On approximating the shape of one-dimensional posterior marginals using the low discrepancy points

A method to approximate Bayesian posterior by evaluating it on a low discrepancy sequence (LDS) point set has recently been proposed. However, this method does not focus on finding the posterior marginals. Finding posterior marginals when the posterior approximation is obtained using LDS is not straightforward, and as yet, there is no method to approximate one dimensional marginals using an LDS. We propose an approximation method for this problem. This method is based on an s-dimensional integration rule together with fitting a polynomial smoothing function. We state and prove results showing conditions under which this polynomial smoothing function will converge to the true one-dimensional function. We also demonstrate the computational efficiency of the new approach compared to a grid based approach.