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world:coupling
Table of Contents
A new coupling technique
Let , be two probability measures on the same measurable space .
We draw jointly the couple of random variables according to the following procedure:
- Draw .
- Draw and set with probability where . Otherwise set .
Proposition. is a coupling of .
- What is nice is that we are able to couple these random variables whereas their densities are known only up to a multiplicative constant. I wonder if it is interesting to couple in that way: and . Up to some tricks, can we deduce a way of coupling two MH starting from different initial distributions, maybe with delayed coupling? Can we compare it to the coupling of Pierre Jacob et al.?
- Moreover, if we look at the problem today, if then, is a coupling of , no?
Proof
Obviously, where
We now show that is a coupling of . To do so, it is sufficient to check that for any bounded or non-negative function , .
Indeed, write, using the detailed balance condition in the second line : which completes the proof.
Some comments
The coupling probability is given by:
Question: we know that . But I can't see how to prove
world/coupling.txt · Last modified: 2024/06/20 15:02 by rdouc
