## Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications (1997)

Venue: | Journal of the American Statistical Association |

Citations: | 63 - 1 self |

### BibTeX

@ARTICLE{Higdon97auxiliaryvariable,

author = {David M. Higdon},

title = {Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications},

journal = {Journal of the American Statistical Association},

year = {1997},

volume = {93},

pages = {585--595}

}

### Years of Citing Articles

### OpenURL

### Abstract

Suppose one wishes to sample from the density ß(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution ß(ujx) can be defined, giving the joint distribution ß(x; u) = ß(x)ß(ujx). A MCMC scheme which samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches. The revolutionary algorithm of Swendsen and Wang (1987) is one such example. In addition to reviewing the Swendsen-Wang algorithm and its generalizations, this paper introduces a new auxiliary variable method called partial decoupling. Two applications in Bayesian image analysis are considered. The first is a binary classification problem in which partial decoupling out performs SW and single site Metropolis. The second is a PET reconstruction which uses the gray level prior of Geman and McClure (1987). A generalized Swendsen-Wang algorithm is developed for this problem, which reduces the computing time to the point that MCMC is a viabl...

### Citations

2237 | Equation of State Calculation by Fast Computing Machines - Metropolis, Rosenbluth, et al. - 1953 |

1216 | Monte Carlo sampling methods using Markov chains and their applications - Hastings - 1970 |

752 | Markov chains for exploring posterior distributions - Tierney - 1994 |

257 | Maximum likelihood reconstruction for emission tomography - Shepp, Vardi - 1982 |

254 |
Nonuniversal critical dynamics in Monte Carlo simulations
- Swendsen, Wang
- 1987
(Show Context)
Citation Context ...how how auxiliary variable methods can lend insight into simple MCMC algorithms, and how they can lead to improvements over standard algorithms. In particular, we discuss the Swendsen-Wang algorithm (=-=Swendsen and Wang, 1987-=-) for Ising (1925) models and introduce partial decoupling, a modification which is more relevant to statistical inference. Section 3 gives two applications in Bayesian image analysis where standard M... |

161 | Beitrag zur Theorie des Ferromagnetismus. Zeitschrift fr Physik 31 - Ising - 1925 |

156 | Statistical methods for tomographic image reconstruction - Geman, McClure - 1987 |

145 | Simulating normalizing constants: From importance sampling to bridge sampling to path sampling
- Gelman, Meng
- 1997
(Show Context)
Citation Context ...ior simulation via MCMC is required to estimate Z(fi) over S using reverse logistic regression (Geyer 1991, 1997). Alternativly, thermodynamic integration (Ogata and Tanemura, 1984) or path sampling (=-=Gelman and Meng, 1996-=-) could have been employed. All of these methods require draws from the prior (8) which could not be done in a reasonable amount of time without the SW algorithm devised in Section 2.2. The resulting ... |

136 |
Generalization of the Fortuin-KasteleynSwendsen-Wang representation and Monte Carlo algorithm, Phys
- Edwards, Sokal
- 1988
(Show Context)
Citation Context ...veloped for this problem, which reduces the computing time to the point that MCMC is a viable method of posterior exploration. 1 Introduction The introduction of auxiliary variables to a MCMC scheme (=-=Edwards and Sokal, 1988-=-; Besag and Green, 1993) may allow one to construct Markov chains which are faster mixing and easier to simulate than standard single site algorithms. The idea is given formally in the above reference... |

133 | Some Generalized Order-Disorder Transformations - Potts - 1952 |

124 | Carlo methods in Statistical Mechanics: Foundations and new algorithms. Lecture Notes at the Cargese summer school on ”Functional Integration: basis and applications - Monte - 1996 |

84 | Likelihood inference for spatial point processes - Geyer - 1999 |

51 | Convergence of slice sampler Markov chains
- Roberts, Rosenthal
- 1999
(Show Context)
Citation Context .... (1997). Although one of the main motivations for its use is ease in coding, recent work has shown that theoretical properties of slice samplers are very good (Fishman, 1996; Mira and Tierney, 1997; =-=Roberts and Rosenthal, 1997-=-). In fact, Mira and Tierney (1997) show the slice sampler is superior to an independence Metoroplis sampler. This isn't surprising in light of the reexpression of the Metropolis algorithm as an auxil... |

43 | Ice floe identification in satellite images using mathematical morphoiogy and clustering about principal curves - Banfield, Raftery - 1992 |

40 | Estimating normalizing constants and reweighting mixtures in Markov chain Monte Carlo
- GEYER
- 1994
(Show Context)
Citation Context ...contains equally spaced values for fi between 0.8 and 1.2. Because Z(fi) is analytically intractable, prior simulation via MCMC is required to estimate Z(fi) over S using reverse logistic regression (=-=Geyer 1991-=-, 1997). Alternativly, thermodynamic integration (Ogata and Tanemura, 1984) or path sampling (Gelman and Meng, 1996) could have been employed. All of these methods require draws from the prior (8) whi... |

33 |
Likelihood analysis of spatial point patterns
- Ogata, Tanemura
- 1984
(Show Context)
Citation Context ...ecause Z(fi) is analytically intractable, prior simulation via MCMC is required to estimate Z(fi) over S using reverse logistic regression (Geyer 1991, 1997). Alternativly, thermodynamic integration (=-=Ogata and Tanemura, 1984-=-) or path sampling (Gelman and Meng, 1996) could have been employed. All of these methods require draws from the prior (8) which could not be done in a reasonable amount of time without the SW algorit... |

19 |
Spatial Applications of Markov Chain Monte Carlo for Bayesian Inference
- Higdon
- 1994
(Show Context)
Citation Context ...ting ffi ij = 0 8j 2 @i, and leaving the remaining ffi's at 1 will keep any neighboring site from bonding with the fixed site, while using the SW scheme away from the fixed site. For an example, see (=-=Higdon, 1994-=-). Another setting in which the SW algorithm can lead to slow mixing is when the external field term of the Ising model results results in a multimodal distribution for x. The original SW algorithm fa... |

16 | On the use of auxiliary variables in Markov chain Monte Carlo sampling
- Mira, Tierney
- 1997
(Show Context)
Citation Context ...ulations in Damien et al. (1997). Although one of the main motivations for its use is ease in coding, recent work has shown that theoretical properties of slice samplers are very good (Fishman, 1996; =-=Mira and Tierney, 1997-=-; Roberts and Rosenthal, 1997). In fact, Mira and Tierney (1997) show the slice sampler is superior to an independence Metoroplis sampler. This isn't surprising in light of the reexpression of the Met... |

15 |
An analysis of Swendsen±Wang and related sampling methods
- Fishman
- 1999
(Show Context)
Citation Context ...nconjugate formulations in Damien et al. (1997). Although one of the main motivations for its use is ease in coding, recent work has shown that theoretical properties of slice samplers are very good (=-=Fishman, 1996-=-; Mira and Tierney, 1997; Roberts and Rosenthal, 1997). In fact, Mira and Tierney (1997) show the slice sampler is superior to an independence Metoroplis sampler. This isn't surprising in light of the... |

9 |
Spatial Statistics and Bayesian Computation" (with discussion
- Besag, Green
- 1993
(Show Context)
Citation Context ... which reduces the computing time to the point that MCMC is a viable method of posterior exploration. 1 Introduction The introduction of auxiliary variables to a MCMC scheme (Edwards and Sokal, 1988; =-=Besag and Green, 1993-=-) may allow one to construct Markov chains which are faster mixing and easier to simulate than standard single site algorithms. The idea is given formally in the above references, but is alluded to in... |

8 | Conditional Monte Carlo for normal samples - TROTTER, TUICEP - 1956 |

6 |
Bayesian Computation and Stochastic Systems" (with discussion
- Besag, Green, et al.
- 1995
(Show Context)
Citation Context ...e patchy as in Figure 3. This suggests that SW may give substantial improvement for gray level priors that also yield patchy realizations. Though the commonly used Gaussian pairwise difference prior (=-=Besag et al. 1995, Sec 3)-=- doesn't exhibit this An illustration of the Swendsen-Wang algorithm for the Ising model on the 8 \Theta 8 lattice. ��(x) / exp 8 ! : fi X i��j I[x i = x j ] 9 = ; d d d d d d d d d d d d d d ... |

5 | Conditional Monte Carlo - Hammersley - 1956 |

4 | Difficulties in the use of auxiliary variables in Markov chains Monte carlo methods - Hurn - 1997 |

4 | A.Brandt, Simulations without Critical Slowing Down: Ising and Three-State Potts Models, Phys - Kandel - 1989 |

2 | Gibbs sampling for Bayesian nonconjugate models using auxiliary variables - Damien, Wakefield, et al. - 1998 |

2 | Contribution: "Spatial statistics and Bayesian computation" (with Discussion) , by - Higdon - 1993 |

1 |
A note on the Swendsen-Wang algorithm for ordered colours
- Green
- 1992
(Show Context)
Citation Context ..., much effort has been spent on generalizing the Swendsen Wang algorithm to a wider class of models. Examples include continuous spin models (Wolff, 1989; De Meo and Oh, 1992) and gray level imaging (=-=Green, 1992). Below we -=-describe the general form of the Swendsen-Wang algorithm due to Edwards and Sokal (1988). Suppose that the distribution of interest ��(x) can be written in the form ��(x) / �� 0 (x) Y k b ... |