A particular case of estimating a probability metric, with especially cruel error properties. There are a few different versions of this problem corresponding to various different information: Mutual information between two variables, KL divergence between two distributions, information of one variable; discrete variables, continuous variable… In the mutual information case this is an independence test.

Say I would like to know the mutual information of the laws of processes generating two streams \(X,Y\) of observations, with weak assumptions on the laws of the generation process. Better, suppose further that each observation from each process is i.i.d. In the case that they have a continuous state space and joint densities \(p_{X,Y}\), marginal densities \(p_{X},p_{Y},\)

\[ \operatorname {I} (X;Y)=\int _{\mathcal {Y}}\int _{\mathcal {X}}{p_{X,Y}X,Y\log {\left({\frac {p_{X,Y}X,Y}{p_{X}(x)\,p_{Y}(y)}}\right)}}\;dx\,dy\]

This is an empirical probability metric estimation problem.

Information is harder than the many metrics, because observations with low frequency have high influence on that value but are by definition rarely observed. It is easy to get a uselessly biased – or even inconsistent – estimator, especially in the nonparametric case.

## Histogram estimator

The obvious one for discrete data. For continuous data where the histogram bins must also be learned, this method is highly sensitive and can be inconsistent if you don’t do it right (Paninski 2003).

## Parametric

🏗

## Monte Carlo parametric

One case you might want to estimate this value which is one where there is no nonparametric estimation problem *per se* but the integral to solve it is inconvenient.
In which case, we might use a Monte Carlo method.

John Schulmann explicates a good trick for estimating KL divergence in the case that you can simulate from \(x_i\sim q\) and calculate \(p(x)\) and \(q(x_i),\) The following estimator is good despite looking unrelated.

\[\begin{aligned} KL[q, p] &= \int_x q(x) \log \frac{q(x)}{p(x)} \mathrm{d}x\\ &= E_{ x \sim q}\left[\log \frac{q(x)}{p(x)} \right]\\ &\approx \frac1N \sum_{i=1}^N \frac12(\log p(x)−\log q(x))^2 \end{aligned}\]

He also introduced a simple debiased one that does even better. The mechanics are interesting. (If you actually want a mutual information this notionally calculates it if we find the KL divergence between joint and product densities; But that is not totally trivial I shall concede.)

Akaike, Hirotogu. 1973. “Information Theory and an Extension of the Maximum Likelihood Principle.” In *Proceeding of the Second International Symposium on Information Theory*, edited by Petrovand F Caski, 199–213. Budapest: Akademiai Kiado. http://link.springer.com/chapter/10.1007/978-1-4612-1694-0_15.

Gao, Shuyang, Greg Ver Steeg, and Aram Galstyan. 2015. “Efficient Estimation of Mutual Information for Strongly Dependent Variables.” In *Journal of Machine Learning Research*, 277–86. http://www.jmlr.org/proceedings/papers/v38/gao15.html.

Grassberger, Peter. 1988. “Finite Sample Corrections to Entropy and Dimension Estimates.” *Physics Letters A* 128 (6–7): 369–73. https://doi.org/10.1016/0375-9601(88)90193-4.

Hausser, Jean, and Korbinian Strimmer. 2009. “Entropy Inference and the James-Stein Estimator, with Application to Nonlinear Gene Association Networks.” *Journal of Machine Learning Research* 10: 1469.

Kandasamy, Kirthevasan, Akshay Krishnamurthy, Barnabas Poczos, Larry Wasserman, and James M. Robins. 2014. “Influence Functions for Machine Learning: Nonparametric Estimators for Entropies, Divergences and Mutual Informations,” November. http://arxiv.org/abs/1411.4342.

Kraskov, Alexander, Harald Stögbauer, and Peter Grassberger. 2004. “Estimating Mutual Information.” *Physical Review E* 69: 066138. https://doi.org/10.1103/PhysRevE.69.066138.

Marzen, S. E., and J. P. Crutchfield. 2020. “Inference, Prediction, and Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes,” May. http://arxiv.org/abs/2005.03750.

Moon, Kevin R., and Alfred O. Hero III. 2014. “Multivariate F-Divergence Estimation with Confidence.” In *NIPS 2014*. http://arxiv.org/abs/1411.2045.

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Paninski, Liam. 2003. “Estimation of Entropy and Mutual Information.” *Neural Computation* 15 (6): 1191–1253. https://doi.org/10.1162/089976603321780272.

Roulston, Mark S. 1999. “Estimating the Errors on Measured Entropy and Mutual Information.” *Physica D: Nonlinear Phenomena* 125 (3-4): 285–94. https://doi.org/10.1016/S0167-2789(98)00269-3.

Schürmann, Thomas. 2015. “A Note on Entropy Estimation.” *Neural Computation* 27 (10): 2097–2106. https://doi.org/10.1162/NECO_a_00775.

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Wolf, David R., and David H. Wolpert. 1994. “Estimating Functions of Distributions from A Finite Set of Samples, Part 2: Bayes Estimators for Mutual Information, Chi-Squared, Covariance and Other Statistics,” March. http://arxiv.org/abs/comp-gas/9403002.

Wolpert, David H., and David R. Wolf. 1994. “Estimating Functions of Probability Distributions from a Finite Set of Samples, Part 1: Bayes Estimators and the Shannon Entropy,” March. http://arxiv.org/abs/comp-gas/9403001.

Zhang, Zhiyi, and Michael Grabchak. 2014. “Nonparametric Estimation of Küllback-Leibler Divergence.” *Neural Computation* 26 (11): 2570–93. https://doi.org/10.1162/NECO_a_00646.