Gaussian distributions are used to model uncertainty in signals and states, and Gaussian mixtures are often used when the underlying distribution is multimodal. Unlike a single Gaussian, a Gaussian mixture generally has no closed-form expression for differential entropy and therefore requires numerical approximation. We propose a Gauss--Hermite quadrature method for evaluating Gaussian mixture differential entropy. The quadrature order controls the numerical resolution of the approximation. The method is evaluated on one- and two-dimensional Gaussian mixture benchmarks against Taylor approximations, analytic entropy bounds, and numerical integration references. For repeated optimization over continuous actions, we also propose a Hermite polynomial surrogate in action space. In a radar pointing benchmark, its second-order form achieves substantially lower surrogate error and optimizer regret than a second-order Taylor surrogate based on local derivatives at the nominal action, while both methods use nine direct objective evaluations per replanning step. The Hermite surrogate also improves pointing performance in the tested benchmark.
Petra Eerikinharju, Marko Tuononen, Ville Hautamäkics.LG
Mutual information is a general measure of statistical dependence that captures both linear and nonlinear relationships between random variables. For continuous and multidimensional variables For continuous multidimensional variables, mutual information must be estimated from samples. Because mutual information is unbounded, its values are not directly comparable across datasets, dimensions, or applications. Normalized mutual information addresses this limitation by converting mutual information into a normalized dependency score. Recent work has demonstrated the practical value of normalized mutual information in applications such as molecular dynamics {arXiv:2405.04980} and interpretable machine learning {arXiv:2409.16768}, but existing estimators remain sensitive to dimensionality and numerical stability {arXiv:2410.07642}. In this paper, we propose a fully neural normalized mutual information estimator for continuous variables. The proposed approach combines a MINE-based neural mutual information estimator {arXiv:1801.04062} with MI-NEE-inspired neural marginal entropy estimators {arXiv:1905.12957}. Mutual information is estimated using the Donsker--Varadhan representation, while marginal entropies are estimated by learning the divergence between each marginal distribution and a uniform reference distribution, from which entropy is recovered. The resulting estimator provides a neural alternative to k-nearest-neighbor-based normalized mutual information estimation {arXiv:2405.04980}. Experiments on Gaussian data from one to eight dimensions show that the proposed estimator improves accuracy over a KSG-based normalized mutual information baseline. These results indicate that neural estimation is a promising direction for normalized dependency measurement in continuous multidimensional settings.