Efficient CRB Estimation for Linear Models via Expectation Propagation and Monte Carlo Sampling

The Cramér–Rao bound (CRB) quantifies the variance lower bound for unbiased estimators, but it is intractable to evaluate in linear hierarchical Bayesian models with non-Gaussian priors due to the intractable marginal likelihood. Existing methods, including variational Bayes and Markov chain Monte Carlo (MCMC)-based approaches, often have high computational cost and slow convergence. We propose an efficient framework to approximate the Fisher information matrix (FIM) and the CRB by expressing the gradient of the log marginal likelihood as a posterior expectation. Expectation propagation (EP) is used to approximate the posterior as a Gaussian, enabling accurate moment estimation compared to pure sampling-based methods. Numerical experiments on small-scale sparse models show that the EP-based CRB approximation achieves lower average normalized mean squared error (NMSE) and faster convergence than classical baselines in non-Gaussian settings.