Asymptotic inference for mixture models by using data‐dependent priors

For certain mixture models, improper priors are undesirable because they yield improper posteriors. However, proper priors may be undesirable because they require subjective input. We propose the use of specially chosen data‐dependent priors. We show that, in some cases, data‐dependent priors are the only priors that produce intervals with second‐order correct frequentist coverage. The resulting posterior also has another interpretation: it is the product of a fixed prior and a pseudolikelihood.

[1]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[2]  B. L. Welch,et al.  On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods , 1963 .

[3]  George E. Policello.  Conditional Maximum Likelihood Estimation in Gaussian Mixtures , 1981 .

[4]  S. Geman,et al.  Nonparametric Maximum Likelihood Estimation by the Method of Sieves , 1982 .

[5]  D. Lindley.  Bayes theory , 1984 .

[6]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[7]  L. Wasserman,et al.  Bayesian analysis of outlier problems using the Gibbs sampler , 1991 .

[8]  Jayaran Sethuramant.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[9]  K. Roeder.  Semiparametric Estimation of Normal Mixture Densities , 1992 .

[10]  A. Barron,et al.  Jeffreys' prior is asymptotically least favorable under entropy risk , 1994 .

[11]  M. Wand,et al.  EXACT MEAN INTEGRATED SQUARED ERROR , 1992 .

[12]  Anna Nicolaou.  Bayesian intervals with good frequency behaviour in the presence of nuisance parameters , 1993 .

[13]  C. Robert,et al.  Estimation of Finite Mixture Distributions Through Bayesian Sampling , 1994 .

[14]  Jiahua Chen.  Optimal Rate of Convergence for Finite Mixture Models , 1995 .

[15]  W. Wong,et al.  Probability inequalities for likelihood ratios and convergence rates of sieve MLEs , 1995 .

[16]  Adrian E. Raftery,et al.  Hypothesis Testing and Model Selection Via Posterior Simulation , 1995 .

[17]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[18]  B. Lindsay.  Mixture models : theory, geometry, and applications , 1995 .

[19]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[20]  L. Wasserman,et al.  Practical Bayesian Density Estimation Using Mixtures of Normals , 1997 .

[21]  P. Green,et al.  Corrigendum: On Bayesian analysis of mixtures with an unknown number of components , 1997 .

[22]  P. Green,et al.  On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion) , 1997 .

[23]  Jayanta K. Ghosh,et al.  Non-informative priors via sieves and packing numbers , 1997 .

[24]  S. MacEachern,et al.  Estimating mixture of dirichlet process models , 1998 .

[25]  L. Wasserman,et al.  RATES OF CONVERGENCE FOR THE GAUSSIAN MIXTURE SIEVE , 2000 .