Bayesian estimation of Dirichlet mixture model with variational inference

• An analytically tractable solution for Bayesian estimation of the Dirichlet mixture model.
• Relative convexity of the multivariate log-inverse-gamma function is proved and utilized.
• The free energy function is approximated by a single lower-bound to guarantee convergence.
• The method outperforms the ML based method and the VI based method, moreover, it is comparable to the sampling based method.
• The performances are demonstrated with important multimedia signal processing applications.

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.