Submitted by
Assigned_Reviewer_4
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)
Paper summary:
The authors propose a Bayesian
inference procedure for conditional bivariate copula models, where the
copula models can have multiple parameters. The copula parameters are
related to conditioning variables by a composition of parameter mapping
functions and arbitrary latent functions with Gaussian process priors. The
inference procedure for this model is based on expectation propagation.
The method is evaluated on synthetic data and on financial time series
data and compared to alternative copulabased models. The paper also
includes an overview of related methods.
Quality:
The
proposed method is sound. The authors include a description of related
methods and list strengths and weaknesses of their method.
Clarity:
The paper is written clearly and concisely.
It makes a very mature impression. I could not find any inconsistencies or
typos. The figures and tables complement the text seamlessly, equations
are clear and well explained and the text is flawless.
Originality:
The model and its inference procedure is
an extension of a similar conditional copula model that was restricted to
copulas with only one parameter. The present work generalizes this
approach to allow more than one copula parameter. To my knowledge, this
extension is novel.
Significance:
The flexibility of
copula families with only one parameter is limited and the applicability
of more flexible families is a huge advantage. Performance gains compared
to the alternative methods are remarkable. This result is very promising.
Q2: Please summarize your review in 12
sentences
The paper extends an inference procedure for
conditional copula models. The contribution is important and well done.
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 an interesting approach for
modeling dependences between variables based on Gaussian process
conditional copulas. It employs Gaussian process priors on the latent
functions to model their interactions. Then an alternating EP algorithm is
proposed for the approximate Bayesian inference. Experimental results on
both synthetic data and two realworld data indicate the better performance
of conditional copula models, especially the one based on student t
copula.
In general, this paper is well written and perhaps useful
for dependence modeling in time series data. The EP inference is also
implementable for the posterior distribution in the paper. A major problem
of this paper is that some details are missing: (1) How the parameters
in the copula model (e.g., \alpha, \beta, \omega for the Symmetrized Joe
Clayton Copula) in are calculated? I assume this paper sets them to
constants. How to choose them for a given time series data? (2) The
details of the parameters the exponential covariance function in Gaussian
process are missing. (3) How to choose the number of latent functions
k?
Q2: Please summarize your review in 12
sentences
This paper discusses an interesting topic and derives
an implementable inference method. However, some details on the parameter
settings are missing. Submitted by
Assigned_Reviewer_7
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 paper tackles the problem of fitting conditional
copula models. The paper goes beyond typical constant copula functions and
aim to model cases where parameters of the copula are functions of other
(time) varying aspects of the problem.
The core solution is
essentially modeling the parameters space as Gaussian Processes. Starting
from a Gaussian Process prior, the information from observed data are
incorporated via Expectation Propagation for approximate inference of the
posterior distribution.
The paper is extremely well written and
easy to follow. The problem is well motivated and most importantly
illustrations with HMM based and multivariate time series copulas are very
nicely written. The experiments are sufficient as
well. Q2: Please summarize your review in 12
sentences
Overall, this is a nice paper that motivates a novel
problem and provides a reasonable solution with excellent experimental
illustrations.
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 would like to thank all of the reviewers for their
time and comments. We are also very pleased to receive unanimously
positive feedback. Below, we answer three specific questions and correct
an error in our submission (which fortunately only strengthens our
conclusions).
Reviewer 6 asked some specific questions which
we answer below  we will update the manuscript to make these points
clearer.
1 ) The parameters alpha, beta and omega in the DSJCC
method are found by maximum likelihood.
2 ) The covariance
function for the Gaussian processes is the squared exponential kernel (for
the parametric form see equation (2) or e.g. Rasmussen and Williams). All
parameters of the kernels were found by maximizing the EP approximation to
the model evidence.
3 ) The number of latent functions k is
determined by the number of parameters of the parametric copula model.
There is one latent function for each parameter.
To all reviewers:
We would also like to correct a “copypaste” error we made just
before submission. The numbers in table 4 were mistakenly set to be
identical to those in table 5. Fortunately, the discussion in the
manuscript was based on the correct figures and the correct values only
strengthen our conclusions. We include the correct figures for Table 4
below. The pattern of bold and underlined numbers is similar to that in
Table 5, which explains why we did not spot the error in our submission.
We include below the average test loglikelihood for each method,
GPCCG, GPCCT, GPCCSJC, HMM, TVC, DSJCC, CONSTG, CONSTT and CONSTSJC,
on each dataset AUD, CAD, JPY, NOK, SEK, EUR, GBP and NZD. The format is
Method Name
results for AUD results for CAD
results for JPY results for NOK results for SEK results
for EUR results for GBP results for NZD
As in the
manuscript, the results of the best method are shown in bold (with a "b"
in front). The results of any other method are underlined (with a "u" in
front) when the differences with respect to the best performing method are
not statistically significant according to a paired ttest at \alpha =
0.05. The best technique is GPCCT (the one with most "b"'s in front),
followed by GPCCG as discussed in the manuscript.
GPCCG
0.1260 u 0.0562 b 0.1221 u 0.4106 u 0.4132
0.8842 u 0.2487 u 0.1045
GPCCT
b 0.1319 b
0.0589 u 0.1201 b 0.4161 b 0.4192 b 0.8995 b 0.2514
b 0.1079
GPCCSJC
0.1168 0.0469 0.1064
0.3941 0.3905 0.8287 0.2404 0.0921
HMM
0.1164 0.0478 0.1009 0.4069 0.3955 0.8700
0.2374 0.0926
TVC
0.1181 0.0524 0.1038
0.3930 0.3878 0.7855 0.2301 0.0974
DSJCC
0.0798 0.0259 0.0891 0.3994 0.3937 0.8335
0.2320 0.0560
CONSTG
0.0925 0.0398 0.0771
0.3413 0.3426 0.6803 0.2085 0.0745
CONSTT
0.1078 0.0463 0.0898 0.3765 0.3760 0.7732
0.2231 0.0875
CONSTSJC
0.1000 0.0425
0.0852 0.3536 0.3544 0.7113 0.2165 0.0796
