing assumptions; C may not be measurable; there may be no predictor whose loss is sublinear, etc. It is shown that the cumulative loss of any possible predictor can be matched by that of a Bayesian predictor whose prior is discrete and is concentrated on C, up to an additive term of order $\log n$, where $n$ is the time step. The bound holds for every $n$ and every measure in C. This is the first non-asymptotic result of this kind. In addition, a non-matching lower bound is established: it goes to infinity with $n$ but may do so arbitrarily slow.