multiple importance sampling

what if we did Importance Sampling, but…. had multiple proposals?!
notation: w_{i}, \tau_{i}, etc. all correspond to stuff that came from proposal q_{i}.
standard multiple importance sampling (s-MIS)
draw samples from current proposals \tau_{i} \sim q_{i}\left(\tau\right) use all of the samples to estimate p_{\text{fail}} \begin{equation} \hat{p}{\text{fail}} = \frac{1}{m} \sum{i=1}^{m} w_{i} 1\left{\tau_{i}\not \in \psi\right} \end{equation}
where

\begin{equation} w_{i} = \left(\frac{p\left(\tau_{i}\right)}{q_{i}\left(\tau_{i}\right)}\right) \end{equation}

deterministic mixture multiple importance sampling (DM-MIS)
draw samples alternating each of the proposals use them to estimate p_{\text{fail}} \begin{equation} w_{i} = \frac{p\left(\tau_{i}\right)}{\frac{1}{m}\sum_{j=1}^{m}q_{j}\left(\tau_{i}\right)} \end{equation}
this version essentially creates a mixture distribution between all of your input distributions.