Simple Monte Carlo integration with respect to Bernoulli convolutions

David M. Gómez; Pablo Dartnell

Applications of Mathematics (2012)

  • Volume: 57, Issue: 6, page 617-626
  • ISSN: 0862-7940

Abstract

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We apply a Markov chain Monte Carlo method to approximate the integral of a continuous function with respect to the asymmetric Bernoulli convolution and, in particular, with respect to a binomial measure. This method---inspired by a cognitive model of memory decay---is extremely easy to implement, because it samples only Bernoulli random variables and combines them in a simple way so as to obtain a sequence of empirical measures converging almost surely to the Bernoulli convolution. We give explicit bounds for the bias and the standard deviation for this approximation, and present numerical simulations showing that it outperforms a general Monte Carlo method using the same number of Bernoulli random samples.

How to cite

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Gómez, David M., and Dartnell, Pablo. "Simple Monte Carlo integration with respect to Bernoulli convolutions." Applications of Mathematics 57.6 (2012): 617-626. <http://eudml.org/doc/246303>.

@article{Gómez2012,
abstract = {We apply a Markov chain Monte Carlo method to approximate the integral of a continuous function with respect to the asymmetric Bernoulli convolution and, in particular, with respect to a binomial measure. This method---inspired by a cognitive model of memory decay---is extremely easy to implement, because it samples only Bernoulli random variables and combines them in a simple way so as to obtain a sequence of empirical measures converging almost surely to the Bernoulli convolution. We give explicit bounds for the bias and the standard deviation for this approximation, and present numerical simulations showing that it outperforms a general Monte Carlo method using the same number of Bernoulli random samples.},
author = {Gómez, David M., Dartnell, Pablo},
journal = {Applications of Mathematics},
keywords = {MCMC; Bernoulli convolution; binomial measure; Monte Carlo integration; empirical measures; Bernoulli convolution; binomial measure; Monte Carlo integration; empirical measures},
language = {eng},
number = {6},
pages = {617-626},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Simple Monte Carlo integration with respect to Bernoulli convolutions},
url = {http://eudml.org/doc/246303},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Gómez, David M.
AU - Dartnell, Pablo
TI - Simple Monte Carlo integration with respect to Bernoulli convolutions
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 6
SP - 617
EP - 626
AB - We apply a Markov chain Monte Carlo method to approximate the integral of a continuous function with respect to the asymmetric Bernoulli convolution and, in particular, with respect to a binomial measure. This method---inspired by a cognitive model of memory decay---is extremely easy to implement, because it samples only Bernoulli random variables and combines them in a simple way so as to obtain a sequence of empirical measures converging almost surely to the Bernoulli convolution. We give explicit bounds for the bias and the standard deviation for this approximation, and present numerical simulations showing that it outperforms a general Monte Carlo method using the same number of Bernoulli random samples.
LA - eng
KW - MCMC; Bernoulli convolution; binomial measure; Monte Carlo integration; empirical measures; Bernoulli convolution; binomial measure; Monte Carlo integration; empirical measures
UR - http://eudml.org/doc/246303
ER -

References

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