Suggest edit — multiple importance sampling

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---
title: "multiple importance sampling"
type: concept
related: [Multiple Importance Sampling, Importance Sampling]
source: https://www.jemoka.com/posts/kbhmultiple_importance_sampling/
confidence: high
status: active
---

what if we did Importance Sampling, but&hellip;. 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 \(\tau_{i} \sim q_{i}\qty(\tau)\) 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\qty {\tau_{i}\not \in \psi} \end{equation}
where
\begin{equation} w_{i} = \qty(\frac{p\qty(\tau_{i})}{q_{i}\qty(\tau_{i})}) \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\qty(\tau_{i})}{\frac{1}{m}\sum_{j=1}^{m}q_{j}\qty(\tau_{i})} \end{equation}
this version essentially creates a mixture distribution between all of your input distributions.