When considering the approximation by Monte Carlo of an expectation of the form$$\mathfrak{I}=\mathbb{E}^X[\log\{\mathbb{E}^{Y|X}[h(X,Y)|X]\}]$$using a resolution of the form$$\hat{\mathfrak{I}}=\frac{1}{N}\sum_{n=1}^N\log\left\{\frac{1}{M}\sum_ {m=1}^M h(x_n,y_{nm})\right\}$$(when the $x_n$'s are generated from the correct marginal on $X$ and the $y_{nm}$'s from the proper conditional of $Y$ given $X=x_n$) produces a convergent but biased estimator. I wonder if there is a genuine way to produce an unbiased estimator in this case, other than using the trick of McLeish (see equation (1) on page 2).

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    $\begingroup$ I don;t see the connection between McLeish's article which is really dealing with Bayesian methods and Markov Chain Monte Carlo and your problem. $\endgroup$ – Michael R. Chernick Dec 8 '16 at 15:10

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