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Presume we have some function $P(X)$ that is either a valid or un-normalized probability density function. Here $X$ is high dimensional. I'd like to get insight about the relative density via quantile contours, or something related to quantiles of the density throughout the support. As just a simple example to help describe it....I've overlaid a plot below....although this is kernel density estimates don't make any assumptions on $P(X)$ except that it's smooth/continuous. Note that I don't want it for plotting -- i want it for online anomaly detection, but without information on the relative density i can't do any kind of inference on the fly from a PDF.

Explanation of below plot:

The contour lines are quantile contours in 5% intervals. This means that about 5% of the points generated from the estimated nonparametric distribution are below the lowest contour, 10% are below the next contour, and so on. The highest contour has about 95% of the points below it.

quantileContours

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  • $\begingroup$ what do you know about P(X)? do you know it's form? is derivable/continuous? or it's just an oracle function? $\endgroup$
    – rapaio
    Commented Nov 28, 2017 at 7:05
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    $\begingroup$ The function, and it's first derivative, are continuous (most likely it's of a neural network form). $\endgroup$
    – JPJ
    Commented Nov 28, 2017 at 7:40

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