## MCMC on zero measure sets Simulating a bivariate normal under the constraint (or conditional to the fact) that x²-y²=1 (a non-linear zero measure curve in the 2-dimensional Euclidean space) is not that easy: if running a random walk along that curve (by running a random walk on y and deducing x as x²=y²+1 and accepting with a Metropolis-Hastings ratio based on the bivariate normal density), the outcome differs from the target predicted by a change of variable and the proper derivation of the conditional. The above graph resulting from the R code below illustrates the discrepancy!

targ=function(y){
exp(-y^2)/(1.52*sqrt(1+y^2))}

T=10^5
Eps=3
ys=xs=rep(runif(1),T)
xs=sqrt(1+ys^2)
for (t in 2:T){
propy=runif(1,-Eps,Eps)+ys[t-1]
propx=sqrt(1+propy^2)
ace=(runif(1)<(dnorm(propy)*dnorm(propx))/
(dnorm(ys[t-1])*dnorm(xs[t-1])))
if (ace){
ys[t]=propy;xs[t]=propx
}else{
ys[t]=ys[t-1];xs[t]=xs[t-1]}}


  ace=(runif(1)<(dnorm(propy)*dnorm(propx)/propx)/
(dnorm(ys[t-1])*dnorm(xs[t-1])/xs[t-1]))


the fit is there. My open question is how to make this derivation generic, i.e. without requiring the (dreaded) computation of the (dreadful) Jacobian. ### 24 Responses to “MCMC on zero measure sets”

1. Joshua Bon Says:

A little late, but this may be of interest?
https://projecteuclid.org/euclid.imsc/1379942050

2. Mike G Says:

This may be obvious to everyone else reading this, but I want to type it just to make sure I’m thinking it out correctly.

The problem is that, while the the proposal jumps are symmetric in “y”, they are not symmetric along the curve that you are sampling from, correct? Because the line integral along the path is not the same length for equal length segments in “y”, drawing a uniform point from y +/- EPS is not equivalent to drawing a uniform point centered at the current point on the x^2-y^2=1 line.

To properly adjust for sampling along the curve, you would want to parameterize the curve in terms of x(t) and y(t), draw a random length from L = runif(-EPS,+EPS), and solve (for T*):
L = line integral( sqrt(dx^2+dy^2)dt) from “current point” to T*
Then get x and y from x(t), y(t)

I think this would be uglier than just doing the variable transformation however…..

(feel free to tell me Im wrong, confusing, or just stating the obvious though)

• xi'an Says:

On the contrary Mike, I find your explanation completely to the point and most helpful! Using a random walk in the wrong space or with respect to the wrong dominating measure kills the validation of the simplified Metropolis step. The random walk one should use need keep invariant the measure on the curve, as you mention. Thank you!

• Mike G Says:

Interestingly enough, while I would think the approach I hinted at up there should work, it doesn’t appear to upon testing it out.

The approach I mean is to draw a length L from a symmetric proposal distribution, then find the (propx,propy) pair that correspond to that move from the current point (x,y).

This should lead to a symmetric move that still satisfies the other conditions of MCMC. Unfortunately, unless I am doing something wrong in implementation, it still fails to return the right distribution for the problem above.

It works for simple test cases, and also appears to return the correct distribution for the “along the path” distribution, but the draws of y do not seem to correctly reflect the conditional distribution of y.

I did some tests in an R-markdown file and put it on RPubs if you are interested (link below). Unfortunately the writeup is not refined (at all, poorly commented, etc), so it may not be that helpful.

The main point is that the approach seems to work for some simple cases, but for reasons that are unclear to me, do not work for the other cases.

http://rpubs.com/mikegros/14906

• xi'an Says:

Dear Mike: did you check the code posted by Zen earlier today? It seems to be doing what you are suggesting, ie parameterising the curve x²-y²=1 to make those symmetric moves around and against the proper measure. I cannot tell from your description where or why things go wrong, but the major issue in this problem is to use the right dominating measure. Best, X.

• Mike G Says:

I did see his solution and noted it did work, I just find it interesting that the method I mentioned did not work (unless I just am implementing it wrong, which is possible).

The code I wrote isn’t so neat as to be easily legible, but I think the images and text around them tell the story. I hid the code so now it only shows the text and images. It’s possible that it is just an error on my end, but it is interesting to me nonetheless.

• Zen Says:

Hi, Mike. This is very interesting. Why don’t you post a question at http://stats.stackexchange.com/ ? Note that I used a parameterization of the hyperbola in which the velocity vector doesn’t have length one for every value of the parameter u. My guess is that we will have best results considering a finite (“big enough”) portion of the hyperbola, and using a natural parameterization (http://en.wikipedia.org/wiki/Differential_geometry_of_curves#Length_and_natural_parametrization). We could test this for conditioning events defined by other curves (ellipsis, parabola, etc).

• Mike G Says:

Hi Zen,

I may post the question there soon.

I reposted the file at the RPubs link with the code for one section so you can see the sampling approach.

I also included my function to calculate the natural parameter iteratively. It appears to work in the tests I ran through it. I also calculated the natural parameter another way utilizing the R uniroot function, but what I show worked faster.

I’ll continue to disclaimer that I could just be doing something dumb or making a silly coding mistake, but if so, I haven’t figured out what it is yet.

• Zen Says:

Dear Prof. Robert and Mike,
I’ve uploaded the lastest version here: http://www.ime.usp.br/~pmarques/R/hyperbola.r
Now I believe we are taking properly into account the geometry of the problem. The random walk is made “intrinsically”, stepping a random length along the hyperbola, in one of the two possible directions, at each iteration. It’s slow because I’m using R’s integrate() and uniroot() to solve the integral equation related to the arc length. The result looks good.
Cheers.

• Hi Zen,
I’m not sure that the fact your move is uniform on the hyperbola is the key point. For example, if I replace your proposal with just a uniform perturbation of u:

candidate <-function(t01,L) {
return(L+t01)
}

the fit of the resulting histogram of MC realizations is still an excellent fit to the target. I think you have implicitly defined a transformation that has constant Jacobian. If you consider the transformation from (X,Y) to (U,Z) such that X=sqrt{Z} cosh(U) and Y=sqrt{Z} sinh(U), then I get a constant Jacobian. This leads us to
f(U|Z) \propto f(U,Z) \propto f(X,Y)
so that the Metropolis-Hastings ration in Christian's original code is correct.

Hope this makes sense. Let me know if you see any flaws in my reasoning.

Best,

• Zen Says:

Hi Brad, that’s what I did in the code contained in my comment of March 27, 2014 at 9:53 pm (see bellow). Thanks.

3. Zen Says:

Hi, Prof! Is the conditioning event just the right sheet of the hyperbola ($x\geq 1$) or both sheets?

• xi'an Says:

sorry, I do not understand your question…

• Zen Says:

$x^2-y^2=1$ defines a hyperbola with two sheets: one with $x\geq 1$, and another with $x\leq -1$.

• xi'an Says:

oh indeed… thanks, I was not aware of the use of “sheet” in English. We use “branch” in French. Now, in the current case, picking either sheet does not make a difference because the Gaussian is centred at zero and hence symmetric in x. As you can read from the R code, I picked the positive side for x. For a non-centred normal, the construct and Jacobian would be a bit more involved. Albeit manageable…

• Zen Says:

Sorry for the sheet! Is it worth flipping a coin every time to decide if we should “warp” to the other branch of the hyperbola?

• xi'an Says:

no really, because we do not simulate the x, simply use the density of x (equal to the density of -x) in the resulting projected density of y. If we were to simulate x instead of y, the same would occur.

• Zen Says:

I see. The warping would be (maybe) useful only if the distribution were not symmetric. Thanks!

• Zen Says:

Revised code:

 N <- 10^5 eps <- 1.5 u <- numeric(N) for (i in 2:N) { v <- u[i - 1] + runif(1, -eps, eps) if (runif(1) < dnorm(sinh(v))*dnorm(cosh(v))/dnorm(sinh(u[i - 1]))/dnorm(cosh(u[i - 1]))) { u[i] <- v } else { u[i] <- u[i - 1] } } hist(sinh(u), breaks = "FD", xlab = "y", freq = FALSE) curve(dnorm(x, mean = mean(sinh(u)), sd = sd(sinh(u))), col = "blue", lwd = 2, add = TRUE) 

• Zen Says:

Using hyperbolic functions to parameterize the hyperbola.

N <- 10^5
eps <- 1.5
u <- numeric(N)
for (i in 2:N) {
v <- u[i – 1] + runif(1, -eps, eps)
if (runif(1) < dnorm(sinh(v))*dnorm(cosh(v))/dnorm(sinh(u[i – 1]))/dnorm(cosh(u[i – 1]))) {
u[i] <- v
} else {
u[i] <- u[i – 1]
}
}
hist(sinh(u), breaks = "FD")

• Luca Martino Says:

Maybe one idea is to draw uniformly a sample on the curve (as in the previous answer, I guess),
Regards,
Luca

4. Rodrigo Targino Says:

Good to read about this problem here, Christian. I’ve been working on a similar problem in a Finance context with Gareth Peters and we will be presenting some of our progress at the MCQMC in Belgium: http://mcqmc2014.cs.kuleuven.be/program/abstract/?aid=121&akey=ac0156c0876059bc08474ed12bc5b1f0

Hope to see you there!

Best,
Rodrigo

• 