I have a simple question. Suppose $X=(X_1,X_2,X_3)$ is multivariate normal. What's the best (quickest) way to draw from the conditional density $X_1\mid \exp(X_1)+\exp(X_2)+X_3$?
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Define a Markov chain like this.
The first coordinate of each value of this Markov chain will have as equilibrium distribution exactly what you need. You should discard the first, say, $5000$ thousand values of the chain, which correspond to the so called burn-in period. You have to find small values for the $\epsilon_i$'s that make the Markov chain mix properly. Large values of the $\epsilon_i$'s will make you reject all the proposals. My sugestion is to target an acceptance rate of about $50\%$ and do a graphical monitoring of the chain.
I've just seen on the bookstore the book "Monte Carlo Statistical Methods" by Robert and Casella that explain all you need to know to justify this Random Walk Metropolis Algorithm.
P.S. As clarified latter by cardinal, Prof. Robert is our fellow user Xi'an, who posted the first answer to this question!