For a MultivariateNormal distribution of dimension (event_shape) N, the samples are vectors in N-dimensional Euclidean space. log_prob called on one such vector x will yield a single scalar – the log of the probability density of the MVN at that x. If your mean vector is m and covariance matrix is C, this log_prob is -1/2 (x - m)^T C^{-1} (x - m) - 1/N log 2π|C|  (pretty sure i got that right). More info here: Multivariate normal distribution - Wikipedia 1. Hopefully this answers some of your question! I know you said you were ok w/ batch_shape, but it may still be worth reading through this tutorial 3.