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In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing forth and back transition probabilities for each Metropolis move.

Can I perform a Monte Carlo estimation with samples that are generated by "inflating" a sequence of independent samples?

As a simple 1D example, if there is a sequence $\{x_i\}$ of values $\{3, 8, 4\}$, then I would like to inflate it into a sequence $\{y^i_j\}$ that looks, e.g., like $\{2.9, 3, 3.1, \quad 7.8, 8.05, 8.07, \quad 4.2, 4.04, 4.01\}$. That is, I take an independent sample and inflate it into multiple samples by perturbing around it.

Given that the sampling probabilities are known for both independent sampling $p(\cdot)$ and for the perturbation of the sampled values $q(\cdot |x_i)$, how can one construct a proper Monte Carlo estimator based on such an "inflated" sequence $\{y^i_j\}$?

I have checked that the ad-hoc estimator

$I=\sum\frac{f(y^i_j)}{p(x_i)q(y^i_j|x_i)}$

does not lead to the correct estimation.

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Just to be clear, exactly what is it that you want to do? I mean, one can potentially view this as a type of importance sampling scheme but then the true distribution/density (your p?) must be known. This should be possible to work out but I'm not quite sure that I understand what you are ultimately after. For example, what is the ad-hoc estimator I supposed to estimate? The mean of the function f (under some distribution)? –  Pierre Jun 12 at 8:17
    
The task is a classical Monte Carlo integration of a function, i.e. the ad-hoc estimator would be $I=\sum\frac{f(x_i)}{p(x_i)}$. What I want to do is some sort of importance sampling. This way I'm trying to efficiently sample a multimodal function. Think about it this way: $p(\cdot)$ can easily sample different modes; $q(\cdot|x_i)$ can efficiently explore a mode where $x_i$ is located. Both distributions $p$ and $q$ are known and normalized. What I cannot understand is how to construct an estimator that would integrate $f$ with this approach. –  Anton Jun 12 at 9:09
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It seems to me as the ad-hoc estimator you propose must indeed give you an erroneous answer since the $X_i$ 's (which are still in the Y sequence) are not connected to the distribution $q$. If you do not start with a sequence and then inflate it, but rather "inflate" from the start then it seems as it should work. This way you still have different modes and explore their neighbourhood but all variables have the same distribution. Or perhaps just remove the first sequence (original $X_i$'s) and use only the remaining values for the estimation. –  Pierre Jun 14 at 21:46
    
Thanks for your notes. Unfortunately, it is crucial to explore each mode for some time. So, if I "inflate" from the start, then the variance is too high, as the samples are all from different modes and there is no benefit in using $q$. In our case, distribution $q$ knows how exactly to explore a single mode, so ideally we'd like to apply some stratified sampling for $y$ given $x_i$ to explore the mode and then generate the new $x_{i+1}$ from the next mode. Given that we are flexible in adjusting this sampling scheme, can we construct a correct estimator with these properties? –  Anton Jun 15 at 13:37
    
Also, regarding removing the sequence $x_i$: the problem is that $q$ cannot jump between modes, i.e. it cannot sample the complete support of $f$. It can explore only a single mode of $f$. –  Anton Jun 15 at 14:18

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