Better Approximate Inference for Partial Likelihood Models with a Latent Structure

Temporal Point Processes (TPP) with partial likelihoods involving a latent structure often entail an intractable marginalization, thus making inference hard. We propose a novel approach to Maximum Likelihood Estimation (MLE) involving approximate inference over the latent variables by minimizing a tight upper bound on the approximation gap. Given a discrete latent variable $Z$, the proposed approximation reduces inference complexity from $O(|Z|^c)$ to $O(|Z|)$. We use convex conjugates to determine this upper bound in a closed form and show that its addition to the optimization objective results in improved results for models assuming proportional hazards as in Survival Analysis.