Guaranteed Deterministic Bounds on the Total Variation Distance between Univariate Mixtures

TL;DR

This paper introduces two methods to compute guaranteed deterministic bounds on the total variation distance for univariate mixture models, addressing the lack of closed-form solutions and improving efficiency and reliability.

Contribution

The paper presents novel guaranteed bounds for total variation distance in univariate mixtures, utilizing information monotonicity and geometric envelopes, which are tighter and more efficient than existing methods.

Findings

The bounds are tight for Gaussian, Gamma, and Rayleigh mixtures.

The methods outperform Monte Carlo in providing deterministic guarantees.

Experimental results demonstrate the effectiveness of the bounds.

Abstract

The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in $[0,1]$. This distance plays a fundamental role in machine learning and signal processing: It is a member of the broader class of $f$-divergences, and it is related to the probability of error in Bayesian hypothesis testing. Since the total variation distance does not admit closed-form expressions for statistical mixtures (like Gaussian mixture models), one often has to rely in practice on costly numerical integrations or on fast Monte Carlo approximations that however do not guarantee deterministic lower and upper bounds. In this work, we consider two methods for bounding the total variation of univariate mixture models: The first method is based on the information monotonicity property of the total variation to design guaranteed nested deterministic lower bounds. The second method relies on computing the geometric lower and upper envelopes of weighted mixture components to derive deterministic bounds based on density ratio. We demonstrate the tightness of our bounds in a series of experiments on Gaussian, Gamma and Rayleigh mixture models.