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.

中文翻译：

---

复合伽马对数正态分布统计特性的变分贝叶斯估计


提出了一种迭代变分贝叶斯方法来估计复合伽玛对数正态分布的统计特性，具体来说，伽玛分量的Nakagami参数以及对数正态分量的均值和方差参数。此外，模型中的随机隐藏信号在参数估计中起着关键作用，因此也估计了其密度。通过评估变分后验密度的变分贝叶斯 Cramer Rao 界限来分析参数估计性能。数值模拟显示界限和估计方差之间具有良好的一致性。为了对所提出的估计器进行基准测试，将参数估计的 MSE 和相对偏差与期望最大化算法的 MSE 和相对偏差进行比较。所提出的估计器总体上优于基准算法。对于 Nakagami 参数，性能改进非常显着，从较大样本量的 9 dB 到较小样本量的 24 dB。同样，随着样本量的增加，方差参数的性能从 1.5 dB 变化到 5.2 dB。所提出的算法还在基于相关潜在变量的模拟数据集上进行了测试，并且发现估计器对于弱相关数据是有效的。最后，对于隐藏信号，发现与期望最大化算法相比，所提出的方法的估计密度具有更低的 Kullback Leibler 散度。