The analytic dually flat space of the mixture family of two prescribed distinct Cauchy distributions (2104.13801v3)

Abstract: A smooth and strictly convex function on an open convex domain induces both (1) a Hessian manifold with respect to the standard flat Euclidean connection, and (2) a dually flat space of information geometry. We first review these constructions and illustrate how to instantiate them for (a) full regular exponential families from their partition functions, (b) regular homogeneous cones from their characteristic functions, and (c) mixture families from their Shannon negentropy functions. Although these structures can be explicitly built for many common examples of the first two classes, the differential entropy of a continuous statistical mixture with distinct prescribed density components sharing the same support is hitherto not known in closed form, hence forcing implementations of mixture family manifolds in practice using Monte Carlo sampling. In this work, we report a notable exception: The family of mixtures defined as the convex combination of two prescribed and distinct Cauchy distributions. As a byproduct, we report closed-form formula for the Jensen-Shannon divergence between two mixtures of two prescribed Cauchy components.

• The paper presents an analytic formulation for the mixture of two distinct Cauchy distributions, establishing a dually flat space with closed-form information-geometric measures.
• It utilizes complex analytic and symbolic computation techniques to derive key metrics such as the Kullback-Leibler divergence, Shannon entropy, and Jensen-Shannon divergence.
• The results offer practical insights for efficient statistical modeling and optimization in applications requiring exact entropy computations for heavy-tailed data.

The Analytic Dually Flat Space of the Mixture Family of Two Prescribed Distinct Cauchy Distributions

This paper by Frank Nielsen addresses a problem in the domain of information geometry by presenting an analytic form for the mixture family of two distinct Cauchy distributions. It focuses on constructing dually flat spaces using information geometric structures, which have applications spanning statistical modeling and data analysis. The work is positioned within the niche of convex analysis and geometric statistics, offering closed-form solutions previously unattainable for such mixture models.

Foundations in Information Geometry

The paper begins with a concise review of constructing geometric structures from smooth, strictly convex functions within open convex domains, leading to Hessian and dually flat manifolds. Specifically, three types of families are discussed: exponential families, regular homogeneous cones, and mixture families. These families are foundational in information geometry, relating geometric structures with statistical models.

Exponential families and the resulting dual structures are well-documented, but mixture families with prescribed continuous density components have lacked closed-form solutions due to the non-analytic nature of their Bregman generators. This paper makes significant progress by presenting one such analytic solution for Cauchy mixtures—a family pivotal in statistical analysis due to its heavy-tailed characteristics.

Mixture of Two Cauchy Distributions

The paper's key contribution is formulating the mixture family of two distinct Cauchy distributions as an analytic dually flat space. This formulation allows direct computation of various information-geometric quantities, such as the Bregman divergence, in closed form.

Through complex analytic techniques, the author derives the Kullback-Leibler divergence between a Cauchy distribution and its mixture, ultimately producing a closed formula for the Shannon entropy of such mixtures. This is extended to compute the Jensen-Shannon divergence, establishing its interpretation in terms of Jensen's inequality—a core concept in convex analysis.

Implications and Computational Techniques

The results open pathways for new applications in statistical modeling where exact computation of mixture entropies is required. For instance, the closed-form solutions provided can serve in optimization problems in statistics and machine learning, where understanding mixtures is critical.

The author uses symbolic computation, specifically employing tools like Maxima, to ensure the correctness and simplify applied processes when deriving analytical solutions. These computational insights pave the way for future research involving continuous mixture models, potentially extending techniques to other distributions lacking closed-form entropy solutions.

Theoretical and Practical Advancements

From a theoretical perspective, this work deepens the understanding of mixture models by exploiting analytic properties of Cauchy distributions. Practically, it allows for more efficient algorithmic implementations in fields requiring precise statistical description of data, such as bioinformatics and aerospace engineering.

In summary, Nielsen's work stands as a crucial bridge in statistical geometry, providing definitive methods to overcome longstanding challenges in computing entropic properties of continuous mixtures. It hints at broader implications, where such methodology could further extend, for developing other statistical models, allowing richer and more precise data analysis tools. The advancement also suggests potential future research directions, like studying similar properties for other distribution families or complex mixture models.