A closed-form expression for the Sharma-Mittal entropy of exponential families (1112.4221v1)

Abstract: The Sharma-Mittal entropies generalize the celebrated Shannon, R\'enyi and Tsallis entropies. We report a closed-form formula for the Sharma-Mittal entropies and relative entropies for arbitrary exponential family distributions. We instantiate explicitly the formula for the case of the multivariate Gaussian distributions and discuss on its estimation.

• The paper derives a closed-form expression for the Sharma-Mittal entropy specifically for exponential family distributions.
• This closed-form solution allows for efficient, analytical computation of entropy, avoiding computationally intensive numerical methods.
• The findings have practical implications, enabling entropy estimation from empirical data using techniques like Maximum Likelihood Estimation and applications in divergences.

A Closed-Form Expression for the Sharma-Mittal Entropy of Exponential Families

The paper by Frank Nielsen and Richard Nock introduces a significant contribution to the field of statistical mechanics and information theory with their derivation of a closed-form expression for the Sharma-Mittal entropy within the framework of exponential family distributions. The Sharma-Mittal entropy is a generalization of several well-known entropy measures, including Shannon, Rényi, and Tsallis entropies. This work is distinguished by its application to exponential families, which encompass a broad array of commonly used statistical distributions.

Theoretical Framework and Main Results

The authors define the Sharma-Mittal entropy for a probability density function $p$ using the parameters $\alpha$ and $\beta$, offering a versatile tool for quantifying information in a way that interpolates between different entropy measures. The exponential families of distributions, which include Gaussian distributions and other pivotal distributions such as multinomials, are expressed in the form:

$$p_F(x|\theta) = \exp (\langle \theta, t(x) \rangle - F(\theta) + k(x))$$

where $\theta$ is the natural parameter, $t(x)$ the sufficient statistic, $F(\theta)$ the log-normalizer, and $k(x)$ an auxiliary carrier measure.

The closed-form formula derived for the Sharma-Mittal entropy is expressed as:

$$H_{\alpha,\beta}(p) = \frac{1}{1-\beta} \left( e^{\frac{1-\beta}{1-\alpha} (F(\alpha\theta)-\alpha F(\theta))} \right)$$

This result simplifies further when the auxiliary measure $k(x) = 0$ which is the case for multivariate Gaussians, allowing a straightforward computation of entropy in practical scenarios.

Implications and Estimations

The implications of this work are manifold. It allows researchers to analytically compute the Sharma-Mittal entropy without having to rely on numerical approximations, which can be computation-intensive especially for high-dimensional data. This permits more efficient and accurate analysis of the information content in datasets modeled by exponential family distributions.

The paper also extends the utility of these expressions to the computation of divergences, specifically the Sharma-Mittal divergence, which can be employed in scenarios such as hypothesis testing and information retrieval, providing a generalized foundation for these applications.

An important practical aspect is the potential to estimate these entropies from empirical data. The authors suggest utilizing the maximum likelihood estimation (MLE) to approximate natural parameters from samples, thus facilitating the extraction of entropy estimates from real-world data consistent with exponential families.

Future Prospects

This work not only bridges gaps between different entropy and divergence measures by showing their interconnectedness through the Sharma-Mittal entropies but also opens up new pathways for experimental and practical applications, particularly in statistical signal processing and machine learning. By providing a parametric and systematic approach to the computation of generalized entropies, the authors contribute a valuable tool for researchers dealing with complex data structures where traditional entropy measures may not suffice.

Anticipated future developments include applying this framework to other non-standard exponential families, extending its utilization in multidimensional data analysis, and refining numerical techniques associated with the estimation of these entropy measures in practice.