# How to compute the entropy of a random variable with values in a metric space? [closed]

I have a cloud of points, and I want to compute its 'diversity'. Variance is not appropriate, because a cloud clustering around few points can still have a large variance.

To that end, I see the cloud of points as the realizations of a random variable.

I want to compute the topological entropy of a random variable, with values in a metric space $(X,d)$.

I only know $N$ (large) realisations of this variable, and I can compute the relative distances between them.

## closed as unclear what you're asking by R W, Ben McKay, Mikhail Katz, Pace Nielsen, Stefan KohlFeb 6 '18 at 20:27

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• see definition 2.5 in this thesis – Carlo Beenakker Feb 2 '18 at 15:53
• Very uninformative comment. It’s the Bowen-Dinaburg definition of entropy.I can't use it here because I don't know the whole map, only N realisations. – Mostafa Feb 2 '18 at 21:25
• Perhaps you should give a bit more of context. Your question is a bit vague. – Rafael Alcaraz Barrera Feb 3 '18 at 5:06
• the context is that I have a cloud of points, and I want to measure its 'diversity' without using variance – Mostafa Feb 3 '18 at 12:59