Say I would like to know the mutual information of the processes generating two streams of observations, with weak assumptions on the form of the generation process. This is a normal sort of empirical probability metric estimation problem
Information is harder than normal, because observations with low frequency have high influence on the estimate. It is easy to get a uselessly biassed – or even inconsistent – estimator, especially in the nonparametric case.
A typical technique, is to construct a joint histogram from your samples, treat the bins as as a finite alphabet and then do the usual calculation. That throws out a lot if information, and it feels clunky and stupid, especially if you suspect your distributions might have some other kind of smoothness that you'd like to exploit.
You cold also estimate the densities. Moreover this method is highly sensitive and can be arbitrarily wrong if you don't do it right (see Paninski, 2003).
So, better alternatives?
One obvious one is asking yourself: Do I really want to know th information? Or do I merely wish to know that something is uninformative, i.e. to estimate some degree of independence? independence is related, but has much more general strategies.
ITE toolbox (estimators)
- ad hominem, KKPW14 is a good place to start.
- Kraskov's (2004) NN-method looks nice, but don't have any guarantees that I know of
- relationship between mutual information and copula entropy.
- those occasional mentions of calculating mutual information from recurrence plots- how do they work?
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- GaVG15: (2015) Efficient Estimation of Mutual Information for Strongly Dependent Variables. In Journal of Machine Learning Research (pp. 277–286).
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- KKPW14: (2014) Influence Functions for Machine Learning: Nonparametric Estimators for Entropies, Divergences and Mutual Informations. ArXiv:1411.4342 [Stat].
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- ZhGr14: (2014) Nonparametric Estimation of Küllback-Leibler Divergence. Neural Computation, 26(11), 2570–2593. DOI