Suppose posterior of $P(\theta,\delta|x,y)$ is given. Gibbs sample is construct as follow
- Start with arbitrary $\theta^{(0)},\delta^{(0)}$ such that $P(\theta^{(0)},\delta^{(0)}|x,y) >0$
- At $t$ iteration , sample each $\theta^{(t)}$ from distribution $P(\theta^{(t)}|\delta^{(t-1)},x,y)$ and $\delta$ from $P(\delta^{(t)}|\theta^{(t)},x,y)$
- Sample of $((\theta^{(m)},\delta^{(m)}),(\theta^{(m+1)},\delta^{(m+1)}),...)$ is taken after burn in.
How can I estimate the $E(\delta|x,y)$ ? noted the expected is with respect to pdf of $P(\delta|x,y)=\int_{\theta}P(\theta,\delta|x,y)d\theta $. Thank in advance.