  Multiple importance sampling estimators are widely used for computing
intractable constants due to its reliability and robustness. The celebrated
balance heuristic estimator belongs to this class of methods and has proved
very successful in computer graphics. The basic ingredients for computing the
estimator are: a set of proposal distributions, indexed by some discrete label,
and a predetermined number of draws from each of these proposals. However, if
the number of available proposals is much larger than the number of permitted
importance points, one needs to select, possibly at random, which of these
distributions will be used. The focus of this work lies within the previous
context, exploring some improvements and variations of the balance heuristic
via a novel extended-space representation of the estimator, leading to
straightforward annealing schemes for variance reduction purposes. In addition,
we also look at the intractable scenario where the proposal density is only
available as a joint function with the discrete label, as may be encountered in
problems where an ordering is imposed. For this case, we look at combinations
of correlated unbiased estimators which also fit into the extended-space
representation and, in turn, will provide other interesting solutions.
