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We have a bivariate random variable $(X,Y)$ for which sampling is challenging.

If we were to know how to sample from the conditionals $(X|Y)$ and $(Y|X)$, we could get samples from the joint using Gibbs sampling by iterating: $$x_{t+1} \sim (X|y_t)$$ $$y_{t+1} \sim (Y|x_t).$$

Assume however that we do not know how to sample from one of the conditionals (say, $Y|X$), but that, for each $x$, we know how to sample from an approximation $\tilde{Y}_x \approx (Y|X=x)$.

If we assume that $\tilde{Y}_x$ is close to $(Y|X=x)$ in some sense (for example, that the Kullback-Leibler divergence between these two distributions is smaller than $\varepsilon$ for all $x$), could we get results about the convergence of the pseudo-Gibbs chain: $$x_{t+1} \sim (X|Y=y_t)$$ $$y_{t+1} \sim \tilde{Y}_{x_t}?$$

PAM
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  • You'll get convergence, of course, it'll just be to the joint distribution of $X$ and $Y$ given the approximate distribution of $Y|X$, not the actual distribution. Given that, the K-L divergence of the converged-to and actual distributions will be smaller than $\epsilon$ too. – jbowman Dec 04 '17 at 15:48
  • What do you precisely mean by the joint distribution given the approximate conditional? It is likely that $X|Y$ and $\tilde{Y}_X$ are not the conditionals of any joint distribution. – PAM Dec 04 '17 at 18:48
  • If they are not the conditionals of any joint distribution, then I have no idea of what, if anything, the pseudo-Gibbs sampler will converge to! Time to upvote the question... – jbowman Dec 04 '17 at 19:18
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    This comment doesn't answer your question, but you could instead run a M-H to sample from the conditional for $Y|X$. That would be a Metropolis within Gibbs sampler, which would converge to the correct distribution. – Greenparker Dec 05 '17 at 11:27
  • Yes, I guess that $\tilde{Y}_x$ could even be used to design a good proposal for M-H ? However, I'm mostly interested in high-dimensional settings where MH will have a hard time. – PAM Dec 05 '17 at 14:21
  • @PAM Yes, it could be used quite well as a proposal I presume. It may be unfair to assume MH will have a hard time in high dimensional situations. Sometimes MH can surprise you. Worth giving a shot if it means asymptotically exact samples. – Greenparker Dec 05 '17 at 15:53

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