12 independent samples are "enough" in MCMC sampling for what? I'm reading Learning in Graphical Models (Jordan 1998) and in the chapter Introduction to MCMC method by D. J. C. Mackay (page 201) it says this:

It's not clear to me to what this is referring to. "We really only need about twelve independent samples" for what precisely?
Add: I've found this particular chapter online here.
 A: It looks like this applies to a situation where you want to predict a future data point.
Let $X$ be a random variable, $\Phi=E[X]$ and $\sigma^2=Var(X)$ (in the notation of your screenshot, this corresponds to $\phi(x)=x$).
You get an estimator $\hat\Phi$ from $R$ independent samples, with $E[\hat\Phi]=\Phi$ and $Var(\hat\Phi)=\frac{\sigma^2}{R}$. You use $\hat\Phi$ as your prediction for the value of a future observation $\tilde X$. Your error can be measured by
$$
\begin{eqnarray}
E[(\tilde X-\hat\Phi)^2]&=&E[\tilde X^2]-2E[\tilde X]E[\hat\Phi]+E[\hat\Phi^2]\\
&=&E[\tilde X^2]-2\Phi^2+E[\hat\Phi^2]\\
&=&Var(\tilde X)+Var(\hat\Phi)\\
&=&\sigma^2\left(1+\frac{1}{R}\right)
\end{eqnarray}
$$
In this situation, there is not much point in spending time and energy on getting $R$ large, since $Var(\tilde X)$ will dominate. The exact value of 12 is rather arbitrary.
This point does not hold if what you want to learn is $\Phi$ itself, in which case a larger sample would be needed (typically 100 or 1000).
