Multidimensional laplace approximation via trotter operator

Abstract

The classical distribution of Laplace, along with the normal one, became one of the most actively used symmetric probabilistic models. A separate task of mathematics is the Laplace approximation, i.e. method of estimating the parameters of the normal distribution in the approximation of a given probability density. In this article the problem of Laplace approximation in d- dimensional space has been investigated. In particular, the rates of convergence in problems of the multidimensional Laplace approximation are studied. The mathematical tool used in this article is the operator method developed by Trotter. It is very elementary and elegant. Two theorems are proved for the evaluation of convergence rate. The convergence rates, proved in the theorems, are expressed using two different types of results, namely: estimates of the convergence rate of the approximation are obtained in terms of “large-O” and “small-o”. The received results in this paper are extensions and generalizations of known results. The results obtained can be used when using the Laplace approximation in machine learning problems. The results in this note present a new approach to the Laplace approximation problems for the d-dimensional independent random variables.