Grapham: Graphical models with adaptive random walk Metropolis algorithms

Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been applied successfully to many problems in Bayesian statistics. Grapham is a new open source implementation covering several such methods, with emphasis on graphical models for directed acyclic graphs. The implemented algorithms include the seminal Adaptive Metropolis algorithm adjusting the proposal covariance according to the history of the chain and a Metropolis algorithm adjusting the proposal scale based on the observed acceptance probability. Different variants of the algorithms allow one, for example, to use these two algorithms together, employ delayed rejection and adjust several parameters of the algorithms. The implemented Metropolis-within-Gibbs update allows arbitrary sampling blocks. The software is written in C and uses a simple extension language Lua in configuration.

[1]  Heikki Haario,et al.  Componentwise adaptation for high dimensional MCMC , 2005, Comput. Stat..

[2]  Heikki Haario,et al.  Adaptive proposal distribution for random walk Metropolis algorithm , 1999, Comput. Stat..

[3]  Michael I. Jordan.  Graphical Models , 2003 .

[4]  E. Saksman,et al.  On the ergodicity of the adaptive Metropolis algorithm on unbounded domains , 2008, 0806.2933.

[5]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[6]  G. W. Snedecor.  Statistical Methods , 1964 .

[7]  C. Robert,et al.  Controlled MCMC for Optimal Sampling , 2001 .

[8]  John W. Eaton,et al.  Gnu Octave Manual , 2002 .

[9]  G. Fort,et al.  Limit theorems for some adaptive MCMC algorithms with subgeometric kernels , 2008, 0807.2952.

[10]  J. Rosenthal,et al.  On adaptive Markov chain Monte Carlo algorithms , 2005 .

[11]  Heikki Haario,et al.  DRAM: Efficient adaptive MCMC , 2006, Stat. Comput..

[12]  Roberto Ierusalimschy,et al.  Lua—An Extensible Extension Language , 1996 .

[13]  Stuart J. Russell,et al.  Dynamic bayesian networks: representation, inference and learning , 2002 .

[14]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[15]  Jeffrey S. Rosenthal,et al.  AMCMC: An R interface for adaptive MCMC , 2007, Comput. Stat. Data Anal..

[16]  Harald Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2006 , 2007 .

[17]  Jeffrey S. Rosenthal,et al.  Analysis of the Gibbs Sampler for a Model Related to James-stein Estimators , 2007 .

[18]  J. Rosenthal,et al.  Coupling and Ergodicity of Adaptive Markov Chain Monte Carlo Algorithms , 2007, Journal of Applied Probability.

[19]  C. Andrieu,et al.  On the ergodicity properties of some adaptive MCMC algorithms , 2006, math/0610317.

[20]  Matti Vihola,et al.  On the stability and ergodicity of adaptive scaling Metropolis algorithms , 2009, 0903.4061.

[21]  Kevin Murphy.  Software for Graphical models : a review by , 2007 .

[22]  J. Rosenthal,et al.  Department of , 1993 .

[23]  Jack Dongarra,et al.  LINPACK Users' Guide , 1987 .

[24]  Walter R. Gilks,et al.  BUGS - Bayesian inference Using Gibbs Sampling Version 0.50 , 1995 .

[25]  Makoto Matsumoto,et al.  SIMD-Oriented Fast Mersenne Twister: a 128-bit Pseudorandom Number Generator , 2008 .

[26]  Jack J. Dongarra,et al.  The Netlib Mathematical Software Repository , 1995, D Lib Mag..

[27]  Gareth O. Roberts,et al.  Examples of Adaptive MCMC , 2009 .

[28]  Christophe Andrieu,et al.  A tutorial on adaptive MCMC , 2008, Stat. Comput..