Approximate Gaussian process inference for the drift of stochastic differential equations
We introduce a nonparametric approach for estimating drift functions in systems of stochastic differential equations from sparse observations of the state vector. Using a Gaussian process prior over the drift as a function of the state vector, we develop an approximate EM algorithm to deal with the unobserved, latent dynamics between observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the MAP estimation of the drift is facilitated by a sparse Gaussian process regression.
This is a solid paper on a topic of potential 
      significant interest. Diffusion processes are widely used in the sciences 
      and engineering, and a data-driven strategy to parametrize the drift 
      function could be potentially useful. Furthermore, the paper also has 
      considerable interest methodologically: the proposal to approximate the 
      intractable problem with a piecewise OU process is interesting, and the 
      approximate quantification of the error is (at least in theory) a nice 
      result. I would say the paper scores highly on quality and originality. It 
      is generally clearly written although some bits feel a bit rushed (see 
      detailed comments below). Re significance, I would say the methodology 
      could be of considerable interest, while the experiments are for the time 
      being a bit less well developed. Detailed comments: - I would 
      have appreciated a bit more motivation for the methodology: SDEs in the 
      sciences are often used as a modelling tool so that the specific form of 
      the drift has a physically interpretable meaning. This is somewhat lost 
      with the GP approximation; - I wonder whether there are some 
      asymptotic results which could be given, at every step you effectively get 
      a measurement of the linearised drift (at a different input value), hence 
      of its value and its derivative. When you increase the number of 
      observations, this should converge to the true posterior (which of course 
      it does as the approach becomes exact for observed paths), but perhaps you 
      can work out the rate at which your approach converges to the truth? - 
      The experimental part is a bit disappointing, there are essentially no 
      comments on the performance of the model on simulated data, and very 
      little context on the real data. I would remove the experiment with 5th 
      degree drift and leave more space to comment on the experiments. Also some 
      runtimes would be useful. Minor comments: - eq(10) worth perhaps 
      saying that the last step uses explicitly the time invariance of the drift 
      function (homogeneous GP); - after (13), it's the two terms on the 
      r.h.s. that obey Ch-K equations (the way it's written is ambiguous); - 
      some confusion with figures, there is a reference to Figure ?? (1a 
      presumably) on page 3; the caption to Figure 2 alludes to a missing third 
      panel; - models that use OU processes with piecewise constant drifts 
      have been considered and shown to be very computationally efficient 
      (Stimberg et al AISTATS11, Opper et al NIPS10 which reported very fast 
      performance on the double well problem). These are different as the 
      changepoints are stochastic processes while here the changes are an 
      approximation strategy, but they seem somewhat related and might be worth 
      referring to.