# Gibbs sampling where I can only find the mode of conditionals?

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 $$p(x_1, x_2, x_3)$$?

• 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. – Xi'an Feb 20 at 18:51