
Submitted by Assigned_Reviewer_1
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)
SUMMARY
Hamiltonian MCMC methods sample from a probability
distribution by treating its log as a "potential energy"
function over the state space, augmenting the space with extra
"momentum variables" and their associated "kinetic energy",
and evolving the state of the Markov process by integrating
the physical Hamiltonian equations of motion of the system.
Each step of the Markov chain is accomplished by numerically
integrating the Hamiltonian equations forward in time.
However, if the energy function is nondifferentiable, the
integral is not welldefined.
The rejection step that is used
to counteract numerical inaccuracies in the integration also
accounts for such nondifferentiable regions, but at the cost
of slowing down the mixing rate of the Markov chain.
This
paper suggests physicallyinspired "reflections" and
"refractions" of the trajectory of the system that occur
whenever the state crosses a discontinuity in the energy
function.
It applies to target distributions that are
differentiable everywhere except on the boundaries of certain
polytopes; the reflection or refraction occurs whenever the
trajectory of the system crosses such a boundary.
Whether a reflection or refraction occurs depends on if the
momentum component of the state is sufficient to "climb" the
gap in energy.
The authors show that the necessary
volumepreservation properties are satisfied by these
reflection and refraction procedures, so that the Markov
chains converges to the required target distribution.
QUALITY
The presented modification to Hamiltonian MCMC is useful and
the case of piecewisedifferentiable target distributions is
quite applicable in practice.
It would, of course, be useful
to have boundaries that are not affine hyperplanes.
CLARITY
The algorithm, its motivation, and the proof of the volume
preservation property is clearly presented.
Some more
(qualitative and/or quantitative) discussion about how the
reflections and refractions affect rejection rates would be
useful.
ORIGINALITY AND SIGNIFICANCE
The idea of applying physicallybased reflection and
refraction to Hamiltonian MCMC methods appears to be novel and
useful.
While the present work proves its correctness under
certain conditions, it is an interesting avenue of research to
extend it further.
Q2: Please summarize your review in 12 sentences
A modification of Hamiltonian MCMC is presented to
handle probability distributions that are not differentiable on
the boundaries of polytopes.
The proposed solution is for the
trajectory of the system to be reflected or refracted when it
encounters such a boundary, thereby reducing rejection rates and
improving mixing speed.
Submitted by Assigned_Reviewer_2
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 and verifies (correctness and performance through simulation) a sensible variation on HMC for pdfs with discontinuities or on a bounded domain.
The method relies on simulating refraction and reflection that result from hamiltonian dynamics, so leads to better proposals around discontinuities (and higher acceptance rates => higher ess).
One thing that would be nice is to see an example of a pdf when it's not worth computing intersections.
What types of high dimensional pdfs is this method not suited for?
Q2: Please summarize your review in 12 sentences
This paper presents and verifies
Submitted by Assigned_Reviewer_3
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 an interesting extension of the standard case and the proof of volume conservation presented is simple but important and was lacking in previous works. The writing is clear.
Typos: l.161:
",Then" l.376: "parameters, are tuned"
Q2: Please summarize your review in 12 sentences
This paper extends the leapfrog discretization used in Hamiltonian Monte Carlo to piecewise smooth distributions. The algorithm is correct and is an interesting extension of the smooth case.
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)
UPDATES AFTER REBUTTAL: Thank you for addressing my questions. The extra numerical results in the supplement are a welcome addition to the paper. I think that this is a good paper, but I will keep my score at 7.
ORIGINAL REVIEW: The paper proposes a modified leapfrog integrator for Hamiltonian Monte Carlo for sampling from piecewise continuous probability distributions, with affine discontinuity boundaries. The simulator makes use of reflection and refraction of the simulated trajectory whenever it hits a boundary. Similar ideas have previously been used for simulating from multivariate Gaussian distributions with constraints, but as far as I can tell this is the first application of this approach to "general" distributions (although, I'm not an expert on HMC). The idea is simple but intuitively appealing (most good ideas are simple!) and I think that this method can be very useful for a restricted class of problems. The numerical evaluation is not very extensive and the authors should consider extending it for a later (re)submission. Furthermore, there are several typos that need to be corrected.
Finally, can you comment on the possibility of using nonaffine discontinuity boundaries? Apart from the computational issues associated with computing intersections, would it be possible to extend the approach to allow for more general boundaries?
Some detailed comments/questions:
 In section 3: in think that it could be useful to mention that K(p') = K(p)  \Delta U to explain the form of the refracted momentum.  In Equation (3): aren't the upper triangle (excluding the four elements in the upper left corner) also zero?  Line 363: Missing normalisation by k in the definition of WMAE.  Footnote 2 on page 7: I think that it would be good if you could elaborate on the sensitivity of the tuning parameters (L and eps) of your method to convince the reader. Perhaps include additional numerical result in a supplementary material?
Q2: Please summarize your review in 12 sentences
The paper proposes a modified leapfrog integrator for Hamiltonian Monte Carlo for sampling from piecewise continuous probability distributions, with affine discontinuity boundaries. The idea is simple but potentially very useful for a restricted class of problems. The numerical evaluation is not very extensive.
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 5000 characters. Note
however, that reviewers and area chairs are 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 the reviewers for their
helpful and constructive comments.
R1> 1. RE More
quantitative/qualitative discussions of the effect of reflections and
refractions on the rejection rate:
We have provided more numerical
results accessible by the following anonymous
link:
https://drive.google.com/file/d/0B9SfUIl9GRZHWWRGOWRwTkx1UmM/view?usp=sharing
or
in case it does not work, by the following anonymous
link:
http://www.keepandshare.com/doc2/85678/supplimentarypdf47meg?da=y
In
the last pages of the uploaded file there are three tables that represent
the rate of reflections and refractions (as long as rejections) with
respect to several different tunings. It also contains a brief discussion
on the key role that reflection plays in the sampler's performance in high
dimensions.
***
R2> 1. We will fix the typos. Thanks
for mentioning them.
***
R3> 1. RE more numerical
evaluation:
In the anonymous link mentioned in R1.1 we have
provided more numerical evaluations (as suggested).
2. RE
elaborating on the sensitivity of the tuning parameters (L and eps) of our
method.
In the uploaded supplementary material, our method and the
baselines are tested on all combinations of parameter L being in {20, 50,
100, 200}; epsilon being in {0.05, 0.1, 0.2, 0.4} and the model dimension
be in {2, 10, 50}. It is already well known that basic HMC is quite
sensitive to tunings (see e.g. the paper's citation [7]). This is
verified by the provided supplementary plots. In contrast, reflective HMC
is not so: On the tested parameters, our method is never worse than HMC
and in most cases (particularly, in high dimensions) it performs
considerably better.
3. RE the possibility of using nonaffine
discontinuity boundaries:
In this case, it can be shown that
analogous to curved mirrors, the volume preservation may be violated (due
to magnification). However, despite this, we hope that (with some
modifications) we can design a reflection/refraction based HMC that even
in the mentioned case, converges to the correct stationary distribution.
We cannot be sure whether that is possible but do consider the prospects
for extending the work beyond affine borders.
4. In Equation (3):
aren't the upper triangle (excluding the four elements in the upper left
corner) also zero?
Not all of them. For instance, in the 3rd row
and 4th column:
(d q'_2) /(dp_2) = tau.
5. Thanks for the
other detailed comments/typos. We will implement/fix
them.
***
R4> 1. RE "the definition of the time
period is not intuitive":
The total leapfrog fullstep time is
epsilon. We divide it into arbitrary smaller periods,
Epsilon =
tau1 + tau2 +... such that in each period tau_i, at most one reflection
or refraction occurs. According to lemmas 1 or 2 (or in case there is no
collision, just a shear transformation) in each period, tau_i, the volume
is preserved. Therefore it is preserved for the total time, epsilon,
also.
2. RE "it is questionable that the total energy is preserved
during the refraction":
Consider a collision to a hyperplane q1 = c
(since any collision can be transformed to this case via rotation). In
this case, only the momentum element associated with the first dimension
changes. Let p1' and p1'' be such momentum elements just before and
after collision. In order to preserve the total energy we equate the
Hamiltonian (that is, H(q,p) = U(q) + K(p) where k(p) = (p^2)/2) right
before and after the collision:
0.5*(p1')^2 = (\Delta U) +
0.5*(p1'')^2
Therefore in order to preserve the total energy, the
momentum just after the collision should be:
(p1'') = SQRT{p1' 
2(\Delta U)}
As used in the paper. We did not provide the proof
in the paper since it is already stated in Pakman's papers [10] and [11].
However, we will either mention it in the paper or provide the above proof
explicitly.
***
R5> 1. What types of highdimensional
pdfs is this method not suited for?
According to the discussion
provided in the anonymous link (mentioned in R1.1), we believe that if the
high dimensional density is truncated or has external boundaries, our
method is always suitable. But if the pdf does not have a finite
support (i.e. is not surrounded by a 0probability space) and it has a
large enough number of internal boundaries and the potential gaps at the
boundaries are sufficiently small, It would eventually be better off just
ignoring them using regular HMC.

