  In this paper, we propose a novel and generic family of multiple importance
sampling estimators. We first revisit the celebrated balance heuristic
estimator, a widely used Monte Carlo technique for the approximation of
intractable integrals. Then, we establish a generalized framework for the
combination of samples simulated from multiple proposals. We show that the
novel framework contains the balance heuristic as a particular case. In
addition, we study the optimal choice of the free parameters in such a way the
variance of the resulting estimator is minimized. A theoretical variance study
shows the optimal solution is always better than the balance heuristic
estimator (except in degenerate cases where both are the same). As a side
result of this analysis, we also provide new upper bounds for the balance
heuristic estimator. Finally, we show the gap in the variance of both
estimators by means of five numerical examples.
