Empirical estimation of information

Informing yourself from your data how informative your data was



This is an empirical probability metric estimation problem, 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 \(p_{X},p_{Y},\)

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

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.)

References

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.
Amigó, José M, Janusz Szczepański, Elek Wajnryb, and Maria V Sanchez-Vives. 2004. “Estimating the Entropy Rate of Spike Trains via Lempel-Ziv Complexity.” Neural Computation 16 (4): 717–36. https://doi.org/10.1162/089976604322860677.
Crumiller, Marshall, Bruce Knight, Yunguo Yu, and Ehud Kaplan. 2011. “Estimating the Amount of Information Conveyed by a Population of Neurons.” Frontiers in Neuroscience 5: 90. https://doi.org/10.3389/fnins.2011.00090.
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.
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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.” arXiv:2005.03750 [cond-Mat, Physics:nlin, Stat], 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.
Nemenman, Ilya, William Bialek, and Rob de Ruyter van Steveninck. 2004. “Entropy and Information in Neural Spike Trains: Progress on the Sampling Problem.” Physical Review E 69 (5): 056111. https://doi.org/10.1103/PhysRevE.69.056111.
Nemenman, Ilya, Fariel Shafee, and William Bialek. 2001. “Entropy and Inference, Revisited.” In arXiv:physics/0108025. http://arxiv.org/abs/physics/0108025.
Paninski, Liam. 2003. “Estimation of Entropy and Mutual Information.” Neural Computation 15 (6): 1191–1253. https://doi.org/10.1162/089976603321780272.
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Schürmann, Thomas. 2015. “A Note on Entropy Estimation.” Neural Computation 27 (10): 2097–2106. https://doi.org/10.1162/NECO_a_00775.
Shibata, Ritei. 1997. “Bootstrap Estimate of Kullback-Leibler Information for Model Selection.” Statistica Sinica 7: 375–94.
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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.” arXiv:comp-Gas/9403001, 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.

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