Today, various technologies that use artificial intelligence or machine learning have been developed and are used around us. This fact has increased the significance of the theoretical research upon which they are based.

The normal mixture model treated in this paper is one of the important models used in various fields including machine learning such as for pattern recognition and clustering analysis. In the analysis using the normal mixture model, it is important to decide the mixture ratio and parameters.
The homogeneity test is one of the important techniques for deciding these values.
However, in the homogeneity test for the normal mixture model, the existence of the singularity makes theoretical analysis difficult.
This paper has adopted the Bayesian hypothesis testing method that assumes the prior distribution of parameters for this problem. In the Bayesian hypothesis test, the logarithm of the mixture of the marginal likelihood ratio with respect to the the prior distribution of the parameters is an important statistic. However, this calculation is generally involved.
This paper has proposed an approach using the variational Bayes method that incorporates latent variables to reduce the amount of calculation and to derive the statistics precisely.
In this paper, an asymptotic analysis has also been made to clarify the behavior of this statistic. In particular, it has been shown that phase transition occurs in this problem setting.

In this paper, several numerical calculations have also been examined.
These results greatly contribute to the development of hypothesis testing problems not only in normal mixture models but also in stochastic models including singular points. The result of this paper is quite basic, and so its application is broad as mentioned above.
For these reasons, this paper is worthy of this award.