unbalanced sampling

A question from X validated on sampling from an unknown density f when given both a sample from the density f restricted to a (known) interval A , f¹ say, and a sample from f restricted to the complement of A, f² say. Or at least on producing an estimate of the mass of A under f, p(A)…

The problem sounds impossible to solve without an ability to compute the density value at a given value, since  any convex combination αf¹+(1-α)f² would return the same two samples. Assuming continuity of the density f at the boundary point a between A and its complement, a desperate solution for p(A)/1-p(A) is to take the ratio of the density estimates at the value a, which turns out not so poor an approximation if seemingly biased. This was surprising to me as kernel density estimates are notoriously bad at boundary points.

If f(x) can be computed [up to a constant] at an arbitrary x, it is obviously feasible to simulate from f and approximate p(A). But the problem is then moot as a resolution would not even need the initial samples. If exploiting those to construct a single kernel density estimate, this estimate can be used as a proposal in an MCMC algorithm. Surprisingly (?), using instead the empirical cdf as proposal does not work.