Smoothness of Metropolis-Hastings algorithm and application to entropy estimation
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 419-431.

The transition kernel of the well-known Metropolis-Hastings (MH) algorithm has a point mass at the chain's current position, which prevent direct smoothness properties to be derived for the successive densities of marginals issued from this algorithm. We show here that under mild smoothness assumption on the MH algorithm “input” densities (the initial, proposal and target distributions), propagation of a Lipschitz condition for the iterative densities can be proved. This allows us to build a consistent nonparametric estimate of the entropy for these iterative densities. This theoretical study can be viewed as a building block for a more general MCMC evaluation tool grounded on such estimates.

DOI : 10.1051/ps/2012004
Classification : 60J22, 62M05, 62G07
Mots clés : entropy, Kullback divergence, Metropolis-Hastings algorithm, nonparametric statistic
@article{PS_2013__17__419_0,
     author = {Chauveau, Didier and Vandekerkhove, Pierre},
     title = {Smoothness of {Metropolis-Hastings} algorithm and application to entropy estimation},
     journal = {ESAIM: Probability and Statistics},
     pages = {419--431},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2012004},
     mrnumber = {3066386},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2012004/}
}
TY  - JOUR
AU  - Chauveau, Didier
AU  - Vandekerkhove, Pierre
TI  - Smoothness of Metropolis-Hastings algorithm and application to entropy estimation
JO  - ESAIM: Probability and Statistics
PY  - 2013
SP  - 419
EP  - 431
VL  - 17
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2012004/
DO  - 10.1051/ps/2012004
LA  - en
ID  - PS_2013__17__419_0
ER  - 
%0 Journal Article
%A Chauveau, Didier
%A Vandekerkhove, Pierre
%T Smoothness of Metropolis-Hastings algorithm and application to entropy estimation
%J ESAIM: Probability and Statistics
%D 2013
%P 419-431
%V 17
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2012004/
%R 10.1051/ps/2012004
%G en
%F PS_2013__17__419_0
Chauveau, Didier; Vandekerkhove, Pierre. Smoothness of Metropolis-Hastings algorithm and application to entropy estimation. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 419-431. doi : 10.1051/ps/2012004. http://archive.numdam.org/articles/10.1051/ps/2012004/

[1] I.A. Ahmad and P.E. Lin, A nonparametric estimation of the entropy for absolutely continuous distributions. IEEE Trans. Inf. Theory 22 (1976) 372-375. | MR | Zbl

[2] I.A. Ahmad and P.E. Lin, A nonparametric estimation of the entropy for absolutely continuous distributions. IEEE Trans. Inf. Theory 36 (1989) 688-692. | Zbl

[3] C. Andrieu and J. Thoms, A tutorial on adaptive MCMC. Stat. Comput. 18 (2008) 343-373. | MR

[4] Y.F. Atchadé and J. Rosenthal, On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 (2005) 815-828. | MR | Zbl

[5] P. Billingsley, Probability and Measure, 3rd edition. Wiley, New York (2005). | Zbl

[6] D. Chauveau and P. Vandekerkhove, Improving convergence of the Hastings-Metropolis algorithm with an adaptive proposal. Scand. J. Stat. 29 (2002) 13-29. | MR | Zbl

[7] D. Chauveau and P. Vandekerkhove, A Monte Carlo estimation of the entropy for Markov chains. Methodol. Comput. Appl. Probab. 9 (2007) 133-149. | MR | Zbl

[8] Y.G. Dmitriev and F.P. Tarasenko, On the estimation of functionals of the probability density and its derivatives. Theory Probab. Appl. 18 (1973) 628-633. | Zbl

[9] Y.G. Dmitriev and F.P. Tarasenko, On a class of non-parametric estimates of non-linear functionals of density. Theory Probab. Appl. 19 (1973) 390-394. | Zbl

[10] R. Douc, A. Guillin, J.M. Marin and C.P. Robert, Convergence of adaptive mixtures of importance sampling schemes. Ann. Statist. 35 (2007) 420-448. | MR | Zbl

[11] E.J. Dudevicz and E.C. Van Der Meulen Entropy-based tests of uniformity. J. Amer. Statist. Assoc. 76 (1981) 967-974. | MR | Zbl

[12] P.P.B. Eggermont and V.N. Lariccia, Best asymptotic normality of the Kernel density entropy estimator for Smooth densities. IEEE Trans. Inf. Theory 45 (1999) 1321-1326. | MR | Zbl

[13] W.R. Gilks, S. Richardson and D.J. Spiegelhalter, Markov Chain Monte Carlo in practice. Chapman & Hall, London (1996) | MR | Zbl

[14] W.R. Gilks, G.O. Roberts and S.K. Sahu, Adaptive Markov chain Monte carlo through regeneration. J. Amer. Statist. Assoc. 93 (1998) 1045-1054. | MR | Zbl

[15] L. Györfi and E.C. Van Der Meulen, Density-free convergence properties of various estimators of the entropy. Comput. Statist. Data Anal. 5 (1987) 425-436. | MR | Zbl

[16] L. Györfi and E.C. Van Der Meulen, An entropy estimate based on a Kernel density estimation, Limit Theorems in Probability and Statistics Pécs (Hungary). Colloquia Mathematica societatis János Bolyai 57 (1989) 229-240. | Zbl

[17] H. Haario, E. Saksman and J. Tamminen, An adaptive metropolis algorithm. Bernouilli 7 (2001) 223-242. | MR | Zbl

[18] W.K. Hastings, Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 (1970) 97-109. | Zbl

[19] L. Holden, Geometric convergence of the Metropolis-Hastings simulation algorithm. Statist. Probab. Lett. 39 (1998). | MR | Zbl

[20] A.V. Ivanov and M.N. Rozhkova, Properties of the statistical estimate of the entropy of a random vector with a probability density (in Russian). Probl. Peredachi Inform. 17 (1981) 33-43. Translated into English in Probl. Inf. Transm. 17 (1981) 171-178. | MR | Zbl

[21] S.F. Jarner and E. Hansen, Geometric ergodicity of metropolis algorithms. Stoc. Proc. Appl. 85 (2000) 341-361. | MR | Zbl

[22] K.L. Mengersen and R.L. Tweedie, Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 (1996) 101-121. | MR | Zbl

[23] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, Equations of state calculations by fast computing machines. J. Chem. Phys. 21 (1953) 1087-1092.

[24] A. Mokkadem, Estimation of the entropy and information of absolutely continuous random variables. IEEE Trans. Inf. Theory 23 (1989) 95-101. | MR | Zbl

[25] R Development Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org (2010), ISBN 3-900051-07-0.

[26] G.O. Roberts and J.S. Rosenthal, Optimal scaling for various Metropolis-Hastings algorithms. Statist. Sci. 16 (2001) 351-367. | MR | Zbl

[27] G.O. Roberts and R.L. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 (1996) 95-110. | MR | Zbl

[28] D. Scott, Multivariate Density Estimation: Theory, Practice and Visualization. John Wiley, New York (1992). | MR | Zbl

[29] F.P. Tarasenko, On the evaluation of an unknown probability density function, the direct estimation of the entropy from independent observations of a continuous random variable and the distribution-free entropy test of goodness-of-fit. Proc. IEEE 56 (1968) 2052-2053.

Cité par Sources :