Conditioning by rare sources

TL;DR

This paper investigates how the posterior probability of rare sources decays exponentially, using L-divergence, and shows that rare sources concentrate on an L-projection, extending large deviations principles to this context.

Contribution

It introduces a framework for analyzing the decay of posterior probabilities for rare sources using L-divergence, linking it to large deviations theory and L-projections.

Findings

Posterior probability of rare sources decays exponentially with rate determined by L-divergence.

Rare sources asymptotically concentrate on an L-projection within a convex, closed set.

Results extend classical large deviations principles to the context of source conditioning.

Abstract

In this paper we study the exponential decay of posterior probability of a set of sources and conditioning by rare sources for both uniform and general prior distributions of sources. The decay rate is determined by $L$-divergence and rare sources from a convex, closed set asymptotically conditionally concentrate on an $L$-projection. $L$-projection on a linear family of sources belongs to $\varLambda$-family of distributions. The results parallel those of Large Deviations for Empirical Measures (Sanov's Theorem and Conditional Limit Theorem).