# Markov chain Monte Carlo sampling using CDFs instead of PDFs

I wonder if there is any MCMC sampling method which uses the definition of the target CDF instead of the target PDF; however, I may use a proposal PDF.

I would like to use Metropolis-Hastings but it is not possible because the calculation of the acceptance ratio is defined in terms of the target PDF.

I say this because it is impossible for me to obtain the PDF associated to certain CDF without doing some kind of numerical approximation which may bias my simulation; also, my CDF might be not continuous, and therefore I cannot differenciate it to obtain the PDF

Regards!

• What about $F^{-1}(U)$, where $U$ is uniform? You'd still likely need to numerically approximate the inverse. – Alex R. May 23 '16 at 21:18
• I assume that the CDF is multivariate? Have a look at this question. – lacerbi May 23 '16 at 21:24
• Yes, it is multivariate. – Diego Andres Alvarez Marin May 24 '16 at 13:17
• Then I think it is an open problem. I suggested a basic solution in the thread that I linked, but it scales badly with the number of dimensions; so it depends on how many dimensions you have. Also, it still requires some numerical approximation (which becomes exact in the limit). – lacerbi May 25 '16 at 17:16