Bayesian estimation of Dirichlet mixture model with variational inference

Abstract

In statistical modeling, parameter estimation is an essential and challengeable task. Estimation of the parameters in the Dirichlet mixture model (DMM) is analytically intractable, due to the integral expressions of the gamma function and its corresponding derivatives. We introduce a Bayesian estimation strategy to estimate the posterior distribution of the parameters in DMM. By assuming the gamma distribution as the prior to each parameter, we approximate both the prior and the posterior distribution of the parameters with a product of several mutually independent gamma distributions. The extended factorized approximation method is applied to introduce a single lower-bound to the variational objective function and an analytically tractable estimation solution is derived. Moreover, there is only one function that is maximized during iterations and, therefore, the convergence of the proposed algorithm is theoretically guaranteed. With synthesized data, the proposed method shows the advantages over the EM-based method and the previously proposed Bayesian estimation method. With two important multimedia signal processing applications, the good performance of the proposed Bayesian estimation method is demonstrated.