Thanks Aki – that makes a lot of sense!

]]>I guess the interest is that most functions are L^2 and almost all of the theory that I’m aware of is L^2 (there are good reasons for this – L^1 is a terrible terrible space of functions).

Perhaps the only exception is low-dimensional QMC, which usually lives in spaces of bounded (H-K) variation, which is not unlike the space of functions with integrable

mixed first derivatives.

So basically, while it may not be unrealistic to want the expectation of a function that is only integrable, I’m not sure we currently (or at the increment of time before these two papers) have any (sampling-based) methods for doing that. I guess I was asking if this was a “build it and they will come”-type situation or if there was a demand in the community.

]]>Uh?! What is so special with square integrable integrands? I may be interested in the expectation of a function that is not square integrable, this does not seem so unrealistic, does it?!

]]>Monte Carlo is a special case of importance sampling where none of these issues really arise. So it’s the whole “change the base measure to improve the computational performance” aspect that I don’t understand the real application of.

If the base measure is the lebesgue measure and you’re hitting infinite variance problems, I would wonder why you are working with L^p (p<2) functions in a statistical context. (It may come up in a problem with lags – the natural space to consider functional volterra equations is some perverted version of L^1, for example)

]]>Uh?! Everything Monte Carlo based “is” importance sampling, Dan, so do you question the relevance of simulation as a whole?!

]]>I always thought that importance sampling was basically a neat trick that was pedagogically useful (it shows everyone what the ||f||^2 term in the variance actually means). But in practice, there was little chance of getting a decent importance distribution for real problems.

But all of these people are putting serious work into it, so what is it that I’m missing? What type of problem is the target here?

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