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Assume we have a d-dimensional vector space.Also, we don't have a parametric distribution function, only we have a set of samples in this space which we assume is sampled from one unknown distribution ,what is the best way to compute the probability that a given new point in this space is sampled from the unknown distribution which data is sampled from?

user85361
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  • I wonder what you mean by "a set of samples". Might you have meant simply a sample, consisting of some finite number of independent observations drawn from the unknown distribution? It seems commonplace that people unschooled in statistics misuse the word "sample" in just that way. – Michael Hardy May 07 '14 at 15:37

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The question itself is incorrectly posed:

"...probability that a given new point ... is sampled from the unknown distribution..."

To answer that, you need to have some alternative set of distributions, as well as a prior probability that your (initial) distribution is true.

For example, suppose you have a distribution that allows large outliers (e.g. Cauchy). Then whenever you get an outlier, it will appear completely incompatible with the points you've had so far.

However, if you'd like to know "how similar is the new point to the ones already observed", you might try kernel density estimators. Based on the previous sample, $x_1,...,x_n$, calculate a KDE, $f(x)$ and then plug in the new observation $x$. Maybe you can detect the outliers this way.

PA6OTA
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  • Thank you. Why we need some alternative set of distributions and prior? Also, I don't understand why question is incorrectly posed, would you please help me understand? – user85361 May 10 '14 at 05:19
  • Also, why in the last paragraph, KDE is about "how similar is the new point to the ones already observed". Is it the meaning of probability distribution? forgive for my silly questions – user85361 May 10 '14 at 05:29