A series of maximum entropy upper bounds of the differential entropy (1612.02954v1)

Abstract: We present a series of closed-form maximum entropy upper bounds for the differential entropy of a continuous univariate random variable and study the properties of that series. We then show how to use those generic bounds for upper bounding the differential entropy of Gaussian mixture models. This requires to calculate the raw moments and raw absolute moments of Gaussian mixtures in closed-form that may also be handy in statistical machine learning and information theory. We report on our experiments and discuss on the tightness of those bounds.

• The paper introduces a novel framework using Jaynes's maximum entropy principle to derive closed-form upper bounds on differential entropy.
• It presents Absolute Monomial Exponential Families (AMEFs) that enable explicit computation of entropy bounds based on raw absolute moments.
• The method is applied to Gaussian Mixture Models, showing that under certain conditions the Laplacian bound can outperform the traditional Gaussian bound.

Maximum Entropy Upper Bounds on Differential Entropy

This paper by Frank Nielsen and Richard Nock addresses a significant theoretical aspect of information theory: establishing upper bounds on the differential entropy of continuous random variables. The authors introduce a series of maximum entropy upper bounds (MEUBs), which provide an elegant mathematical approach for bounding differential entropy — particularly useful when dealing with Gaussian Mixture Models (GMMs).

Summary of Contributions

1. Maximum Entropy Principle: The paper employs Jaynes's Maximum Entropy principle to frame the problem of determining a distribution that maximizes entropy under given moment constraints. This distribution is identified as belonging to an exponential family, with natural expressions derived for its entropy.
2. Absolute Monomial Exponential Families (AMEFs): The authors introduce a new class of maximum entropy distributions called Absolute Monomial Exponential Families. These are key to formulating a series of upper bounds for differential entropy by exploiting the closed-form expressions of these families' entropies.
3. Expression of Upper Bounds: The series of upper bounds $U_l(X)$ are derived based on the raw absolute moments of a random variable. The authors provide a theorem that details how these bounds can be systematically computed for any continuous univariate distribution.
4. Application to Gaussian Mixture Models: The theory is made practical by its application to GMMs. Closed-form expressions for raw absolute moments of Gaussian mixtures are provided, facilitating the computation of these bounds for GMMs, where closed-form differential entropy calculations are generally intractable.
5. Comparative Analysis of MEUBs: Experimentation highlights that under certain conditions, the Laplacian MEUB can outperform the Gaussian MEUB, contrary to typical expectations. This insight prompts the recommendation to practitioners to choose the tighter of these bounds in applied settings.

Implications and Future Work

The implications of this research extend into multiple domains within AI and statistical machine learning, particularly wherein modeling involves complex mixtures of distributions. By providing efficient entropy upper bounds, the authors offer robust tools for tasks like density estimation, model selection, and anomaly detection, where entropy plays a pivotal role.

From a theoretical perspective, the introduction of the AMEFs not only broadens our understanding of entropy maximization but also sets a foundation for future exploration into other parametric families. Additionally, the methodological framework may inspire analogous developments in multivariate contexts or under different statistical constraints.

In conclusion, this work provides a substantial contribution to the theoretical landscape of information theory, with numerous potential applications in AI and data science. The mathematical rigor and practical insights delineated in this paper are poised to influence both ongoing research and applications requiring robust entropy estimation methods.