Likelihood based inference for multi-state latent factor intensity models is hindered by the fact that exact closed-form expressions for the implied data density are not available. This is a common and well-known problem for most parameter driven dynamic econometric models. This paper reviews, adapts and compares three different approaches for solving this problem. For evaluating the likelihood, two of the methods rely on Monte Carlo integration with importance sampling techniques. The third method, in contrast, is based on fully deterministic numerical procedures. A Monte Carlo study is conducted to illustrate the use of each method, and assess its corresponding finite sample performance.