Measure-Valued Differentiation for Markov Chains |
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Authors: | B Heidergott F J Vázquez-Abad |
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Institution: | (1) Department of Econometrics and Operations Research, Vrije Universiteit and Tinbergen Institute, Amsterdam, Netherlands;(2) Department of Mathematics and Statistics, and ARC Special Research Centre for Ultra-Broadband Information Networks, University of Melbourne, Melbourne, Australia |
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Abstract: | This paper addresses the problem of sensitivity analysis for finite-horizon performance measures of general Markov chains.
We derive closed-form expressions and associated unbiased gradient estimators for the derivatives of finite products of Markov
kernels by measure-valued differentiation (MVD). In the MVD setting, the derivatives of Markov kernels, called
-derivatives, are defined with respect to a class of performance functions
such that, for any performance measure
, the derivative of the integral of g with respect to the one-step transition probability of the Markov chain exists. The MVD approach (i) yields results that
can be applied to performance functions out of a predefined class, (ii) allows for a product rule of differentiation, that
is, analyzing the derivative of the transition kernel immediately yields finite-horizon results, (iii) provides an operator
language approach to the differentiation of Markov chains and (iv) clearly identifies the trade-off between the generality
of the performance classes that can be analyzed and the generality of the classes of measures (Markov kernels). The
-derivative of a measure can be interpreted in terms of various (unbiased) gradient estimators and the product rule for
-differentiation yields a product-rule for various gradient estimators.
Part of this work was done while the first author was with EURANDOM, Eindhoven, Netherlands, where he was supported by Deutsche
Forschungsgemeinschaft under Grant He3139/1-1. The work of the second author was partially supported by NSERC and FCAR grants
of the Government of Canada and Québec. |
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Keywords: | Gradient estimation Simulation Perturbation analysis Measure-valued differentiation |
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