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I'm evaluating several methods to estimate the density of an unknown distribution $f$ from observed data, among which kernel density estimation with distinct kernel functions, a mixture density network, and a simple histogram distribution.

Currently I'm using the maximum likelihood leave-one-out method to tune the bandwidths of my models, and in the end I compare them based on maximum likelihood I obtain on a set of test data (i.e. data that was not use to fit them).

I've also read about the Mean Integrated Square Error, but I cannot understand how to use it for model selection since it apparently requires knowing the true density function, which is what I'm trying to approximate in the first place (looks like a chicken-and-egg problem). I think I'm failing to understand something basic.

Can someone point me in the right direction?

Thanks! Marco

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You're right, you don't know the true density. That's why we use cross-validation or a rule of thumb (which, for example, assumes the true density is normal) to get a bandwidth. Check out these slides for more details.

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