Timeline for Inflated independent samples for Monte Carlo estimation
Current License: CC BY-SA 3.0
8 events
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| Jun 15, 2014 at 14:18 | comment | added | Anton | 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 $f$. | |
| Jun 15, 2014 at 14:11 | history | edited | Anton |
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| Jun 15, 2014 at 13:37 | comment | added | Anton | 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? | |
| Jun 14, 2014 at 21:46 | comment | added | Pierre | 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. | |
| Jun 12, 2014 at 9:09 | comment | added | Anton | 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. | |
| Jun 12, 2014 at 8:17 | comment | added | Pierre | 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)? | |
| Jun 12, 2014 at 8:10 | history | edited | Anton | CC BY-SA 3.0 |
edited body; edited title
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| Jun 10, 2014 at 21:36 | history | asked | Anton | CC BY-SA 3.0 |