## Computing Normalizing Constants for Finite Mixture Models via Incremental Mixture Importance Sampling (IMIS) (2003)

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Citations: | 14 - 5 self |

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@MISC{Steele03computingnormalizing,

author = {Russell J. Steele and Adrian E. Raftery and Mary J. Emond},

title = {Computing Normalizing Constants for Finite Mixture Models via Incremental Mixture Importance Sampling (IMIS)},

year = {2003}

}

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### Abstract

We propose a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive importance sampling function which is itself a mixture, with two types of component distributions, one concentrated and one diffuse. The more concentrated type of component serves the usual purpose of an importance sampling function, sampling mostly group assignments of high posterior probability. The less concentrated type of component allows for the importance sampling function to explore the space in a controlled way to find other, unvisited assignments with high posterior probability. Components are added adaptively, one at a time, to cover areas of high posterior probability not well covered by the current important sampling function. The method is called Incremental Mixture Importance Sampling (IMIS). IMIS is easy to implement and to monitor for convergence. It scales easily for higher dimensional

### Citations

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983 | Bayes factors
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Citation Context ...ce approximate 95% confidence bands for Î within 1.0 of the “gold” standard long EMC estimate. This is adequate for interpretation on the standard scale for interpreting Bayes factors (Jeffreys 1961; =-=Kass and Raftery 1995-=-), which views a Bayes factor of three or less as weak evidence or, in Jeffreys’s words, “evidence not worth more than a bare mention.” Sampling from the prior gives a reasonably good answer when aver... |

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Citation Context ... addressed difficulties with the multiple likelihood modes due to label-switching by specifying ordering contraints on the parameters by, for example, constraining θ1 >θ2 for a two-component mixture (=-=Richardson and Green 1997-=-). There are two drawbacks to this sort of prior specification. First, ordering components can become complicated as the dimensionality of θi and the number of groups both increase. Second, other rese... |

412 | Monte Carlo methods - Hammersley, Handscomb - 1964 |

324 | Marginal Likelihood from the Gibbs Output
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Citation Context ...Laplace method does not work in this situation. Markov chain Monte Carlo (MCMC) can be used to estimate mixture models, and associated methods can be used to approximate integrated likelihoods (e.g., =-=Chib 1995-=-;s714 R. J. STEELE, A.E.RAFTERY, AND M. J. EMOND Raftery 1996b). However, in addition to the usual problems with MCMC methods (dependent samples, convergence issues, complexity of programming and impl... |

281 |
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Citation Context ...and Raftery (1994), Raftery (1996b), and Lewis and Raftery (1997). The Bayesian information criterion (BIC) can be used as the basis for an asymptotic approximation to the Bayes factor (Schwarz 1978; =-=Kass and Wasserman 1995-=-; Raftery 1995). For finite mixture models, however, none of these methods is fully satisfactory. Two features of mixture models make many current methods for approximating the integrated likelihood p... |

118 |
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Citation Context ... hold in finite mixture models whenever one estimates a model with G components but the true number of components is smaller, so that the true parameter values lie on the edge of the parameter space (=-=Lindsay 1995-=-). A second feature is the “label-switching” problem, namely that the likelihood is invariant to relabeling of the mixture components, and so has G! modes of the same height. Additional local modes ar... |

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Citation Context ... as the dimensionality of θi and the number of groups both increase. Second, other researchers have found that ordering the components can cause computational and inferential difficulties (see, e.g., =-=Celeux et al. 2000-=- and Stephens 2000). Another difficulty with sampling from p(z|ˆτ,y) is that p(z|ˆτ,y) often contains many values close to 1, which does not allow the importance sampling function to explore much of t... |

109 | Simulating ratios of normalizing constants via a simple identity: A theoretical exploration
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Citation Context ...ikewise in order to facilitate comparisons with other methods. Liang and Wong (2001) suggested a simulated annealing MCMC approach for calculating normalizing constants combined with bridge sampling (=-=Meng and Wong 1996-=-). Their method, called evolutionary Monte Carlo (EMC), requires running several (in their examples, 20) Markov chains, each of which samples from fi(τ) =(p(y|τ,G)) ui p(τ|G), where ui =0, 0.05,...,1.... |

109 | Dealing with label switching in mixture models - Stephens |

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75 |
Bayesian Model Selection
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Citation Context ...ry (1996b), and Lewis and Raftery (1997). The Bayesian information criterion (BIC) can be used as the basis for an asymptotic approximation to the Bayes factor (Schwarz 1978; Kass and Wasserman 1995; =-=Raftery 1995-=-). For finite mixture models, however, none of these methods is fully satisfactory. Two features of mixture models make many current methods for approximating the integrated likelihood problematic. Th... |

67 | Sequential importance sampling for nonparametric Bayes models: the next generation, Canadian - MacEachern, Clyde, et al. - 1999 |

67 |
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Citation Context ...ce. Hesterberg (1995) suggested a simple fix for this particular drawback of importance sampling. Although mixtures of importance sampling functions had been proposed in the past (Oh and Berger 1993; =-=West 1993-=-; Givens and Raftery 1996), Hesterberg (1995) was the first to suggest using the Monte Carlo sampling function, p(z), as a component of the mixture importance sampling function δp(z)+(1− δ)g(z), givin... |

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54 |
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(Show Context)
Citation Context ...er than the parameters of the component densities) that are themselves mixtures and are specified adaptively. We propose two approaches to this. The first takes defensive mixture importance sampling (=-=Hesterberg 1995-=-; Raghavan and Cox 1998) as a starting point, and the second is based on sampling via perturbation of an initial grouping that has high posterior probability. One key advantage of our approach is that... |

45 | Real parameter evolutionary Monte Carlo with applications to Bayesian mixture models - Liang, Wong - 2001 |

45 | Theory of Probability (3rd edition - Jeffreys - 1960 |

40 |
Theory of Probability (3rd
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Citation Context ... all runs produce approximate 95% confidence bands for Î within 1.0 of the “gold” standard long EMC estimate. This is adequate for interpretation on the standard scale for interpreting Bayes factors (=-=Jeffreys 1961-=-; Kass and Raftery 1995), which views a Bayes factor of three or less as weak evidence or, in Jeffreys’s words, “evidence not worth more than a bare mention.” Sampling from the prior gives a reasonabl... |

40 | Inference in molecular population genetics - Stephens, Donnelly - 2000 |

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Citation Context ...y elliptically contoured (e.g., Raftery 1996a), and when this assumption holds it can provide approximations of remarkable quality (e.g., Tierney and Kadane 1986; Grunwald, Guttorp, and Raftery 1993; =-=Lewis and Raftery 1997-=-). However, for mixture models this assumption fails when the model being fit has G components and the actual number of components is smaller (Lindsay 1995), which is a situation of great interest for... |

33 | Methods for approximating integrals in statistics with special emphasis on Bayesian integration - Evans, Swartz - 1995 |

30 |
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Citation Context ...erg (1995) suggested a simple fix for this particular drawback of importance sampling. Although mixtures of importance sampling functions had been proposed in the past (Oh and Berger 1993; West 1993; =-=Givens and Raftery 1996-=-), Hesterberg (1995) was the first to suggest using the Monte Carlo sampling function, p(z), as a component of the mixture importance sampling function δp(z)+(1− δ)g(z), giving the following importanc... |

28 | Safe and effective importance sampling - Owen, Zhou - 2000 |

27 | Rejection control and sequential importance sampling - Liu, Chen, et al. - 1998 |

25 |
Integration of multimodal functions by Monte Carlo importance sampling
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Citation Context ... a very large variance. Hesterberg (1995) suggested a simple fix for this particular drawback of importance sampling. Although mixtures of importance sampling functions had been proposed in the past (=-=Oh and Berger 1993-=-; West 1993; Givens and Raftery 1996), Hesterberg (1995) was the first to suggest using the Monte Carlo sampling function, p(z), as a component of the mixture importance sampling function δp(z)+(1− δ)... |

19 | Monte Carlo Methods - Chen, Shao, et al. - 2000 |

19 |
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- Stephens
(Show Context)
Citation Context ...ent samples, convergence issues, complexity of programming and implementation), in mixture models they can easily fall foul of the label-switching problem (Celeux 1997; Celeux, Hurn, and Robert 2000; =-=Stephens 1997-=-, 2000b). For example, Neal (1998) pointed out that Chib’s (1995) results for a mixture model were in error for this reason. The problem could be solved correctly using the methods of Chib and Jeliazk... |

16 |
An attempt to define the nature of chemical diabetes using a multidimensional analysis
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Citation Context ...onte Carlo integration for this example makes it undesirable, and it is included here only for comparison. 3.2 DIABETES DATA Next we consider a higher-dimensional example from the medical literature (=-=Reaven and Miller 1979-=-). The dataset consists of blood measures of insulin, glucose, and insulin resistance levels (SSPG) for 145 diabetes patients; the pairs plot of the data is shown in Figure 4. Fraley and Raftery (1998... |

15 | Erroneous results in “Marginal likelihood from the Gibbs output - Neal - 1999 |

14 | Time Series of Continuous Proportions - Grunwald, Raftery, et al. - 1993 |

12 | Determination of the frequency of loss of heterozygosity in esophageal adeno-carcinoma nu cell sorting, whole genome amplification and microsatellite polymorphisms,” Oncogene - Barrett, Galipeau, et al. - 1996 |

12 | Probes of large-scale structure in the Corona Borealis region - POSTMAN, HUCHRA, et al. - 1986 |

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10 | Mixture Models for Genetic changes in cancer cells - Desai - 2000 |

9 | Reweighting Monte Carlo Mixtures
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Citation Context ...ng the mixture importance sampling function by incrementally adding components to the mixture to capture parts of the space that have been missed. We use a mixture importance sampling function (as in =-=Geyer 1991-=-), based on several τ ∗ j ’s, where each τ ∗ j =(θ∗ ,π ∗ ) corresponds to a local posterior mode in the parameter space. In the notation of Section 2.2, we will be adaptively constructing a function o... |

7 | T: Methods for Approximating Integrals - Evans, Swartz - 1995 |

5 | Covariate selection in hierarchical models of hospital admission counts: A Bayes factor approach - Rozenkranz, Raftery - 1994 |

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Citation Context ...eters of the component densities) that are themselves mixtures and are specified adaptively. We propose two approaches to this. The first takes defensive mixture importance sampling (Hesterberg 1995; =-=Raghavan and Cox 1998-=-) as a starting point, and the second is based on sampling via perturbation of an initial grouping that has high posterior probability. One key advantage of our approach is that the algorithm does not... |