The magic of score matching estimators and approximations for distributions on manifolds and some cutting edge applications to molecular biology

It is well known that one of the major problems for maximum likelihood estimation (MLE) in the well established directional models is that the normalising constants can be difficult to evaluate. A new general method of “score matching estimation” (SME) will be presented on a compact oriented Riemannian manifold following Mardia et al. (2016). Important applications include von Mises-Fisher, Bingham and joint models on the sphere and related spaces. The estimator is found to be consistent and asymptotically normally distributed under mild regularity conditions. Further, it is easy to compute as a solution of a linear set of equations and requires no knowledge of the normalizing constant. Some examples will be given to demonstrate its good performance. To highlight its properties, in this paper we give another proof of the important result that the SME is the exact MLE for the multivariate normal distribution. Also in this paper, we show how we can approximate in general a given distribution to a member of the exponential family; we call this method ”score matching approximation” (SMA). Research for such types of approximations is common, for example for the wrapped normal distribution, and we introduce here the SMA for the multivariate wrapped normal using a multivariate von Mises distribution. Practical examples from structural molecular biology related to protein and RNA will be presented in the talk.