### Applied Aspects of Information Technology

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.

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DOI
//10.15276/aait.01.2018.4
References
1. MacKay, D. J. C. (2005). “Information Theory, Inference, and Learning Algorithms”. Cambridge University Press, 628 pp.
2. Kotz, S., Kozubowski, T. J., & Podgo'rski, K. (2001). “The Laplace distribution and generalizations: A revisit with applications to communications economics, engineering, and finance”, Birkhauser Boston, Inc., Boston, MA.
3. Sakalauskas, V. (1977). “On an estimate in the multidimensional limit theorems”, Liet. matem. Rink, 17 (4), pp. 195-201.
4. Hung, T. L., & Giang, L. T. (2014).“On bounds in Poisson approximation for integer-valued independent random variables”, Journal of Inequalities and Applications, 291 p.
5. Trotter, H. F. (1959). “An elementary proof of the central limit theorem”, Arch. Math (Basel), 10, pp. 226-234.
6. Kirschfink, H. (1989). “The generalized Trotter operator and weak convergence of dependent random variables in different probability metrics”, Results in Mathematics, Vol. 15, pp. 294-323.
7. Butzer, P. L., & Hahn, L. (1978). “General theorems on rates of convergence in distribution of random variables I. General limit theorems”, Journal of multivariate analysis 8, pp. 181-201.
8. Prakasa Rao, B. L. S. (1977). “On the rate of approximation in the multidimensional central limit theorem”, Liet. matem. rink, 17, pp. 187-194.
9. Khuri, A. I. (2003), “Advanced calculus with applications in statistics”, Wiley, Inc., Hoboken, New Jersey.
10. Kalashnikov, V. (1997). “Geometric sums: bounds for rare events with applications. Risk analysis, reliability, and queueing. Mathematics and its Applications”, 413. Kluwer Academic Publishers Group, Dordrecht, 265 pp.
11. Nghiem, T. H., & Giang, L. T. (2016). “Laplace approximation with the method of Trotter operator”, Can the University Journal of science, 47(1), pp. 120-126.
12. Toda, A. A. (2012). “Weak limit of the geometric sum of independent but not identically distributed random variables”. – Available at: https://arxiv.org/abs/1111.1786.
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