Address concerns in gh-12999 for ks_2samp in exact mode.

[pvanmulbregt]

Return an estimate of the relative error of the probability computed by
stats._compute_prob_inside_method(). As the desired probability is the
complement of the one computed, subtractive cancellation can occur.
Check if it may have, and declare the attempt a failure. This prevents
a probability ~ 1e-16 being reported when the actual probability is many orders
of magnitude lower.

Reference issue

Closes gh-12999

What does this implement/fix?

Checks if the computation of a probability in stats._compute_prob_inside_method(), using exact algorithm, may have lost too much precision to be used as a complementary probability. If so, back off to the asymptotic computation.
In stats. _compute_prob_outside_square(), do a pre-check to determine if the computation will definitely overflow. If so, return immediately.

[pvanmulbregt]

Actually this cure is starting to look worse than the problem it is addressing...

[mdhaber]

Actually this cure is starting to look worse than the problem it is addressing...

Shall I hold off on review, then?

[pvanmulbregt]

yes, hold off on any review, the "fix" affects too many probability calculations.
A better approach may be to compute an outside probability, scaling with every iteration to avoid underflow/ overflow.

[t-vi]

@pvanmulbregt I wrote up a more stable recursion for the inner method / 2-sided 2-sample in
https://arxiv.org/abs/2102.08037

The basic trick that you can do instead of doing 1-A(m,n)/Binom(m+n,n) at the end, you can do the entire recursion on values of this form 1-A(i,j)/Binom(i+j,j). It seems to be about 4x slower than the current implementation, which I think it is due to the extra compute, but then both the current implementation and the stable recursion can be moved to numba / other ways to compile to speed them up.

[rgommers]

This was superceded by gh-14432, closing. Thanks @pvanmulbregt, all