The Monte Carlo Computation Error of Transition Probabilities

Please always quote using this URN: urn:nbn:de:0297-zib-59933
  • In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the $L^2$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains.

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Metadaten
Author:Adam Nielsen
Document Type:ZIB-Report
Tag:Computation error; Markov operator; Measurable state space; Monte Carlo methods; Reversible Markov chain
MSC-Classification:37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
65-XX NUMERICAL ANALYSIS
CCS-Classification:G. Mathematics of Computing / G.1 NUMERICAL ANALYSIS
G. Mathematics of Computing / G.3 PROBABILITY AND STATISTICS
PACS-Classification:00.00.00 GENERAL / 02.00.00 Mathematical methods in physics
Date of first Publication:2016/07/22
Series (Serial Number):ZIB-Report (16-37)
ISSN:1438-0064
Published in:appeared in: Statistics & Probability Letters 118 (2016) pp. 163-170.