Weighted sampling without replacement and negative weights

Question

I have an unusual sampling problem that I'm trying to implement for a Monte Carlo technique. I am aware there are related questions and answers regarding the fully-positive problem.

I have a list of n weights w_1,...,w_n and I need to choose k elements, labelled s_1,...,s_k say. The probability distribution that I want to sample from is

p(s_1,...,s_k) = |w_s_1 + ... + w_s_k| / P_total

where P_total is a normalization factor (the sum of all possible p(s,...) without P_total). I don't really care about how the elements are ordered for my purpose.

Note that some of the w_i may be less than zero and the absolute magnitude signs above. With purely non-negative w_i this distribution is relatively straightforward by sampling without replacement - a tree method being the most efficient as far as I can tell. With some negative weights, though, I feel like I must resort to explicitly writing out each possibility and sampling from this exponentially large set. Any suggestions or insights would be appreciated!

Answer 1

Rejection sampling is worth a try. Compute the maximum weight of a sample (max of the abs of each of the k least and k greatest). Repeatedly generate a uniform random sample and accept it with probability equal to its weight over the maximum weight until a sample is accepted.