# Finite-Sample Concentration of the Multinomial in Relative Entropy

## TL;DR

This paper derives finite-sample concentration bounds for the divergence of empirical distributions from the true distribution in multinomial settings, improving existing bounds and providing moment estimates.

## Contribution

It introduces a gamma distribution-based bound for the moment generating function of the KL divergence in multinomial samples, enhancing finite-sample analysis.

## Key findings

- Exponential concentration inequality improves over standard bounds for large divergence values.
- Finite-sample bounds on moments are close to asymptotic values, within constant factors.
- Reduction to binary case allows for straightforward generalization to multinomial distributions.

## Abstract

We show that the moment generating function of the Kullback-Leibler divergence (relative entropy) between the empirical distribution of $n$ independent samples from a distribution $P$ over a finite alphabet of size $k$ (i.e. a multinomial distribution) and $P$ itself is no more than that of a gamma distribution with shape $k - 1$ and rate $n$. The resulting exponential concentration inequality becomes meaningful (less than 1) when the divergence $\varepsilon$ is larger than $(k-1)/n$, whereas the standard method of types bound requires $\varepsilon > \frac{1}{n} \cdot \log{\binom{n+k-1}{k-1}} \geq (k-1)/n \cdot \log(1 + n/(k-1))$, thus saving a factor of order $\log(n/k)$ in the standard regime of parameters where $n\gg k$. As a consequence, we also obtain finite-sample bounds on all the moments of the empirical divergence (equivalently, the discrete likelihood-ratio statistic), which are within constant factors (depending on the moment) of their asymptotic values. Our proof proceeds via a simple reduction to the case $k = 2$ of a binary alphabet (i.e. a binomial distribution), and has the property that improvements in the case of $k = 2$ directly translate to improvements for general $k$. In particular, we conjecture a bound on the binomial moment generating function that would almost close the quadratic gap between our finite-sample bound and the asymptotic moment generating function bound from Wilks' theorem (which does not hold for finite samples).