A mixture of logistic skew-normal multinomial models

The logistic normal multinomial distribution is gaining interest in modelling microbiome data. It utilizes a hierarchical structure such that the observed counts conditional on the compositions are assumed to be multinomial random variables and the log -ratio transformed compositions are assumed to be from a Gaussian distribution. While multinomial distribution accounts for the compositional nature of the data, and a Gaussian prior offers flexibility in the structure of covariance matrices, the log -ratio transformed compositions of the microbiome data can be highly skewed, especially at a lower taxonomic level. Thus, a Gaussian distribution may not be an ideal prior for the log -ratio transformed compositions. A novel mixture of logistic skew -normal multinomial (LSNM) distribution is proposed in which a multivariate skew -normal distribution is utilized as a prior for the log -ratio transformed compositions. A variational Gaussian approximation in conjunction with the EM algorithm is utilized for parameter estimation.