## Bayesian Density Estimation and Inference Using Mixtures (1994)

Venue: | Journal of the American Statistical Association |

Citations: | 454 - 18 self |

### BibTeX

@ARTICLE{Escobar94bayesiandensity,

author = {Michael D. Escobar and Mike West},

title = {Bayesian Density Estimation and Inference Using Mixtures},

journal = {Journal of the American Statistical Association},

year = {1994},

volume = {90},

pages = {577--588}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe and illustrate Bayesian inference in models for density estimation using mixtures of Dirichlet processes. These models provide natural settings for density estimation, and are exemplified by special cases where data are modelled as a sample from mixtures of normal distributions. Efficient simulation methods are used to approximate various prior, posterior and predictive distributions. This allows for direct inference on a variety of practical issues, including problems of local versus global smoothing, uncertainty about density estimates, assessment of modality, and the inference on the numbers of components. Also, convergence results are established for a general class of normal mixture models. Keywords: Kernel estimation; Mixtures of Dirichlet processes; Multimodality; Normal mixtures; Posterior sampling; Smoothing parameter estimation * Michael D. Escobar is Assistant Professor, Department of Statistics and Department of Preventive Medicine and Biostatistics, University ...

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Citation Context ... n ) = Z P (Y n+1 j��)dP (��jD n ): (6) Direct evaluation of (6) is computationally extremely involved for even rather small sample size n due to the inherent complexity of the posterior P (��=-=��jD n ) (Antoniak 1974-=-; Escobar 1992; Lo 1984; West 1990). Fortunately, Monte Carlo approximation is possible using extensions of the iterative technique in Escobar (1988, 1994), now described. 3. COMPUTATIONS Recall that,... |

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Citation Context ...er of modes of a population distribution. Roeder (1990), for example, nonparametric inference on the number of modes in a mixture. Various methods exists for inference about the modality of mixtures (=-=Hartigan and Hartigan 1985-=-; Silverman 1981; Roeder 1990), though approaches to direct inference on numbers of components are less well-developed. In our framework, prior and posterior distributions for Bayesian density estimat... |

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Citation Context ...d depending on the form of the prior p(ff): Alternatively, we may discretise the range of ff so that (11) provides a discrete approximation to the posteriors -- the so-called `griddy Gibbs' approach (=-=Ritter and Tanner 1991-=-). More attractively, sampling from the exact, continuous posterior (11) is possible in the Gibbs iterations when the prior p(ff) comes from the class of mixtures of gamma distributions. We develop th... |

26 |
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Citation Context ...on functions appears in Figures 1(c) and 1(d). A nice way to exhibit uncertainties about density and distribution functions is via `live' animated graphical display of sequentially sampled functions (=-=Tierney 1991-=-). Restricted to static plots, we prefer displaying sampled curves to bands mapping pointwise interval estimates of the functions since the latter do not define density or distribution functions. The ... |

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Citation Context ...+1 jD n ): If the common prior distribution for the �� i is uncertain and modelled, in whole or in part, as a Dirichlet process, then the data come from a Dirichlet mixture of normals (Ferguson 19=-=83; Escobar 1988, 19-=-94; West 1990). The important special case in which V i = V has been studied widely; references appear in West (1990, 1992) who considers the common setup in which the �� i have a uncertain prior ... |

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Citation Context ...D n ): (6) Direct evaluation of (6) is computationally extremely involved for even rather small sample size n due to the inherent complexity of the posterior P (��jD n ) (Antoniak 1974; Escobar 19=-=92; Lo 1984; We-=-st 1990). Fortunately, Monte Carlo approximation is possible using extensions of the iterative technique in Escobar (1988, 1994), now described. 3. COMPUTATIONS Recall that, for each i, �� (i) = f... |

4 |
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Citation Context ...based on mixtures of standard components, such as normal mixtures, underly mainstream approaches to density estimation, including kernel techniques (Silverman 1986), nonparametric maximum likelihood (=-=Lindsay 1983-=-), and Bayesian approaches using mixtures of Dirichlet processes (Ferguson 1983). The latter provide theoretical bases for more traditional, nonparametric methods, such as kernel techniques, and hence... |

2 |
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Citation Context ...eder (1990), for example, nonparametric inference on the number of modes in a mixture. Various methods exists for inference about the modality of mixtures (Hartigan and Hartigan 1985; Silverman 1981; =-=Roeder 1990-=-), though approaches to direct inference on numbers of components are less well-developed. In our framework, prior and posterior distributions for Bayesian density estimation July 5, 1994 the number o... |

2 |
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- West
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Citation Context ...mmon prior distribution for the �� i is uncertain and modelled, in whole or in part, as a Dirichlet process, then the data come from a Dirichlet mixture of normals (Ferguson 1983; Escobar 1988, 19=-=94; West 1990). T-=-he important special case in which V i = V has been studied widely; references appear in West (1990, 1992) who considers the common setup in which the �� i have a uncertain prior which is modelled... |

1 | Bayesian density estimation July 5 - Silverman - 1994 |