Background
In my exploration of Gaussian Mixture Models (GMMs) within the scope of statistical learning, I have encountered the concept of entropy as a measure of uncertainty or randomness in a probability distribution. For single Gaussian distributions, the calculation of entropy is straightforward and is known to be associated with the distribution's variance.
Given that GMMs comprise several Gaussian distributions, each with distinct means, variances, and mixing coefficients, I understand that these parameters significantly influence the GMM's entropy. My goal is to delve deeper into this topic and gain a more formal understanding of the entropy of GMMs.
Questions
What is the entropy of a Gaussian Mixture Model (GMM)? I am looking for a formal definition or formula that captures the entropy of a GMM, considering its composite nature of multiple Gaussian distributions.
Under what conditions does a GMM achieve maximum entropy? Specifically, are there known relationships or formulas that connect the entropy of a GMM with its components' variances, means, and mixing coefficients? Does an increase in the number of components with equal mixing coefficients and variances necessarily lead to higher entropy, or are the interactions between the components' means and variances more critical?
Additionally, I am interested in understanding how the arrangement and characteristics of the Gaussian components, such as their separation in means and overlap in variances, affect the model's overall entropy. Are there particular configurations of these parameters that are known to maximize entropy for a GMM with a set number of components?
Seeking Guidance
Any theoretical insights, references to scholarly articles, or illustrative examples that shed light on these questions would be greatly appreciated. Formal derivations or discussions that elaborate on these aspects of GMMs and their entropy would be particularly helpful.
Thank you in advance for your contributions to clarifying these complex topics.