On learning parametric distributions from quantized samples

TL;DR

This paper investigates the fundamental limits of estimating unknown parametric distributions from quantized samples in a network, providing theoretical bounds on estimation error using generalized inequalities and optimal transport metrics.

Contribution

It extends the van Trees inequality to $L_p$-norms with $p > 1$ and derives minimax lower bounds for distribution estimation from quantized data.

Findings

Generalized van Trees inequality for $L_p$-norms established.

Minimax lower bounds derived for estimation error under $L_p$ and Wasserstein losses.

Theoretical framework applicable to distributed sensor networks and quantized data scenarios.

Abstract

We consider the problem of learning parametric distributions from their quantized samples in a network. Specifically, $n$ agents or sensors observe independent samples of an unknown parametric distribution; and each of them uses $k$ bits to describe its observed sample to a central processor whose goal is to estimate the unknown distribution. First, we establish a generalization of the well-known van Trees inequality to general $L_p$-norms, with $p > 1$, in terms of Generalized Fisher information. Then, we develop minimax lower bounds on the estimation error for two losses: general $L_p$-norms and the related Wasserstein loss from optimal transport.