Guaranteed Deterministic Bounds on the Total Variation Distance between Univariate Mixtures (1806.11311v1)

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.

• The paper introduces two deterministic methods—information monotonicity-based and geometric envelope-based—to compute tight total variation distance bounds for univariate mixtures.
• It employs nested coarse-grained quantization and density ratio envelopes to achieve efficient and reliable approximations.
• Experiments with Gaussian, Gamma, and Rayleigh mixtures show that the proposed bounds outperform traditional Monte Carlo and Pinsker inequality approaches.

Analyzing Deterministic Bounds on Total Variation Distance in Univariate Mixture Models

The paper "Guaranteed Deterministic Bounds on the Total Variation Distance between Univariate Mixtures" by Frank Nielsen and Ke Sun addresses a fundamental problem in machine learning and signal processing: the computation of the total variation distance between statistical mixtures. This core statistical distance is crucial for Bayesian hypothesis testing, yet calculating it often necessitates costly numerical approaches or swift Monte Carlo approximations, which lack deterministic bounds. The authors introduce two methodologies to establish such bounds for univariate mixture models.

Methodological Approaches

The paper presents two methods for bounding the total variation distance—an integral metric that measures the disparity between probability distributions—between univariate mixture models:

1. Information Monotonicity-Based Lower Bounds: The authors leverage the information monotonicity property of the total variation distance to construct nested coarse-grained quantized lower bounds (CGQLB). This technique decomposes the sample space into a finite partition of intervals, yielding a hierarchical series of bounds. These bounds utilize a telescopic inequality across nested partitions of the mixture distributions, thereby ensuring a conservative approximation of the TV distance.
2. Geometric Envelope-Based Bounds: The second method computes the geometric lower and upper envelopes of component distributions, resulting in combinatorial envelope lower and upper bounds (CELB and CEUB). This approach employs density ratio bounds within pre-defined intervals, using cumulative distribution functions to quickly approximate bounds. The geometric interpretation provides a computationally efficient strategy for securing tight bounds.

Experimental Evaluation

The authors evaluated the proposed bounds through a series of experiments involving Gaussian, Gamma, and Rayleigh mixture models. The results indicated that the deterministic bounds are notably tight, positioning them as preferred alternatives to traditional Monte Carlo methods. Specifically, the CGQLB demonstrated superior performance as the sample size increased, offering an efficient means of obtaining reliable bounds with minimal computational overhead compared to Monte Carlo simulations.

In random Gaussian mixture model (GMM) experiments, the CELB and CEUB outperformed conventional bounds derived from the Pinsker inequality. The deterministic nature of these bounds makes them advantageous for settings where stochastic approximations are unsuitable or lack precision.

Theoretical and Practical Implications

The deterministic bounds described have significant implications for both theoretical understanding and practical applications in fields that rely on mixture models. The proven information monotonicity property for total variation distance affirms the robustness of these bounds, providing a foundation for extending the bounds to other $f$-divergences. Practically, this research offers substantial efficiency improvements for applications that require precise measurement of statistical distances, such as model evaluation, hypothesis testing, and data similarity assessments.

Additionally, this paper suggests avenues for future research in generalized total variation distances, pointing to the potential extension of these techniques to other bounded statistical distances. This could broaden the scope and applicability of deterministic bounds in machine learning and signal processing domains.

In conclusion, this paper marks a definitive contribution to the understanding and practical computation of bounds on the total variation distance for univariate mixture models. The deterministic, tight, and computationally efficient nature of the methods proposed underscores their potential utility and relevance in a variety of research and application contexts.