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Dirichlet Prior Sieves in Finite Normal Mixtures
 Statistica Sinica
, 2002
"... Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive ..."
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Cited by 40 (1 self)
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Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive use of the popular uniform Dirichlet prior leads to an inconsistent posterior. However, a simple adjustment to the parameters in the prior induces a random probability measure that approximates the Dirichlet process and yields a posterior that is strongly consistent for the density and weakly consistent for the unknown mixing distribution. The dimension of the resulting sieve can be selected easily in practice and a simple and efficient Gibbs sampler can be used to sample the posterior of the mixing distribution. Key words and phrases: BoseEinstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small
, 2003
"... We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that the maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(n ), 0 < d < 1, where n is the sample size. ..."
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Cited by 7 (1 self)
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We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that the maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(n ), 0 < d < 1, where n is the sample size. 1
BOOTSTRAPPING FINITE MIXTURE MODELS
 COMPSTAT’2004 SYMPOSIUM
, 2004
"... Finite mixture regression models are used for modelling unobserved heterogeneity in the population. However, depending on the specifications these models need not be identifiable, which is especially of concern if the parameters are interpreted. As bootstrap methods are already used as a diagnostic ..."
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Cited by 6 (4 self)
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Finite mixture regression models are used for modelling unobserved heterogeneity in the population. However, depending on the specifications these models need not be identifiable, which is especially of concern if the parameters are interpreted. As bootstrap methods are already used as a diagnostic tool for linear regression models, we investigate their use for finite mixture models. We show that bootstrapping helps in revealing identifiability problems and that parametric bootstrapping can be used for analyzing the reliability of coefficient estimates.
On the Identifiability of MixturesofExperts
 Neural Networks
, 1999
"... In mixturesofexperts (ME) models, "experts" of generalized linear models are combined, according to a set of local weights called the "gating function". The invariant transformations of the ME probability density functions include the permutations of the expert labels and the translations of the p ..."
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Cited by 5 (2 self)
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In mixturesofexperts (ME) models, "experts" of generalized linear models are combined, according to a set of local weights called the "gating function". The invariant transformations of the ME probability density functions include the permutations of the expert labels and the translations of the parameters in the gating functions. Under certain conditions, we show that the ME systems are identifiable if the experts are ordered and the gating parameters are initialized. The conditions are validated for Poisson, gamma, normal and binomial experts. KeywordsGeneralized linear models, identifiability, invariant transformations, mixturesofexperts. 1 INTRODUCTION MixturesofExperts (ME) (Jacobs et. al. 1991) and Hierarchical MixturesofExperts (HME) (Jordan and Jacobs 1994) originated from the neural network literature, and have had wide applications for examining relationships among variables [Cacciatore and Nowlan (1994), Meila and Jordan (1995), Ghahramani and Hinton (1996), Tip...
Limit Theorems and Estimation for Structural and Aggregate Teletraffic Models
, 2003
"... The thesis proposes models for aggregate data network traffic which incorporate the additional randomness arising from the randomness in the number of data sources. A conditionallyGaussian scale mixture process is shown to be a limit for the cumulative work from a random superposition of alternatin ..."
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Cited by 4 (1 self)
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The thesis proposes models for aggregate data network traffic which incorporate the additional randomness arising from the randomness in the number of data sources. A conditionallyGaussian scale mixture process is shown to be a limit for the cumulative work from a random superposition of alternating onoff processes. SubFractional Brownian Motion is shown to be the limit in a particular case. Queueing and estimation results for processes which are conditionally Fractional Gaussian Noise are included. A model with a superposition of alternating onoff processes with independent lifetimes is also considered.
A new approach to fitting linear models in high dimensional spaces
, 2000
"... This thesis presents a new approach to fitting linear models, called “pace regression”, which also overcomes the dimensionality determination problem. Its optimality in minimizing the expected prediction loss is theoretically established, when the number of free parameters is infinitely large. In th ..."
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Cited by 2 (0 self)
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This thesis presents a new approach to fitting linear models, called “pace regression”, which also overcomes the dimensionality determination problem. Its optimality in minimizing the expected prediction loss is theoretically established, when the number of free parameters is infinitely large. In this sense, pace regression outperforms existing procedures for fitting linear models. Dimensionality determination, a special case of fitting linear models, turns out to be a natural byproduct. A range of simulation studies are conducted; the results support the theoretical analysis. Through the thesis, a deeper understanding is gained of the problem of fitting linear models. Many key issues are discussed. Existing procedures, namely OLS, AIC, BIC, RIC, CIC, CV(d), BS(m), RIDGE, NNGAROTTE and LASSO, are reviewed and compared, both theoretically and empirically, with the new methods. Estimating a mixing distribution is an indispensable part of pace regression. A measurebased minimum distance approach, including probability measures and nonnegative measures, is proposed, and strongly consistent estimators are produced. Of all minimum distance methods for estimating a mixing distribution, only the
Pace Regression
, 1999
"... This paper articulates a new method of linear regression, \pace regression," that addresses many drawbacks of standard regression reported in the literatureparticularly the subset selection problem. Pace regression improves on classical ordinary least squares (ols) regression by evaluating the ee ..."
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Cited by 2 (0 self)
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This paper articulates a new method of linear regression, \pace regression," that addresses many drawbacks of standard regression reported in the literatureparticularly the subset selection problem. Pace regression improves on classical ordinary least squares (ols) regression by evaluating the eect of each variable and using a clustering analysis to improve the statistical basis for estimating their contribution to the overall regression. As well as outperforming ols, it also outperformsin a remarkably general senseother linear modeling techniques in the literature, including subset selection procedures, which seek a reduction in dimensionality that falls out as a natural byproduct of pace regression. The paper denes six procedures that share the fundamental idea of pace regression, all of which are theoretically justied in terms of asymptotic performance. Experiments conrm the performance improvement over other techniques. Keywords: Linear regression; subset model sele...
CONVERGENCE OF LATENT MIXING MEASURES IN FINITE AND INFINITE MIXTURE MODELS
"... This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and fdivergence functionals such as Hellinge ..."
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Cited by 1 (0 self)
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This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and fdivergence functionals such as Hellinger and Kullback–Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process. 1. Introduction. A
Understanding Choice Intensity: A Poisson Mixture Model with Logitbased Random Utility Selective Mixing
, 2009
"... In this paper we introduce a new Poisson mixture model for count panel data where the underlying Poisson process intensity is determined endogenously by consumer latent utility maximization over a set of choice alternatives. This formulation accommodates the choice and count in a single random utili ..."
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In this paper we introduce a new Poisson mixture model for count panel data where the underlying Poisson process intensity is determined endogenously by consumer latent utility maximization over a set of choice alternatives. This formulation accommodates the choice and count in a single random utility framework with desirable theoretical properties. Individual heterogeneity is introduced through the random coefficient framework with a flexible semiparametric distribution. We deal with the analytical intractability of the resulting mixture by recasting the model as an embedding of infinite sequences of scaled moments of the mixing distribution, and newly derive their cumulant representations along with bounds on their rate of numerical convergence. We further develop an efficient recursive algorithm for fast evaluation of the model likelihood within a Bayesian Gibbs sampling scheme. We apply our model to a recent household panel of supermarket visit counts. We estimate the nonparametric density of three key variables of interest – price, driving distance, and total expenditure – while controlling for a range of consumer demographic characteristics. We use this econometric framework to assess the opportunity cost of time and analyze the interaction between store choice, trip frequency, household characteristics and store characteristics.