We consider the inverse problem of reconstructing the posterior measure over the trajec- tories of a diffusion process from discrete time observations and continuous time constraints. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference. We then show how the approximation can be extended to a wide class of discrete-state Markov jump pro- cesses by making use of the chemical Langevin equation. Our empirical results show that the proposed method is computationally efficient and provides good approximations for these classes of inverse problems.
In @cite_5 we propose an expectation propagation method for diffusion process models where the prior process is Gaussian--Markov. In this paper, we extend this method to models with non-nonlinear prior processes. Here we use expectation propagation only to approximate the likelihood terms. We avoid approximating the prior process by using moment-closure approximations on the process resulting from the prior and the likelihood approximations. When we choose a Gaussian--Markov prior process, the method proposed in this paper is identical to the one proposed in @cite_5 .
@cite_5
2117804315
We propose an approximate inference algorithm for continuous time Gaussian Markov process models with both discrete and continuous time likelihoods. We show that the continuous time limit of the expectation propagation algorithm exists and results in a hybrid fixed point iteration consisting of (1) expectation propagation updates for discrete time terms and (2) variational updates for the continuous time term. We introduce post-inference corrections methods that improve on the marginals of the approximation. This approach extends the classical Kalman-Bucy smoothing procedure to non-Gaussian observations, enabling continuous-time inference in a variety of models, including spiking neuronal models (state-space models with point process observations) and box likelihood models. Experimental results on real and simulated data demonstrate high distributional accuracy and significant computational savings compared to discrete-time approaches in a neural application.