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Suppose we know that random vectors $x, y$ have joint density $p(x, y) \propto \exp(-U(x_1, \ldots, x_m, y_1, \ldots, y_n))$, and we want to draw a random sample from the marginal $p(x)$ (i.e. we want a random sample of $x$ but are not interested in $y$.)

Naively, we can do an MCMC on the jointly density $p(x, y)$, getting a joint vector $(x, y)$ and taking the $x$ part as the sample. But when $n \gg m$, I speculate that this is not very efficient since we also draw a 'useless' sample of $y$, which we do not want. (e.g. papers like this indicate that the complexity of MCMC sampling scales polynomially with dimension)

Question: Are there better techniques to sample from $p(x)$ when $n \gg m$?

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  • $\begingroup$ I am not sure I understand your $p(x, y) \propto \exp(-U(x_1, \ldots, x_m, y_1, \ldots, y_n))$. If that is something special, you might be able to separate out $x$ and $y$ $\endgroup$
    – Henry
    Commented May 4, 2021 at 8:47
  • $\begingroup$ $U$ is not "something special". It is called the potential function of the distribution from which we intend to draw our sample. Ideally, $U$ is a strongly convex function $\mathbb R^{m+n} \to \mathbb R$; but there is no separability guarantee on $U$. $\endgroup$
    – Dromeda
    Commented May 4, 2021 at 14:56
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    $\begingroup$ That is useful information: some people might otherwise read $U$ for uniform $\endgroup$
    – Henry
    Commented May 4, 2021 at 14:59
  • $\begingroup$ If no separation or factorisation are available, I do not see a way out since the marginal is only defined by the joint and the $y$ component is not useless. Some savings may be found from Rao-Blackwellisation, though: Once a sequence of $(x_t,y_t)_{1\le t\le T}$ has been simulated from the joint, using the $y_t$ to approximate the marginal by$$\frac{1}{T}\sum_{t=1}^Tp(x|y_t)$$would work. $\endgroup$
    – Xi'an
    Commented May 5, 2021 at 7:34

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