In this note we introduce a new family of entropy powers which are related to generalized entropies, called Sharma-Mittal entropies, and we prove their concavity along diffusion processes generated by$L^{2}$-Wasserstein gradient flows of corresponding entropy functionals. This result extends the result of Savaré and Toscani (2014) on the concavity of Rényi entropy powers and reveals a connection to Rényi entropy power inequalities by Bobkov and Marsiglietti (2017).

Bayesian filters are often used in statistical inference and consist of recursively alternating between two steps: prediction and correction. Most commonly the Gaussian distribution is used within the Bayes filtering framework, but other distributions, which could model better the nature of the estimated phenomenon like the von Mises-Fisher distribution on the unit sphere, have also been subject of research interest. However, the von Mises-Fisher filter requires approximations since the prediction step does not yield an another von Mises-Fisher distribution. Furthermore, other advanced filtering methods require approximating a mixture of distributions with just a single component. In this paper we propose to use the score matching within the context of Bayesian assumed density filtering inlieu of the more common moment matching. Moment matching functions by assuming the type of the resulting distribution and then matching its moments with the prior distribution, which in the end minimizes the Kullback-Leibler divergence. Score matching also assumes the resulting distribution type, but finds optimal parameters by minimizing the relative Fisher information. In the paper we show that the score matching procedure results with identical performance, but with simpler equations that, unlike moment matching, do not require tedious numerical methods. In the end, we corroborate theoretical results by running the moment and score matching based filters for single and multiple object tracking on a large number of randomly generated trajectories on the unit sphere.