2018/12/09 by Natsuki Kariya, Sumio Watanabe, Kariya, Natsuki +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistical Distribution Estimation and Applications #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1812.03510
openalex publication_date 2018/12/09 · openalex created_date 2019/05/03 · openalex updated_date 2026/07/28
When we use the normal mixture model, the optimal number of the components describing the data should be determined. Testing homogeneity is good for this purpose; however, to construct its theory is challenging, since the test statistic does not converge to the χ2 distribution even asymptotically. The reason for such asymptotic behavior is that the parameter set describing the null hypothesis (N.H.) contains singularities in the space of the alternative hypothesis (A.H.). Recently, a \itBayesian theory for singular models was developed, and it has elucidated various problems of statistical inference. However, its application to hypothesis tests for singular models has been limited. In this paper, we introduce a scaling technique that greatly simplifies the derivation and study testing of homogeneity for the first time the basis of Bayesian theory. We derive the asymptotic distributions of the marginal likelihood ratios in three cases: (1) only the mixture ratio is a variable in the A.H. ; (2) the mixture ratio and the mean of the mixed distribution are variables; And (3) the mixture ratio, the mean, and the variance of the mixed distribution are variables.; In all cases, the results are complex, but can be described as functions of random variables obeying normal distributions. A testing scheme based on them was constructed, and their validity was confirmed through numerical experiments.