Bayesian analysis of finite mixture models of distributions from exponential families

This paper deals with the Bayesian analysis of finite mixture models with a fixed number of component distributions from natural exponential families with quadratic variance function (NEF-QVF). A unified Bayesian framework addressing the two main difficulties in this context is presented, i.e., the prior distribution choice and the parameter unidentifiability problem. In order to deal with the first issue, conjugate prior distributions are used. An algorithm to calculate the parameters in the prior distribution to obtain the least informative one into the class of conjugate distributions is developed. Regarding the second issue, a general algorithm to solve the label-switching problem is presented. These techniques are easily applied in practice as it is shown with an illustrative example.

[1]  Don H. Johnson,et al.  Information-Theoretic Analysis of Neural Coding , 2004, Journal of Computational Neuroscience.

[2]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[3]  C. Morris,et al.  Exponential and bayesian conjugate families: Review and extensions , 1997 .

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

[5]  P. Green,et al.  Bayesian Analysis of Poisson Mixtures , 2002 .

[6]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[7]  J. F. C. Kingman,et al.  Information and Exponential Families in Statistical Theory , 1980 .

[8]  C. Morris.  Natural Exponential Families with Quadratic Variance Functions: Statistical Theory , 1983 .

[9]  P. Green,et al.  Modelling spatially correlated data via mixtures: a Bayesian approach , 2002 .

[10]  Guido Consonni,et al.  Conjugate Priors for Exponential Families Having Quadratic Variance Functions , 1992 .

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

[12]  Wilfried Seidel,et al.  Editorial: recent developments in mixture models , 2003, Comput. Stat. Data Anal..

[13]  Student,et al.  ON THE ERROR OF COUNTING WITH A HAEMACYTOMETER , 1907 .

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

[15]  M. Stephens.  Dealing with label switching in mixture models , 2000 .

[16]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[17]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[18]  C. Morris.  Natural Exponential Families with Quadratic Variance Functions , 1982 .

[19]  C. Robert,et al.  Computational and Inferential Difficulties with Mixture Posterior Distributions , 2000 .

[20]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .