Approximate inference in latent Gaussian-Markov models from continuous time observations
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 postinference 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.
-- Update Just a few other thoughts. Simo Saarka 
      has more recent work on continuous-discrete time systems (as you 
      referenced) that might be interesting to contrast against - there, using 
      Gaussian cubature that provides a nice alternative deterministic 
      approximation method. This might be useful for additional discussion and 
      future work. In addition I also now wondered if it is possible to derive 
      an algorithm directly using the variational Gaussian approach. This would 
      be more appealing from the point of having a well defined objective 
      function with which to optimise, potentially fewer numerical issues and 
      interpretation directly in terms of the marginal likelihood. We could 
      afterwards add low-order marginal corrections using cumulant perturbations 
      (like those of Opper for EP) - the only place I think shows this is Barber 
      and van de Laar (http://arxiv.org/pdf/1105.5455.pdf). I do look forward to 
      reading the final version of the paper. -- Original The paper 
      presents an algorithm for approximate Bayesian inference in models 
      with continuous and discrete time observations. The model can be 
      cast in the framework of latent Gaussian models and a parallel 
      expectation propagation algorithm can be used to derive a principled 
      approach for inference and learning dealing with both continuous and 
      discrete time. This EP inference algorithm is embedded within an EM 
      algorithm to both learn parameters of the model as well as marginal 
      distributions. The algorithm is shown to be effective in the number of 
      experimental settings. Overall I enjoyed the paper and thoughts 
      that it extends the applicability of approximately message passing to 
      a wider class of models. In particular I thought it was 
      interesting that EP updates for a continuous time limit collapsed to 
      the variational Gaussian updates. This is related to the latent 
      Gaussian structure but I wondered if there is a deeper reason 
      underlying this connection. The algorithm seems robust due to be 
      implied fractional updating but I wondered if you could comment on any 
      experienced difficulties in implementation, such as issues of slow 
      convergence of parameter learning, numerical stability, etc. 
      The algorithm is still cubic due to the inverses is in the 
      inference as well as the M-step updating - could comment on approaches 
      to scaling up such algorithms. In the experimental section it 
      would be nice to see plots giving insight into the convergence of the 
      algorithm. Can we also demonstrate the advantages obtained by of 
      having an estimate of the marginal likelihood. For example, it could 
      be possible in figure 3C to plot each of the individual points with a 
      size proportional the marginal likelihood value.