Key insight: suppose you have some fairly rare event and you want the likelihood of it. We can do this by drawing normal samples and reweighing them.
Suppose we want \(p_{\text{fail}}\); and we have \(q\) the proposal distribution and \(p\) the nominal distribution:
\(\tau \sim q\qty(\cdot)\), \(p_{\text{fail}} = \int 1 \qty {\tau \not\in \psi} p\qty(\tau) \dd{\tau }\)
What if we define a weird \(1\) such that:
\begin{equation} 1 = \frac{q\qty(\tau)}{q\qty(\tau)} \end{equation}
let’s see:
\begin{align} \int 1 \qty {\tau \not\in \psi} p\qty(\tau) \dd{\tau } &= \int \frac{q\qty(\tau)}{q\qty(\tau)}1 \qty {\tau \not\in \psi} p\qty(\tau) \dd{\tau } \\ &= \int q\qty(\tau) \qty(\frac{p\qty(\tau)}{q\qty(\tau)} \mathbb{1}\qty{\tau \not \in \psi}) \dd{\tau} \\ &= \mathbb{E}_{\tau \sim q\qty(\cdot)} \qty [\frac{p\qty(\tau)}{q\qty(\tau)} \mathbb{1}\qty{\tau \not \in \psi}] \end{align}
we can now estimate this by using an average of what we observed:
\begin{equation} \hat{p}_{\text{fail}} = \frac{1}{m}\sum_{i=1}^{m} \frac{p\qty(\tau_{i})}{q\qty(\tau_{i})} \mathbb{1} \qty {\tau_{i}\not \in \psi} \end{equation}
Suppose you have a function \(f(s)\) which isn’t super well integrate-able, yet you want:
\begin{equation} \mu = \mathbb{E}(f(s)) = \int_{0}^{1} f(s)p(s) \dd{s} \end{equation}
how would you sample various \(f(s)\) effectively such that you end up with \(\hat{\mu}\) that’s close enough?
Well, what if you have an importance distribution \(q(s): S \to \mathbb{R}^{[0,1]}\), which tells you how “important” to the expected value of the distribution a particular state is? Then, we can formulate a new, better normalization function called the “importance weight”:
\begin{equation} w(s) = \frac{p(s)}{q(s)} \end{equation}
Therefore, this would make our estimator:
\begin{equation} \hat{\mu} = \frac{\sum_{n} f(s_{n}) w(s_{n})}{\sum_{n} w(s_{n})} \end{equation}
Theoretic grantees
So, there’s a distribution over \(f\):
\begin{equation} q(s) = \frac{b(s)}{w_{\pi}(s)} \end{equation}
where
\begin{equation} w(s) = \frac{\mathbb{E}_{b} \qty( \sqrt{[\mathbb{E}(v|s, \pi )]^{2} + [Var(v|s, \pi )]})}{[\mathbb{E}(v|s, \pi )]^{2} + [Var(v|s, \pi )]} \end{equation}
which measures how important a state is, where \(\pi\) is the total discounted reward.