Geodesic
Hastings-Metropolis algorithm on Markov chains for small-probability estimation
Francois Bachoc1, 2 ; Achref Bachouch3 ; Lionel Lenôtre4, 5
1 Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna
2 When the work presented in this manuscript was carried out, the author was affiliated to CEA-Saclay, DEN, DM2S, STMF, LGLS, F-91191 Gif-Sur-Yvette, France and to the Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VII
3 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany
4 Inria, Research Centre Rennes-Bretagne Atlantique, Campus de Beaulieu 35042 Rennes Cedex France
5 Université de Rennes 1, Campus de Beaulieu 35042 Rennes Cedex France
ESAIM. Proceedings, Tome 48 (2015), pp. 276-307.

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Shielding studies in neutron transport, with Monte Carlo codes, yield challenging problems of small-probability estimation. The particularity of these studies is that the small probability to estimate is formulated in terms of the distribution of a Markov chain, instead of that of a random vector in more classical cases. Thus, it is not straightforward to adapt classical statistical methods, for estimating small probabilities involving random vectors, to these neutron-transport problems. A recent interacting-particle method for small-probability estimation, relying on the Hastings-Metropolis algorithm, is presented. It is shown how to adapt the Hastings-Metropolis algorithm when dealing with Markov chains. A convergence result is also shown. Then, the practical implementation of the resulting method for small-probability estimation is treated in details, for a Monte Carlo shielding study. Finally, it is shown, for this study, that the proposed interacting-particle method considerably outperforms a simple Monte Carlo method, when the probability to estimate is small.
DOI : 10.1051/proc/201448013

Francois Bachoc 1, 2 ; Achref Bachouch 3 ; Lionel Lenôtre 4, 5

1 Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna
2 When the work presented in this manuscript was carried out, the author was affiliated to CEA-Saclay, DEN, DM2S, STMF, LGLS, F-91191 Gif-Sur-Yvette, France and to the Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VII
3 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany
4 Inria, Research Centre Rennes-Bretagne Atlantique, Campus de Beaulieu 35042 Rennes Cedex France
5 Université de Rennes 1, Campus de Beaulieu 35042 Rennes Cedex France 
@article{EP_2015_48_a13,
     author = {Francois Bachoc and Achref Bachouch and Lionel Len\^otre},
     title = {Hastings-Metropolis algorithm on {Markov} chains for small-probability estimation},
     journal = {ESAIM. Proceedings},
     pages = {276--307},
     publisher = {mathdoc},
     volume = {48},
     year = {2015},
     doi = {10.1051/proc/201448013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201448013/}
}
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Francois Bachoc; Achref Bachouch; Lionel Lenôtre. Hastings-Metropolis algorithm on Markov chains for small-probability estimation. ESAIM. Proceedings, Tome 48 (2015), pp. 276-307. doi : 10.1051/proc/201448013. http://geodesic.mathdoc.fr/articles/10.1051/proc/201448013/

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