0
$\begingroup$

I'm trying to solve a problem with Gibbs Sampling, so I'm trying to do:

$$ x_1^1 \sim p(x_1 | x_2^0, x_3^0)\\ x_2^1 \sim p(x_2 | x_1^1, x_3^0)\\ x_3^1 \sim p(x_3 | x_1^1, x_2^1)\\ x_1^2 \sim p(x_1 | x_2^1, x_3^1)\\ ... $$

But I can't compute the full distribution of each conditional, eg. $p(x_1 | x_2, x_3)$. However, I can find the MAP estimate of each conditional. So I'm thinking about doing this:

$$ x_1^1 = \arg\max_{x_1} p(x_1 | x_2^0, x_3^0)\\ x_2^1 \sim p(x_2 | x_1^1, x_3^0)\\ x_3^1 \sim p(x_3 | x_1^1, x_2^1)\\ x_1^2 \sim p(x_1 | x_2^1, x_3^1)\\ ... $$

I couldn't find any theoretical results for this (or maybe I'm not searching the right terms). If I "sample" like this, would I end up with a mode for joint $p(x_1, x_2, x_3)$?

$\endgroup$
  • 2
    $\begingroup$ There is no theoretical result to validate this method if the target is not unimodal . If it is unimodal, then each maximisation increases the value of the target and the sequence of $x$'s can only converge to the mode. $\endgroup$ – Xi'an Feb 20 at 18:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.