Question:

1. A prior distribution of exogenous variable $U_i$
2. A factual variable $x_i$, and the posterior distribution of exogenous variable $\mathbb P(U_i|x_i)$
3. Change the distribution of $U_i$, such that $F(U_i)=x_i$

Question:

1. Are the posterior of independent variable still independent?(Inverse function theorem, sometimes finite, sometimes infinite. Finite variable to finite variable?
2. If it does, I think Langevin is enough, cause the transport of independent samples must be independent.

Hardness:

For an optimal transport from a measure which marginal distributions are independent to another measure is #P-hard with the input bits. That means, for a Cartesian product of $K$ independent distribution, although the input is $O(K)$ while the complexity will be $O(n^K)$.

While, usually the point may not very large:

A potential direction: maintain $K$ discretized marginal distribution, then update the potential function whenever a particle comes.

entropy: Independent marginal distribution has largest entropy

To create another independent case, and calculate the posterior