The inverse of the covariance matrix for a distribution can be a good value for the mass matrix of a Hamiltonian monte carlo distribution.
If the distribution in question is the posterior of a Bayesian graphical model, many or most of the variables will be conditionally independent of each other. Thus, the inverse of the covariance matrix will have zeros for those pairs of variables. You can figure out the sparsity pattern of the matrix by examining the model graph.
I would like to efficiently estimate the inverse of the covariance matrix for a distribution given some IID samples from the distribution and assuming that we know its sparsity pattern.
It appears this problem is well studied for the more general case where we're trying to infer the sparsity pattern. For example, this paper is well cited and draws upon a number of other papers. However, I don't know of papers that deal with the case where the sparsity pattern is known. Any pointers?