Variational Bayesian Estimation of Statistical Properties of Composite Gamma Log-normal Distribution

An iterative variational Bayesian method is proposed for estimation of the statistical properties of the composite gamma log-normal distribution, specifically, the Nakagami parameter of the gamma component and the mean and variance parameters of the log-normal component. Moreover, the random hidden signal in the model plays a key role in the parameter estimation, hence, its density is also estimated. The parameter estimation performance is analyzed by evaluation of the variational Bayesian Cramer Rao bound from the variational posterior densities. The numerical simulations show a good agreement between the bounds and the estimator variances. In order to benchmark the proposed estimators, the MSE and relative bias of the parameter estimates are compared with those of the expectation maximization algorithm. The proposed estimator has generally outperformed the benchmark algorithm. The performance improvement is significant for the Nakagami parameter varying from 9 dB for a larger sample size to 24 dB for a smaller sample size. Similarly, the performance for the variance parameter varies from 1.5 dB to 5.2 dB with increasing the sample size. The proposed algorithm is also tested on a simulated data-set based on correlated latent variables and the estimator is found to be effective for weakly correlated data. Finally, for the hidden signal, the estimated density from the proposed method is found to be having lower Kullback Leibler divergence in comparison to that of the expectation maximization algorithm.