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world:coupling
This is an old revision of the document!
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 a candidate and accept the candidate with probability with . Otherwise reject the candidate and 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 better to couple in that way: and . These two densities are known only up to multiplicative constant. Up to some tricks, we can deduce a way of coupling two MH starting from different initial distributions? Can we compare it to the coupling of Pierre Jacob et al.?
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: which completes the proof.
Some comments
The probability of coupling is given by:
Question: we know that . But I can't see how to prove
world/coupling.1718400697.txt.gz · Last modified: 2024/06/14 23:31 by rdouc
