Variance–covariance from a metropolis chain on a curved, singular manifold

We consider estimation of variance and covariance from a point cloud that are draws from a posterior distribution that lie on a curved, singular manifold. The motivating application is Bayesian inference regarding a likelihood subject to overidentified moment equations using MCMC (Markov Chain Monte Carlo). The MCMC draws lie on a singular manifold that typically is curved. Variance and covariance are Euclidean concepts. A curved, singular manifold is not typically a Euclidean space. We explore some suggestions on how to adapt a Euclidean concept to a non-Euclidean space then build on them to propose and illustrate appropriate methods.