# Pareto optimality in Metropolis sampling

In the Metropolis sampling algorithm, we have some function $$f(x)$$ proportional to a probability distribution $$P(x)$$. To generate a random walk with stationary distribution $$P(x)$$, we generate a candidate $$x'$$ from a (symmetric) proposal distribution at previous step $$x$$ and keep it with probability $$\min\left(\frac{f(x')}{f(x)},1\right)$$.

Now, let's say we have multiple $$f$$'s, encoded as a function $$F:x\mapsto \mathbb R^k$$. We again generate a proposal point $$x'$$, and we keep it with probability $$\min\left(1,\min_{\ell\in\{1,\ldots,k\}} \frac{F_\ell(x')}{F_\ell(x)}\right).$$ This random walk has two properties: First, when $$k=1$$ this algorithm reduces to the Metropolis algorithm. Also, it keeps $$x'$$ any time $$x'$$ Pareto dominates $$x$$.

Can we say anything about the stationary distribution of this walk? Is it more likely to find points in the Pareto set of $$F$$?

I'm also OK with asking the same question about any random walk with a similar Pareto property, if the particular acceptance probability above has some flaws.

Thanks!