Large deviation principles for the Ewens-Pitman sampling model

TL;DR

This paper establishes large deviation principles for the distribution of block frequencies in the Ewens-Pitman sampling model, revealing that initial samples do not influence the asymptotic behavior of future sample partitions.

Contribution

It provides the first large deviation principles for block frequency counts in the Ewens-Pitman model, including a conditional version, enhancing understanding of Bayesian nonparametric inference.

Findings

Large deviation principle for $n^{-1}M_{l,n}$ as $n$ grows

Conditional large deviation principle given initial sample

Conditional and unconditional principles coincide, showing no long-term initial sample impact

Abstract

Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable random partition induced by a sample of size $n$ from the Ewens-Pitman sampling model. We show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a large deviation principle and we characterize the corresponding rate function. A conditional counterpart of this large deviation principle is also presented. Specifically, given an initial sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional sample of size $m$. For any fixed $n$ and as $m$ tends to infinity, we establish a large deviation principle for the conditional number of blocks with frequency $l$ in the enlarged sample, given the initial sample. Interestingly, the conditional and unconditional large deviation principles coincide, namely there is no long lasting impact of the given initial sample. Potential applications of our results are discussed in the context of Bayesian nonparametric inference for discovery probabilities.