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.



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