## Detecting Poor Convergence of Posterior Samplers due to Multimodality

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Citations: | 3 - 1 self |

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@MISC{Woodard_detectingpoor,

author = {Dawn B. Woodard},

title = {Detecting Poor Convergence of Posterior Samplers due to Multimodality},

year = {}

}

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### Abstract

Computation in Bayesian statistical models is often performed us-ing sampling techniques such as Markov chain Monte Carlo (MCMC) or adaptive Monte Carlo methods. The convergence of the sampler to the posterior distribution is typically assessed using a set of standard diag-nostics; recent draft Food and Drug Administration guidelines for the use of Bayesian statistics in medical device trials, for instance, advocate this approach for validating computations. We give several examples showing that this approach may be in-sufficient when the posterior distribution is multimodal–that lack of convergence due to posterior multimodality can be undetected using the standard convergence diagnostics, including the Gelman-Rubin di-agnostic that was introduced for exactly this problem. We show that the poor convergence can be detected by modifying a validation technique that was originally proposed for detecting coding errors in MCMC soft-

### Citations

3889 |
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...y for multimodal distributions. For example, the Potts model is a multimodal 3smodel used in statistical physics as well as spatial statistics and image processing (Banerjee, Carlin and Gelfand 2004; =-=Geman and Geman 1984-=-). A Gibbs sampler based on only local moves has been proven to have a convergence rate that is exponentially small as a function of the parameter dimension, for some types of Potts models (Borgs et a... |

726 |
Inference from iterative simulation using multiple sequences with discussion
- Gelman, Rubin
- 1992
(Show Context)
Citation Context ...e posterior distribution. Convergence diagnostics such as time series and autocorrelation plots that are based on a single chain also fail to diagnose that there are modes that have never been found (=-=Gelman and Rubin 1992-=-b). In recognition of this fact, the diagnostic by Gelman and Rubin (1992) was introduced to detect such problems. This diagnostic is applied by simulating several Markov chains, started from initial ... |

343 |
Variable selection via Gibbs sampling
- McCulloch, George
- 1993
(Show Context)
Citation Context ... that there is an undetected narrow mode then leads to overestimation of the posterior variance. We show that the same effect can occur for the popular stochastic search variable selection technique (=-=George and McCulloch 1993-=-), leading to incorrect inferences. We then argue that the modified validation technique should be widely applied when using sampling methods such as MCMC for computation in models where posterior uni... |

293 | Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments - Geweke - 1992 |

248 | Markov chain Monte Carlo convergence diagnostics: a comparative review - COWLES, CARLIN - 1996 |

235 | General methods for monitoring convergence of iterative simulations - Brooks, Gelman - 1998 |

210 | Hierarchical Modeling and Analysis for Spatial Data - Banerjee, Carlin, et al. - 2004 |

115 | Computational and inferential difficulties with mixture posterior distributions - Celeux, Hurn, et al. - 2000 |

57 | Bayesian variable selection with related predictors,” The Canadian - Chipman - 1996 |

55 | Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling - Jasra, Holmes, et al. - 2005 |

55 | Bayesian curve fitting using multivariate normal mixtures - Muller, Erkanli, et al. - 1996 |

46 |
W.: Real-Parameter Evolutionary Monte Carlo With Applications to Bayesian Mixture Models
- Liang, Wong
(Show Context)
Citation Context ...irable properties. The example that we use is that of posterior multimodality, which can occur in mixture models, nonparametric models, and model selection problems among others (Liang and Wong 2000; =-=Liang and Wong 2001-=-; Jasra, Holmes and Stephens 2005). Many common sampling methods use only local moves in the parameter space. For this reason, they can become trapped for extended periods within a single mode of the ... |

33 |
Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models
- ISHWARAN, ZAREPOUR
- 2000
(Show Context)
Citation Context ...ihood is invariant to certain changes in the labeling of the parameters. These models included, for instance, several nonparametric Dirich17slet process mixture models (Muller, Erkanli and West 1996; =-=Ishwaran and Zarepour 2000-=-). This choice was due to the well-known multimodal nature of these models and poor mixing properties of their Gibbs samplers. However, for these examples the validation method did not find lack of co... |

29 |
A single series from the Gibbs sampler provides a false sense of security
- GELMAN, RUBIN
- 1992
(Show Context)
Citation Context ...e posterior distribution. Convergence diagnostics such as time series and autocorrelation plots that are based on a single chain also fail to diagnose that there are modes that have never been found (=-=Gelman and Rubin 1992-=-b). In recognition of this fact, the diagnostic by Gelman and Rubin (1992) was introduced to detect such problems. This diagnostic is applied by simulating several Markov chains, started from initial ... |

29 | The Bayesian Choice, 2nd ed - Robert - 2001 |

25 | Evolutionary monte carlo: Applications to cp model sampling and change point problem
- Liang, Wong
(Show Context)
Citation Context ...s, or their other desirable properties. The example that we use is that of posterior multimodality, which can occur in mixture models, nonparametric models, and model selection problems among others (=-=Liang and Wong 2000-=-; Liang and Wong 2001; Jasra, Holmes and Stephens 2005). Many common sampling methods use only local moves in the parameter space. For this reason, they can become trapped for extended periods within ... |

15 |
V.Vu, Torpid mixing of some mcmc algorithms in statistical physics
- Borgs, Frieze, et al.
- 1999
(Show Context)
Citation Context ...Geman 1984). A Gibbs sampler based on only local moves has been proven to have a convergence rate that is exponentially small as a function of the parameter dimension, for some types of Potts models (=-=Borgs et al. 1999-=-). Slow convergence has also been noted empirically for many MCMC techniques applied to multimodal examples (Celeux, Hurn and Robert 2000). If only one or a few modes of a multimodal distribution is e... |

11 | Validation of software for bayesian models using posterior quantiles
- Cook, Gelman, et al.
(Show Context)
Citation Context ... of convergence. The narrower mode or modes are simply too difficult to detect. However, we show that a validation technique originally introduced for detecting coding errors in MCMC implementations (=-=Cook et al. 2006-=-) can be modified to effectively detect lack of convergence of any posterior sampling technique. The resulting method assesses the convergence properties of the posterior sampler for many data sets dr... |

10 | Interpretation and inference in mixture models: simple MCMC works - Geweke |

6 |
Discussion on the Meeting on the Gibbs sampler and other Markov Chain Monte Carlo Methods
- Clifford
- 1993
(Show Context)
Citation Context ...tribution. It has been observed that if the features of the posterior distribution are completely unknown, then convergence diagnosis for a single Markov chain 2is inherently an intractable problem (=-=Clifford 1993-=-). The possible presence of modes with very small “basins of attraction” (Section 3.1) implies that a convergence diagnostic would have to exhaustively enumerate the space in order to guarantee that t... |

3 |
Discussion of ’Equi-energy sampler’ by
- Atchadé, Liu
- 2006
(Show Context)
Citation Context ...lidation method can be applied to any such posterior sampler. For instance, many adaptive Monte Carlo methods are known to have this property, such as the equi-energy sampler on a finite state space (=-=Atchade and Liu 2006-=-). After n independent replications of steps 1-3, obtaining empirical quantiles {qj : j = 1, . . . , n}, Cook et al. (2006) suggest testing uniformity using the function f({qj : j = 1, . . . , n}) = n... |

2 | The computational complexity of estimating convergence time - Bhatnagar, Bogdanov, et al. - 2010 |

2 | Convergence of adaptive MCMC algorithms: Ergodicity and law of large numbers,. Available at: http://perso. telecom-paristech.fr/~gfort/Publications.html - Fort, Moulines, et al. - 2010 |

2 | Interpretation and inference in mixture models - Geweke - 2007 |