<p>Compound Gaussian (CG) models are widely used to characterize high-resolution radar sea clutter due to their ability to represent heavy-tailed fluctuations. This paper addresses the estimation of the texture variance parameter in a CG clutter model with log-normal texture using a Bayesian minimum mean square error (MMSE) framework. By transforming the squared clutter envelope into the logarithmic domain, the estimation problem is reformulated as a Gaussian variance estimation task. An inverse-gamma prior is assigned to the augmented variance parameter, leading to a closed-form MMSE estimator from the posterior distribution. Unlike conventional maximum likelihood and moment-based approaches, the proposed method avoids iterative optimization while naturally allowing the incorporation of prior information and uncertainty quantification. Its performance is evaluated through extensive Monte Carlo simulations and comparisons with the maximum likelihood estimator (MLE), the [z log(z)] estimator, and higher- and fractional-order moment estimators. Validation using real sea-clutter measurements from the IPIX X-band radar database further confirms the effectiveness of the approach. Results from both simulated and real data demonstrate that the proposed estimator provides lower bias and mean square error, particularly in small sample and heterogeneous clutter environments, making it a practical and robust solution for radar clutter parameter estimation.</p>