
Submitted by
Assigned_Reviewer_6
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
 Update Just a few other thoughts. Simo Saarka
has more recent work on continuousdiscrete 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 loworder 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 Mstep 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.
Q2: Please summarize your review in 12
sentences
Overall the paper is well written and extend the
applicability of approximate Bayesian inference methods to the class
of continuous and discrete time settings, which many will find
interesting. Submitted by
Assigned_Reviewer_10
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper applies approximate inference to nonlinear
diffusion equations by taking the continuoustime limit of the
expectationpropagation technique. The result is a tracking algorithm
which is, naturally, much faster than sampling methods, and on the
experiments shown is rather accurate. I'm curious if a linearization of
the loss function + the application of KalmanBucy (ie, extended KB),
possibly applied iteratively, would lead to a more/less effective
algorithm. It would also be interesting to see more substantial
experiments, for instance with highfrequency financial data where this
framework is often use and existing benchmarks are available.
Quality: the paper is technically solid. Clarity: the paper is
clearly written and well organized. Originality: relatively high.
Significance: the speedup over MC achieved here is potentially
important.
Q2: Please summarize your review in 12
sentences
This paper applies an approximate inference method,
namely, expectationpropagation, to nonlinear diffusion processes and
obtains a significant speedup over sampling methods. The paper is clear
and straightforward to follow. Submitted by
Assigned_Reviewer_11
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This is a theoretically strong and interesting paper
that proposes a novel EPtype inference approach for continuoustime
stochastic dynamical systems. The paper is well structured and the theory
clearly presented. However, the authors should have better motivated the
approach with a stronger experimental section. The only comparison with
other approaches is in the first example in Section 3.1, where the authors
use a MCMC approach as a benchmark. A part from not completely agreeing in
using MCMC approaches as a benchmark, as the proposed method performs as
well as the MCMC approach, what is the advantage in using it? The authors
should discuss this point in detail. In the third example, the authors
seem to suggest that the results are similar to those in Zammit Mangion et
al. They then justify the use of their approach from a computational
viewpoint. The authors should give some quantification of the advantage 
it is not useful for the community to introduce a new approach without
giving an idea of its characteristics with respect to existing ones. I
would really appreciate some quantitative answers on this point in the
rebuttal period. Q2: Please summarize your review in 12
sentences
Very interesting paper but the experimental section is
not completely satisfactory.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank the reviewers for their appreciative comments
and constructive criticism, please find the corresponding replies below.
Assigned reviewer 10: The suggested algorithm is interesting,
as would be interesting comparisons with a local Laplace approximation
(which is clearly related); implementation of these approaches in the
continuous time setting is nontrivial and best left for further future
comparisons. From experience in multivariate and discrete time models, EP
and variational tend to perform better than either extended Kalman
filtering or Laplace approximations (Ypma&Heskes, 2005;
Zoeter&Ypma&Heskes, 2006), which leads to suspect these
comparisons would also hold in continuous time. We appreciate the
suggestion of financial data as a good arena for testing the method, but,
as we are not domain experts, we do not feel we can properly address that
at the moment. However, we will definitely consider such models and fields
of application in the future as well as various other inference algorithms
and correction schemes.
Assigned reviewer 11: We agree with
the reviewer that MCMC is also an approximation, however it is frequently
used in ML as a benchmark, so we followed that approach. The advantages of
the proposed method are both in computational speed and in the retained
continuous time nature of inference (as we point out, MCMC needed time
discretisation and sampling at small timelag can be computationally
demanding (Golightly&Wilkinson, 2008; Kou et al.,2012). Similar
considerations also hold for the comparison with ZammitMangion et al; we
emphasised the similarity of some results because of their biological
interpretability (clustering of parameters). Some of the advantaged of
continuous time approach are: (i) numerical stability when timelag tends
to zero (ii) a natural way of dealing with nonequidistant observations or
observations that are too close to each other in time (iii) a natural
interpretability of parameters. We will expand the discussion of the
computational advantages in the final version; however, we emphasise that
the contributions of the paper are a novel methodology for continuous time
systems.
Assigned reviewer 6: We also wonder whether large
scale averaging effects underlie the collapse of EP to variational, but we
have no further proof of this intuition. Formally, the collapse happens
due to the timelags limiting procedure (or the cavity distribution being
equal in limit to the marginal). There are indeed issues with slow EM
convergence on the realworld data set, these can be remedied using an
expected conjugate gradient framework (Salakhutdinov et al. ICML2013),
however, we opted for EM because of the ease in exposition, suitability to
the framework of the VBEM. The convergence of the method is typical to
the fixed point iteration or message passing methods, with an sensible
initialisation around 1020 iterations (forwardbackwards) were
sufficient. Clearly, as with any EP type algorithms, multimodal and
nonlogconcave losses and likelihoods can cause issues and other avenues
such as double loop (Yuille et al., 2002 or Heskes&Zoeter, 2002) and
direct minimisation (Welling&Tech2001, Archambeau 2007) have to be
explored. The cubic nature of the algorithm is inherent in the Riccati
differential equations for the covariance parameters (even with sparse A_t
and a diagonal B_t) and we are not aware of any methods in the literature
that could do a decent job in continuos time models as for example Cseke
et al. (2013) (http://arxiv.org/abs/1305.4152) in discrete time models.
Nonetheless, we are actively exploring alternatives, one possibility is
using pathwise decompositions and LBP type approaches over paths in
distributed systems. Thank you the interesting suggestion w.r.t. panel 3c,
we will try to find a way to scale the points in the final version so that
the panel remains visually pleasing.
 