Philatelic Mixtures and Multimodal Densities

Izenman, Alan Julian; Sommer, Charles J.

doi:10.2307/2290118

Cited by 53 publications

(75 citation statements)

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“…12. Indicators of convergence for a ®xed number of components for the data set ofIzenman and Sommer (1988), discussed inRobert and Mengersen (1995): (a) successive ®ts of the mixture with estimated parameters; (b) convergence of the estimated parameters; (c) allocation maps with grey level indicating the current component at a given iteration; (d) evaluation of the central limit theorem for the standardized sample of allocation averagesFig. 13.…”

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confidence: 99%

On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)

Richardson

Green

1997

Journal of the Royal Statistical Society Series B: Statistical Methodology

1,748

1,533

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New methodology for fully Bayesian mixture analysis is developed, making use of reversible jump Markov chain Monte Carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture. A sample from the full joint distribution of all unknown variables is thereby generated, and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution. The methodology is applied here to the analysis of univariate normal mixtures, using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context.

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mentioning

confidence: 99%

On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)

Richardson

Green

1997

Journal of the Royal Statistical Society Series B: Statistical Methodology

1,748

1,533

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“…The excess variance of the second component is caused by the underlying skewness, which GMM fails to capture by design. Fig 11(B) analyzes the distribution of the thicknesses of Hidalgo Stamp data measured by Walton Van Winkle and analyzed first by Wilson (1983) and then by many researchers including Izenman and Sommer (1988).…”

Section: Biological Sciencementioning

confidence: 99%

Large-scale mode identification and data-driven sciences

Mukhopadhyay¹

2017

Electron. J. Statist.

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Bump-hunting or mode identification is a fundamental problem that arises in almost every scientific field of data-driven discovery. Surprisingly, very few data modeling tools are available for automatic (not requiring manual case-by-case investigation), objective (not subjective), and nonparametric (not based on restrictive parametric model assumptions) mode discovery, which can scale to large data sets. This article introduces LPMode-an algorithm based on a new theory for detecting multimodality of a probability density. We apply LPMode to answer important research questions arising in various fields from environmental science, ecology, econometrics, analytical chemistry to astronomy and cancer genomics.

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“…Another well-known benchmark for mixtures is the Hidalgo stamps dataset, first studied by Izenman and Sommer (1988) (see also e.g. Basford et al (1997)), which consists of the thickness (in mm) of n = 485 stamps from a given Mexican stamp issue; see Figure 1 for a histogram.…”

Section: A Second Example: the Hidalgo Stamps Datamentioning

confidence: 99%

Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors

2011

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Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a "reaction coordinate", that is, a "direction" in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called "free energy" in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered.A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate.We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets.

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Philatelic Mixtures and Multimodal Densities

Cited by 53 publications

References 0 publications

On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)

On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)

Large-scale mode identification and data-driven sciences

Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors

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