MCMC on zero measure sets

March 23, 2014

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

zeromesSimulating 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!


for (t in 2:T){
  if (ace){

If instead we add the proper Jacobian as in


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.


Filed under: R, Statistics Tagged: conditional density, Hastings-Metropolis sampler, Jacobian, MCMC, measure theory, measure zero set, projected measure, random walk

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