  Bayesian Experimental Design (BED), which aims to find the optimal
experimental conditions for Bayesian inference, is usually posed as to optimize
the expected information gain (EIG). The gradient information is often needed
for efficient EIG optimization, and as a result the ability to estimate the
gradient of EIG is essential for BED problems. The primary goal of this work is
to develop methods for estimating the gradient of EIG, which, combined with the
stochastic gradient descent algorithms, result in efficient optimization of
EIG. Specifically, we first introduce a posterior expected representation of
the EIG gradient with respect to the design variables. Based on this, we
propose two methods for estimating the EIG gradient, UEEG-MCMC that leverages
posterior samples generated through Markov Chain Monte Carlo (MCMC) to estimate
the EIG gradient, and BEEG-AP that focuses on achieving high simulation
efficiency by repeatedly using parameter samples. Theoretical analysis and
numerical studies illustrate that UEEG-MCMC is robust agains the actual EIG
value, while BEEG-AP is more efficient when the EIG value to be optimized is
small. Moreover, both methods show superior performance compared to several
popular benchmarks in our numerical experiments.
