\item If we obtain $\{x_i\}_{i=1}^N$ from another distribution $q(x)$:
\item If we obtain $\{x_i\}_{i=1}^N$ from another distribution with density $q(x)$:
\[
\mathbb E_p\left[\varphi(X)\right]=\int q(x)\frac{p(x)}{q(x)}\varphi(x)\mathrm d x =\int q(x)\tilde{\omega}(x)\varphi(x)\mathrm d x =\mathbb E_q\left[\tilde{\omega}(X)\varphi(X)\right]
\]
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@@ -147,7 +147,7 @@
\begin{itemize}
\item IS works as long as the support of the targetted distribution ($p$) is included in the support of the instrumental distribution ($q$).\newline
\item IS works as long as the support of the targeted distribution ($p$) is included in the support of the instrumental distribution ($q$).\newline
\item How do we choose distribution $q(x)$?\newline
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@@ -161,7 +161,7 @@
% ⇒ We are better off if the unormalized weight variance is low, i.e. if f/g is approximately constant.
% The closer g is to f, the better we are.
\item Is it possible to find a reasonable intrumental distribution for a posterior distribution?\newline
\item Is it possible to find a reasonable intrumental distribution for a posterior distribution? The prior?\newline
\item Probably better not to ``jump'' directly to the posterior distribution...
\item Consider the ''simplified`` object $p\left(\theta\right)p\left(\mathcal Y_T\right)^{\phi}$\newline
\item Consider the ''simplified`` object $p\left(\theta\right)p\left(\mathcal Y_T| \theta\right)^{\phi}$\newline
\item$p\left(\theta\right)$ is a good instrumental distribution for $p\left(\theta\right)p\left(\mathcal Y_T\right)^{\phi}$ for small values of $\phi$.\newline
\item$p\left(\theta\right)$ is a good instrumental distribution for $p\left(\theta\right)p\left(\mathcal Y_T| \theta\right)^{\phi}$ for small values of $\phi$.\newline
\item Consider a sequence of $\{\phi_i\}_{i=1}^M$ with $\phi_i<\phi_j$ for all $j>i$ and $\phi_M =1$.\newline
\item Use $p\left(\theta\right)$ as an intrumental distribution for $p\left(\theta\right)p\left(\mathcal Y_T\right)^{\phi_1}$ and $p\left(\theta\right)p\left(\mathcal Y_T\right)^{\phi_i}$
as an instrumental distribution for $p\left(\theta\right)p\left(\mathcal Y_T\right)^{\phi_{i+1}}$.\newline
\item Use $p\left(\theta\right)$ as an intrumental distribution for $p\left(\theta\right)p\left(\mathcal Y_T| \theta\right)^{\phi_1}$ and $p\left(\theta\right)p\left(\mathcal Y_T| \theta\right)^{\phi_i}$
as an instrumental distribution for $p\left(\theta\right)p\left(\mathcal Y_T| \theta\right)^{\phi_{i+1}}$.\newline
\item Herbst and Schofheide consider $\phi_n =\left(\frac{n}{M}\right)^\lambda$ with $\lambda>1$ (convexity).
