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)
This paper presents a Bayesian approach to state and
parameter estimation in nonlinear statespace models, while also learning
the transition dynamics through the use of a Gaussian process (GP) prior.
The inference mechanism is based on particle Markov chain Monte Carlo
(PMCMC) with the recentlyintroduced idea of ancestor sampling. The paper
also discusses computational efficiencies to be had with respect to
sparsity and lowrank Cholesky updates.
This is a technically
sound and strong paper with clear and accessible presentation. The online
marginalisation of the transition dynamics and the use of ancestor
sampling to achieve this is novel. The consideration of computational
issues such as sparsity and lowrank updates/downdates to the Cholesky
factors of covariance matrices strengthens the paper further. The
empirical results, while brief, are sufficient (further suggestions
below).
In addition to its stated aims, the paper will likely
stimulate discussion around inference methods for nonMarkovian
statespace models and the potential advantages/disadvantages of learning
the transition dynamics in this way rather than specifying a parametric
model a priori.
While space is slight, the authors may like to
consider some further discussion around the differences between using a
parametric transition model given a priori against the use of a similar
model as the mean function of the GP. For example in out of sample
prediction (e.g. forecasting).
The results in Table 1 and the
description in the preceding paragraph are slightly unclear to me. I am
unsure as to whether the RMSE is against a withheld set of data points or
the same set of data points that is conditioned upon (the *data in the
column headings). My main interest would be an RMSE against an
outofsample prediction, especially a forecast forward in time against
withheld data. It is in this scenario that I would expect to see the
largest differences between the learnt dynamics and the ground truth
model. If Table 1 is not already showing this, an extra column that does
so would be a great addition.
One minor point: the abbreviation
CPFAS is used in Algorithm 1 before being defined in the first paragraph
of Section 3.3.1 below. Q2: Please summarize your review
in 12 sentences
A strong and novel paper that should stimulate some
interesting discussion. Submitted by
Assigned_Reviewer_9
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)
The authors propose to apply particle MCMC to perform
inference in Gaussian process statespace models. In particular, they
focus on the recent ancestral sampling particle Gibbs algorithm of
Lindsten et al. The paper is clear and it is an interesting and original
application of particle MCMC. There are also some useful modelspecific
methodology developed in the paper, namely sparse GPSSM.
One
thing I find truly regrettable is the lack of comparisons to other
particle MCMC schemes, in particular the particle marginal MH (PMMH)
scheme and the particle Gibbs with backward sampling (as in Whiteley et
al.). They could have been straightforwardly implemented and it would be
of interest to know how those variants compared to the proposed scheme
(and it would not be much work for the authors either).
Additionally I would like to see graphs displaying the performance
of the algorithms (e.g. in terms of ACF or ESS) as a function of N and T.
As they stand the results are not very informative. Do I need to scale N
linearly with T, sublinearly? I believe that for such models the PMMH
would require a number of particles increasing quadratically with T as
observed in Whiteley et al. whereas both particle Gibbs require a number
of particle growing sublinearly with T.
Q2: Please summarize your review in 12
sentences
A wellwritten application of particle MCMC to GP
statespace models. The paper could be significantly improved if the
proposed algorithm was compared to the PMMH and the particle Gibbs with
backward sampling. 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)
PMCMC sampling is exploited in an ssm with GP process
prior to extend to actual parameters rather than just the usual filtering
and smoothing. Nice straightforward application of PMCMC methodology. Pity
that a proper evaluation of the challenge tp get the PMCMC scheme to work
is not described and evaluated in more detail as this would really have
been the important contribution. A more detailed and critical evaluation
of the strengths and weaknesses of the approach would have made the paper
of value given it is an application of PMCMC methodology. It is probably
too much to ask that the experimental section be revised to provide more
evaluation than demonstration. Q2: Please summarize your
review in 12 sentences
Application of PMCMC methodology to ssm
would
have been more useful assessing the practicial difficulties in getting
such a scheme to work  and how well it actually works
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 positive comments.
We would like to emphasize that PMCMC has allowed us to learn
Bayesian GP statespace models while keeping alive the whole nonparametric
richness of this kind of model. We believe that it is the first time that
this has been achieved irrespective of the inference method used. In our
opinion, this is a valuable result in Bayesian Nonparametrics in its own
right besides being a demonstration of the power of PMCMC.
The
practical difficulties in getting PMCMC to work efficiently were solved
through the marginalization of the latent function f(x), the use of a
sparse covariance function (FIC) and the careful sequential construction
of Cholesky factorizations of the covariance matrix. Those (non trivial to
us) contributions allowed Particle Gibbs with Ancestor sampling (PGAS) to
perform very well "out of the box". In our opinion this was possible
thanks to: 1) all the work done in adapting the model for efficient
sampling and 2) the inherent power of PGAS to sample from nonMarkovian
models such as the one induced by the marginalization of the GP.
We agree that providing more experimental evaluation would improve
the paper. However, severe space limitations did not allow us to present
in much detail very important computational aspects of our method such as
the sparse GPSSM or the sequential updating of factorizations of
covariance matrices. As a consequence, we felt that the small amount of
space left would be better used in providing an illustrative demonstration
of the capabilities of our approach to Bayesian inference in GPSSMs. In
particular, the figures emphasized the particles from the smoothing
distribution since an unconventional property of this statespace model is
that any prediction made by the model uses the particles of the smoothing
distribution. This is in contrast with parametric models where the learned
parameters contain all that is needed to make predictions.
Although comparison with other PMCMC methods would undoubtedly
make the paper stronger, our choice of Particle Gibbs with Ancestor
sampling (PGAS) was motivated by its particularly good performance for
nonMarkovian models such as the one obtained when marginalizing the
Gaussian process latent function. In the original PGAS paper (Lindsten,
Jordan and Schön, 2012), the authors showed how PGAS consistently
performed better than Particle Gibbs with Backward Simulation when applied
to nonMarkovian models.
The RMSEs reported in Table 1 are indeed
outofsample predictions on long data records. We will make sure to
update the text to try to remove any ambiguity regarding these
outofsample predictions.
