I've found many ways to estimate the pdf of the distribution you take samples from. However I did not find any ways to calculate THE probability density of the samples itself.

How can I do that?

Some more information (after comment): In my case a single Monte Carlo run takes quite some time. Let's say we have done a simulation consisting of n runs with a standard normal distribution for the input position_x. This gives us n inputs and n outputs (simplified one input, one output case). If we now would like to have the average output for input position_x if it had been a Normal Distribution with mean 0 and std 0.5, we could use importance sampling to reweight the outputs generated by our initial simulation. This would give the result without lengthy simulation.

For importance sampling you need the pdf of the input. But for relatively small n, the created distribution of position_x might not be a close match to the standard normal one.

  • Couldn't we use THE sample density as the pdf?

  • Couldn't we also use THE sample density to correct the standard Normal input case for the effect of realised sample density mismatch by importance sampling using the ratio between the standard normal density and the realised sample density?

  • $\begingroup$ Not enough information. (1) Can you give an example? What kind of data do you have for estimating the density. You will not be able to find the exact PDF from data. But if you're looking for an 'envelope' for importance sampling, that might not matter. (2) What are you trying to simulate or integrate with importance sampling? $\endgroup$ – BruceET May 30 '15 at 20:36
  • $\begingroup$ @Bruce Trumbo: More information added. $\endgroup$ – Martijn Pot May 31 '15 at 10:21

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