Markov Bridges, Bisection and Variance Reduction

Time-continuous Markov jump processes are popular modeling tools in disciplines ranging from computational finance and operations research to human genetics and genomics. The data is often sampled at discrete points in time, and it can be useful to simulate sample paths between the datapoints. In this paper we firstly consider the problem of generating sample paths from a continuous-time Markov chain conditioned on the endpoints using a new algorithm based on the idea of bisection. Secondly we study the potentials of the bisection algorithm for variance reduction. In particular, examples are presented where the methods of stratification, importance sampling and quasi Monte Carlo are investigated.

[1]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[2]  Miriam Hodge.  Valuing path-dependent options in the variance-gamma model by Monte Carlo with a gamma bridge , 2004 .

[3]  M. Kimura.  A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences , 1980, Journal of Molecular Evolution.

[4]  Cláudia Ribeiro,et al.  Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Lévy Processes , 2006 .

[5]  Jim Pitman,et al.  Markovian Bridges: Construction, Palm Interpretation, and Splicing , 1993 .

[6]  P. Green,et al.  Bayesian Markov chain Monte Carlo sequence analysis reveals varying neutral substitution patterns in mammalian evolution. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Asger Hobolth,et al.  SIMULATION FROM ENDPOINT-CONDITIONED, CONTINUOUS-TIME MARKOV CHAINS ON A FINITE STATE SPACE, WITH APPLICATIONS TO MOLECULAR EVOLUTION. , 2009, The annals of applied statistics.

[8]  D. Hunter.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 2000 .

[9]  Wanjun Gu,et al.  Rapid likelihood analysis on large phylogenies using partial sampling of substitution histories. , 2010, Molecular biology and evolution.

[10]  Gunther Leobacher,et al.  Stratified sampling and quasi-Monte Carlo simulation of Lévy processes , 2006, Monte Carlo Methods Appl..

[11]  Steffen L. Lauritzen,et al.  The estimation of phase-type related functionals using Markov chain Monte Carlo methods , 2003 .

[12]  G. Roberts,et al.  Retrospective exact simulation of diffusion sample paths with applications , 2006 .

[13]  Christof Schütte,et al.  Generator estimation of Markov jump processes , 2007, J. Comput. Phys..

[14]  Athanassios N. Avramidis,et al.  Efficient simulation of gamma and variance-gamma processes , 2003, Proceedings of the 2003 Winter Simulation Conference, 2003..

[15]  Gregory R. Grant,et al.  Statistical Methods in Bioinformatics , 2001 .

[16]  Alexandros Beskos,et al.  A Factorisation of Diffusion Measure and Finite Sample Path Constructions , 2008 .

[17]  P. Fearnhead,et al.  An exact Gibbs sampler for the Markov‐modulated Poisson process , 2006 .

[18]  G. Roberts,et al.  MCMC methods for diffusion bridges , 2008 .

[19]  G. Roberts,et al.  On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm , 2001 .

[20]  J. Felsenstein.  Evolutionary trees from DNA sequences: A maximum likelihood approach , 2005, Journal of Molecular Evolution.

[21]  Mogens Bladt,et al.  Corrigendum to “Simple simulation of diffusion bridges with application to likelihood inference for diffusions” , 2010, Bernoulli.

[22]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[23]  R. Caflisch,et al.  Modified Monte Carlo Methods Using Quasi-Random Sequences , 1995 .

[24]  J. L. Jensen,et al.  Probabilistic models of DNA sequence evolution with context dependent rates of substitution , 2000, Advances in Applied Probability.

[25]  David Haussler,et al.  New Methods for Detecting Lineage-Specific Selection , 2006, RECOMB.

[26]  Asger Hobolth,et al.  A Markov chain Monte Carlo Expectation Maximization Algorithm for Statistical Analysis of DNA Sequence Evolution with Neighbor-Dependent Substitution Rates , 2008 .

[27]  R. Caflisch,et al.  Smoothness and dimension reduction in Quasi-Monte Carlo methods , 1996 .

[28]  S. Asmussen,et al.  Applied Probability and Queues , 1989 .

[29]  Rasmus Nielsen,et al.  Mapping mutations on phylogenies , 2005 .

[30]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..