Informations

Entropies and other measures of surprise



TODO: explain this diagram which I ripped off from Wikipedia.

Informations as measure, Venn diagram

Not: what you hope to get from the newspaper. (Although…) Rather: Different types of (formally defined) entropy/information and their disambiguation. The seductive power of the logarithm, and convex functions rather like it.

A proven path to publication is to find or reinvent a derived measure based on Shannon information, and apply it to something provocative-sounding. (Qualia! Stock markets! Evolution! Language! The qualia of evolving stock market languages!)

This is purely about the analytic definition given random variables, not the estimation theory, which is a different problem

Connected also to functional equations and yes, statistical mechanics, and quantum information physics.

Shannon Information

“I have are given a discrete random process, how many bits of information do I need to reconstruct it?”

Vanilla information, thanks be to Claude Shannon. A thing related to coding of random processes.

Given a random variable \(X\) taking values \(x \in \mathcal{X}\) from some discrete alphabet \(\mathcal{X}\), with probability mass function \(p(x)\).

\[ \begin{array}{ccc} H(X) & := & -\sum_{x \in \mathcal{X}} p(x) \log p(x) \\ & \equiv & E( \log 1/p(x) ) \end{array} \]

More generally if we have a measure \(P\) over some Borel space \[ H(X)=-\int _{X}\log {\frac {\mathrm {d} P}{\mathrm {d} \mu }}\,dP \]

Over at the functional equations page is a link to Tom Leinster’s clever proof of the optimality of Shannon information via functional equations.

One interesting aspect of the proof is where the difficulty lies. Let \(I:\Delta_n \to \mathbb{R}^+\) be continuous functions satisfying the chain rule; we have to show that \(I\) is proportional to \(H\). All the effort and ingenuity goes into showing that \(I\) is proportional to \(H\) when restricted to the uniform distributions. In other words, the hard part is to show that there exists a constant \(c\) such that

\[ I(1/n, \ldots, 1/n) = c H(1/n, \ldots, 1/n) \]

for all \(n \geq 1\).

Venkatesan Guruswami, Atri Rudra and Madhu Sudan, Essential Coding Theory.

K-L divergence

Because “Kullback-Leibler divergence” is a lot of syllables for something you use so often. Or you could call it the “relative entropy”, if you want to sound fancy and/or mysterious.

KL divergence is defined between the probability mass functions of two discrete random variables, \(X,Y\) over the same space, where those probability mass functions are given \(p(x)\) and \(q(x)\) respectively.

\[ \begin{array}{cccc} D(P \parallel Q) & := & -\sum_{x \in \mathcal{X}} p(x) \log p(x) \frac{p(x)}{q(x)} \\ & \equiv & E \log p(x) \frac{p(x)}{q(x)} \end{array} \]

More generally, if the random variables have laws, respectively \(P\) and \(Q\): \[ {\displaystyle D_{\operatorname {KL} }(P\|Q)=\int _{\operatorname {supp} P}{\frac {\mathrm {d} P}{\mathrm {d} Q}}\log {\frac {\mathrm {d} P}{\mathrm {d} Q}}\,dQ=\int _{\operatorname {supp} P}\log {\frac {\mathrm {d} P}{\mathrm {d} Q}}\,dP,} \]

Jenson-Shannon divergence

Symmetrized version. Have never used.

Mutual information

The “informativeness” of one variable given another… Most simply, the K-L divergence between the product distribution and the joint distribution of two random variables. (That is, it vanishes if the two variables are independent).

Now, take \(X\) and \(Y\) with joint probability mass distribution \(p_{XY}(x,y)\) and, for clarity, marginal distributions \(p_X\) and \(p_Y\).

Then the mutual information \(I\) is given

\[ I(X; Y) = H(X) - H(X|Y) \]

Estimating this one has been giving me grief lately, so I’ll be happy when I get to this section and solve it forever. See nonparametric mutual information.

Getting an intuition of what this measure does is handy, so I’ll expound some equivalent definitions that emphasises different characteristics:

\[ \begin{array}{cccc} I(X; Y) & := & \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p_{XY}(x, y) \log p(x, y) \frac{p_{XY}(x,y)}{p_X(x)p_Y(y)} \\ & = & D( p_{XY} \parallel p_X p_Y) \\ & = & E \log \frac{p_{XY}(x,y)}{p_X(x)p_Y(y)} \end{array} \]

More usually we want the Conditional Mutual information.

\[I(X;Y|Z)=\int _{\mathcal {Z}}D_{\mathrm {KL} }(P_{(X,Y)|Z}\|P_{X|Z}\otimes P_{Y|Z})dP_{Z}\]

See Christopher Olah’s excellent visual explanation.

Kolmogorov-Sinai entropy

Schreiber says:

If \(I\) is obtained by coarse graining a continuous system \(X\) at resolution \(\epsilon\), the entropy \(HX(\epsilon)\) and entropy rate \(hX(\epsilon)\) will depend on the partitioning and in general diverge like \(\log(\epsilon)\) when \(\epsilon \to 0\). However, for the special case of a deterministic dynamical system, \(\lim_{\epsilon\to 0} hX (\epsilon) = hKS\) may exist and is then called the Kolmogorov-Sinai entropy. (For non-Markov systems, also the limit \(k \to \infty\) needs to be taken.)

That is, it is a special case of the entropy rate for a dynamical system - Cue connection to algorithmic complexity. Also metric entropy?

Relatives

Rényi Information

Also, the Hartley measure.

You don’t need to use a logarithm in your information summation. Free energy, something something. (?)

The observation that many of the attractive features of information measures are simply due to the concavity of the logarithm term in the function. So, why not whack another concave function with even more handy features in there? Bam, you are now working on Rényi information. How do you feel?

Tsallis statistics

Attempting to make information measures “non-extensive”. “q-entropy”. Seems to have made a big splash in Brazil, but less in other countries. Non-extensive measures are an intriguing idea, though. I wonder if it’s parochialism that keeps everyone off Tsallis statistics, or a lack of demonstrated usefulness?

Fisher information

See maximum likelihood and information criteria.

Estimating information

Wait, you don’t know the exact parameters of your generating process a priori? You need to estimate it from data.

Connection to statistical mechanics

Informational entropy versus thermodynamics entropy.

  • Shalizi and Moore (2003):

    We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables.

  • John Baez’s A Characterisation of Entropy etc. See also

  • David H. Wolpert (2006a)

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.
Amari, Shunʼichi. 2001. Information Geometry on Hierarchy of Probability Distributions.” IEEE Transactions on Information Theory 47: 1701–11.
Ay, N., N. Bertschinger, R. Der, F. Güttler, and E. Olbrich. 2008. Predictive Information and Explorative Behavior of Autonomous Robots.” The European Physical Journal B - Condensed Matter and Complex Systems 63 (3): 329–39.
Baez, John C., Tobias Fritz, and Tom Leinster. 2011. A Characterization of Entropy in Terms of Information Loss.” Entropy 13 (11): 1945–57.
Barnett, Lionel, Adam B. Barrett, and Anil K. Seth. 2009. Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables.” Physical Review Letters 103 (23): 238701.
Barrett, Adam B, Lionel Barnett, and Anil K Seth. 2010. Multivariate Granger Causality and Generalized Variance.” Phys. Rev. E 81 (4): 041907.
Bialek, William, Ilya Nemenman, and Naftali Tishby. 2001. Complexity Through Nonextensivity.” Physica A: Statistical and Theoretical Physics 302 (1-4): 89–99.
———. 2006. Predictability, Complexity, and Learning.” Neural Computation 13 (11): 2409–63.
Bieniawski, Stefan, and David H. Wolpert. 2004. Adaptive, Distributed Control of Constrained Multi-Agent Systems.” In Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems-Volume 3, 4:1230–31. IEEE Computer Society.
Calude, Cristian S. 2002. Information and Randomness : An Algorithmic Perspective. Springer.
Cappé, Olivier, Eric Moulines, and Tobias Ryden. 2007. Inference in Hidden Markov Models. 1st ed. 2005. Corr. 2nd printing 2007 edition. New York ; London: Springer.
Cerf, Nicolas J., and Chris Adami. 1998. “Information Theory of Quantum Entanglement and Measurement.” Physica D: Nonlinear Phenomena 120 (1-2): 62–81.
Cerf, Nicolas J., and Christoph Adami. 1997. “Entropic Bell Inequalities.” Physical Review A 55 (5): 3371.
Chaitin, Gregory J. 1977. “Algorithmic Information Theory.” IBM Journal of Research and Development.
———. 2002. “The Intelligibility of the Universe and the Notions of Simplicity, Complexity and Irreducibility.”
Chiribella, G, G M d’Ariano, and P Perinotti. 2011. Informational Derivation of Quantum Theory.” Physical Review A 84 (1): 012311.
Chow, C K, and C N Liu. 1968. Approximating Discrete Probability Distributions with Dependence Trees.” IEEE Transactions on Information Theory 14: 462–67.
Cohen, Joel E. 1962. Information Theory and Music.” Behavioral Science 7 (2): 137–63.
Cover, Thomas M., Péter Gács, and Robert M. Gray. 1989. Kolmogorov’s Contributions to Information Theory and Algorithmic Complexity.” The Annals of Probability 17 (3): 840–65.
Cover, Thomas M, and Joy A Thomas. 2006. Elements of Information Theory. Wiley-Interscience.
Crooks, Gavin E. 2007. Measuring Thermodynamic Length.” Physical Review Letters 99 (10): 100602.
Csiszár, Imre, and Paul C Shields. 2004. Information Theory and Statistics: A Tutorial. Vol. 1. Foundations and Trends in Communications and Information Theory.
Dahlhaus, R. 1996. On the Kullback-Leibler Information Divergence of Locally Stationary Processes.” Stochastic Processes and Their Applications 62 (1): 139–68.
Dewar, Roderick C. 2003. Information Theory Explanation of the Fluctuation Theorem, Maximum Entropy Production and Self-Organized Criticality in Non-Equilibrium Stationary States.” Journal of Physics A: Mathematical and General 36: 631–41.
Earman, John, and John D Norton. 1998. Exorcist XIV: The Wrath of Maxwell’s Demon. Part I. From Maxwell to Szilard.” Studies in History and Philosophy of Modern Physics 29 (4): 435–71.
———. 1999. Exorcist XIV: The Wrath of Maxwell’s Demon. Part II. From Szilard to Landauer and Beyond.” Studies in History and Philosophy of Modern Physics 30 (1): 1–40.
Eichler, Michael. 2001. Granger-Causality Graphs for Multivariate Time Series.” Granger-Causality Graphs for Multivariate Time Series.
Elidan, Gal, and Nir Friedman. 2005. “Learning Hidden Variable Networks: The Information Bottleneck Approach.” Journal of Machine Learning Research 6: 81–127.
Ellerman, David. 2017. Logical Information Theory: New Foundations for Information Theory.” arXiv:1707.04728 [Quant-Ph], May.
Feldman, David P. 2002. A Brief Introduction to: Information Theory, Excess Entropy and Computational Mechanics.”
Feldman, David P, and James P Crutchfield. 2004. Synchronizing to Periodicity: The Transient Information and Synchronization Time of Periodic Sequences.” Advances in Complex Systems 7 (03): 329–55.
Frazier, Peter I, and Warren B Powell. 2010. Paradoxes in Learning and the Marginal Value of Information.” Decision Analysis 7 (4): 378–403.
Friedman, Nir, Ori Mosenzon, Noam Slonim, and Naftali Tishby. 2001. Multivariate Information Bottleneck.” In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, 152–61. UAI’01. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.
Friston, Karl. 2010. The Free-Energy Principle: A Unified Brain Theory? Nature Reviews Neuroscience 11 (2): 127.
Gács, Péter, JohnT. Tromp, and Paul M.B. Vitányi. 2001. Algorithmic Statistics.” IEEE Transactions on Information Theory 47 (6): 2443–63.
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.
Goldfeld, Ziv, and Yury Polyanskiy. 2020. The Information Bottleneck Problem and Its Applications in Machine Learning.” arXiv:2004.14941 [Cs, Stat], May.
Granger, Clive W J. 1963. Economic Processes Involving Feedback.” Information and Control 6 (1): 28–48.
Grassberger, Peter. 1988. Finite Sample Corrections to Entropy and Dimension Estimates.” Physics Letters A 128 (6–7): 369–73.
Gray, Robert M. 1991. Entropy and Information Theory. New York: Springer-Verlag.
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.
Haussler, David, and Manfred Opper. 1997. Mutual Information, Metric Entropy and Cumulative Relative Entropy Risk.” The Annals of Statistics 25 (6): 2451–92.
Hirata, Hironori, and Robert E Ulanowicz. 1985. Information Theoretical Analysis of the Aggregation and Hierarchical Structure of Ecological Networks.” Journal of Theoretical Biology 116 (3): 321–41.
Ho, Yu-Chi, M. Kastner, and E. Wong. 1978. Teams, Signaling, and Information Theory.” IEEE Transactions on Automatic Control 23 (2): 305–12.
Jaynes, Edwin Thompson. 1963. “Information Theory and Statistical Mechanics.” In Statistical Physics. Vol. 3. Brandeis University Summer Institute Lectures in Theoretical Physics.
———. 1965. Gibbs Vs Boltzmann Entropies.” American Journal of Physics 33: 391–98.
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.” arXiv:1411.4342 [Stat], November.
Keane, M. S., and George L. O’Brien. 1994. A Bernoulli Factory.” ACM Trans. Model. Comput. Simul. 4 (2): 213–19.
Kelly Jr, J L. 1956. A New Interpretation of Information Rate.” Bell System Technical Journal 35 (3): 917–26.
Klir, George J. 1998. Uncertainty and Information: Foundations of Generalized Information Theory. Wiley-IEEE Press.
Kolmogorov, A N. 1968. “Three Approaches to the Quantitative Definition of Information.” International Journal of Computer Mathematics 2 (1): 157–68.
Kraskov, Alexander, Harald Stögbauer, and Peter Grassberger. 2004. Estimating Mutual Information.” Physical Review E 69: 066138.
Krause, Andreas, and Carlos Guestrin. 2009. “Optimal Value of Information in Graphical Models.” J. Artif. Int. Res. 35 (1): 557–91.
Kullback, S, and R A Leibler. 1951. On Information and Sufficiency.” The Annals of Mathematical Statistics 22 (1): 79–86.
Langton, Chris G. 1990. Computation at the Edge of Chaos: Phase Transitions and Emergent Computation.” Physica D: Nonlinear Phenomena 42 (1–3): 12–37.
Lecomte, V., C. Appert-Rolland, and F. van Wijland. 2007. Thermodynamic Formalism for Systems with Markov Dynamics.” Journal of Statistical Physics 127 (1): 51–106.
Leonenko, Nikolai. 2008. A Class of Rényi Information Estimators for Multidimensional Densities.” The Annals of Statistics 36 (5): 2153–82.
Leskovec, Jure. 2012. Information Diffusion and External Influence in Networks.” Eprint arXiv:1206.1331, June.
Liese, F, and I Vajda. 2006. On Divergences and Informations in Statistics and Information Theory.” IEEE Transactions on Information Theory 52 (10): 4394–4412.
Lin, Jianhua. 1991. Divergence Measures Based on the Shannon Entropy.” IEEE Transactions on Information Theory 37 (1): 145–51.
Lizier, Joseph T, and Mikhail Prokopenko. 2010. Differentiating Information Transfer and Causal Effect.” The European Physical Journal B - Condensed Matter and Complex Systems 73 (4): 605–15.
Lizier, Joseph T, Mikhail Prokopenko, and Albert Y Zomaya. 2008. Local Information Transfer as a Spatiotemporal Filter for Complex Systems.” Physical Review E 77: 026110.
Marton, Katalin. 2015. Logarithmic Sobolev Inequalities in Discrete Product Spaces: A Proof by a Transportation Cost Distance.” arXiv:1507.02803 [Math], July.
Maynard Smith, John. 2000. The Concept of Information in Biology.” Philosophy of Science 67 (2): 177–94.
Moon, Kevin R., and Alfred O. Hero III. 2014. Multivariate f-Divergence Estimation With Confidence.” In NIPS 2014.
Nemenman, Ilya, Fariel Shafee, and William Bialek. 2001. Entropy and Inference, Revisited.” In arXiv:physics/0108025.
Nielsen, Frank. 2021. On a Variational Definition for the Jensen-Shannon Symmetrization of Distances Based on the Information Radius.” Entropy 23 (4): 464.
Odum, Howard T. 1988. “Self-Organization, Transformity, and Information.” Science, 1988.
Palomar, D.P., and S. Verdu. 2008. Lautum Information.” IEEE Transactions on Information Theory 54 (3): 964–75.
Paninski, Liam. 2003. Estimation of Entropy and Mutual Information.” Neural Computation 15 (6): 1191–1253.
Parry, William. 1964. Intrinsic Markov Chains.” Transactions of the American Mathematical Society 112 (1): 55–66.
Phat, Vu N., Nguyen T. Thanh, and Hieu Trinh. 2014. Full-Order Observer Design for Nonlinear Complex Large-Scale Systems with Unknown Time-Varying Delayed Interactions.” Complexity, August, n/a–.
Pinkerton, Richard C. 1956. Information Theory and Melody.” Scientific American 194 (2): 77–86.
Plotkin, Joshua B, and Martin A Nowak. 2000. Language Evolution and Information Theory.” Journal of Theoretical Biology 205: 147–59.
Raginsky, M. 2011. Directed Information and Pearl’s Causal Calculus.” In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 958–65.
Raginsky, Maxim, and Igal Sason. 2012. Concentration of Measure Inequalities in Information Theory, Communications and Coding.” Foundations and Trends in Communications and Information Theory, December.
Riegler, Erwin, Helmut Bölcskei, and Günther Koliander. 2023. Lossy Compression of General Random Variables.” arXiv.
Rissanen, Jorma. 2007. Information and Complexity in Statistical Modeling. Information Science and Statistics. New York: Springer.
Roulston, Mark S. 1999. Estimating the Errors on Measured Entropy and Mutual Information.” Physica D: Nonlinear Phenomena 125 (3-4): 285–94.
Ryabko, Daniil, and Boris Ryabko. 2010. Nonparametric Statistical Inference for Ergodic Processes.” IEEE Transactions on Information Theory 56 (3): 1430–35.
Schreiber, Thomas. 2000. “Measuring Information Transfer.” Physical Review Letters 85 (2): 461–64.
Schürmann, Thomas. 2015. A Note on Entropy Estimation.” Neural Computation 27 (10): 2097–2106.
Sethna, James P. 2006. Statistical Mechanics: Entropy, Order Parameters, and Complexity. Oxford University Press, USA.
Shalizi, Cosma Rohilla. n.d.a. “Information Theory.”
———. n.d.b. “Optimal Prediction.”
Shalizi, Cosma Rohilla, and James P. Crutchfield. 2002. Information Bottlenecks, Causal States, and Statistical Relevance Bases: How to Represent Relevant Information in Memoryless Transduction.” Advances in Complex Systems 05 (01): 91–95.
Shalizi, Cosma Rohilla, Robert Haslinger, Jean-Baptiste Rouquier, Kristina L Klinkner, and Cristopher Moore. 2006. Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems.” Physical Review E 73 (3).
Shalizi, Cosma Rohilla, and Cristopher Moore. 2003. What Is a Macrostate? Subjective Observations and Objective Dynamics.” arXiv.
Shannon, Claude E. 1948. “A Mathematical Theory of Communication.” The Bell Syst Tech J 27: 379–423.
Shibata, Ritei. 1997. “Bootstrap Estimate of Kullback-Leibler Information for Model Selection.” Statistica Sinica 7: 375–94.
Shields, P C. 1998. The Interactions Between Ergodic Theory and Information Theory.” IEEE Transactions on Information Theory 44 (6): 2079–93.
Singh, Nirvikar. 1985. Monitoring and Hierarchies: The Marginal Value of Information in a Principal-Agent Model.” Journal of Political Economy 93 (3): 599–609.
Slonim, Noam, Gurinder S Atwal, Gašper Tkačik, and William Bialek. 2005. Information-Based Clustering.” Proceedings of the National Academy of Sciences of the United States of America 102: 18297–302.
Slonim, Noam, Nir Friedman, and Naftali Tishby. 2006. Multivariate Information Bottleneck.” Neural Computation 18 (8): 1739–89.
Smith, D Eric. 2008a. Thermodynamics of Natural Selection I: Energy Flow and the Limits on Organization.” Journal of Theoretical Biology 252: 185–97.
———. 2008b. Thermodynamics of Natural Selection II: Chemical Carnot Cycles.” Journal of Theoretical Biology 252: 198–212.
Spence, Michael. 2002. Signaling in Retrospect and the Informational Structure of Markets.” American Economic Review 92: 434–59.
Steeg, Greg ver, and Aram Galstyan. 2015. The Information Sieve.” arXiv:1507.02284 [Cs, Math, Stat], July.
Strong, Steven P, Roland Koberle, Rob R de Ruyter van Steveninck, and William Bialek. 1998. Entropy and Information in Neural Spike Trains.” Phys. Rev. Lett. 80 (1): 197–200.
Studený, Milan. 2016. Basic Facts Concerning Supermodular Functions.” arXiv:1612.06599 [Math, Stat], December.
Studený, Milan, and Jiřina Vejnarová. 1998. “On Multiinformation Function as a Tool for Measuring Stochastic Dependence.” In Learning in Graphical Models, 261–97. Cambridge, Mass.: MIT Press.
Taylor, Samuel F, Naftali Tishby, and William Bialek. 2007. “Information and Fitness.” Arxiv Preprint arXiv:0712.4382.
Tishby, Naftali, Fernando C Pereira, and William Bialek. 2000. The Information Bottleneck Method.” arXiv:physics/0004057, April.
Tishby, Naftali, and Daniel Polani. 2011. “Information Theory of Decisions and Actions.” In PERCEPTION-ACTION CYCLE, 601–36. Springer.
Tumer, Kagan, and David H Wolpert. 2004. “Coordination in Large Collectives- Chapter 1.” In.
Vereshchagin, N.K., and Paul MB Vitányi. 2004. Kolmogorov’s Structure Functions and Model Selection.” IEEE Transactions on Information Theory 50 (12): 3265–90.
———. 2010. Rate Distortion and Denoising of Individual Data Using Kolmogorov Complexity.” IEEE Transactions on Information Theory 56 (7): 3438–54.
Vitányi, Paul M. 2006. Meaningful Information.” IEEE Transactions on Information Theory 52 (10): 4617–26.
Weidmann, Claudio, and Martin Vetterli. 2012. Rate Distortion Behavior of Sparse Sources.” IEEE Transactions on Information Theory 58 (8): 4969–92.
Weissman, T., Y. H. Kim, and H. H. Permuter. 2013. Directed Information, Causal Estimation, and Communication in Continuous Time.” IEEE Transactions on Information Theory 59 (3): 1271–87.
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.” arXiv:comp-Gas/9403002, March.
Wolpert, David H. 2006a. Information Theory — The Bridge Connecting Bounded Rational Game Theory and Statistical Physics.” In Complex Engineered Systems, 262–90. Understanding Complex Systems. Springer Berlin Heidelberg.
———. 2006b. What Information Theory Says about Bounded Rational Best Response.” In The Complex Networks of Economic Interactions, 293–306. Lecture Notes in Economics and Mathematical Systems 567. Springer.
Wolpert, David H, and John W Lawson. 2002. Designing Agent Collectives for Systems with Markovian Dynamics.” In, 1066–73.
Wolpert, David H, Kevin R Wheeler, and Kagan Tumer. 1999. General Principles of Learning-Based Multi-Agent Systems.” In, 77–83.
———. 2000. Collective Intelligence for Control of Distributed Dynamical Systems.” EPL (Europhysics Letters) 49: 708.
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.
Xu, Aolin, and Maxim Raginsky. 2017. Information-Theoretic Analysis of Generalization Capability of Learning Algorithms.” In Advances In Neural Information Processing Systems.
Zellner, Arnold. 1988. Optimal Information Processing and Bayes’s Theorem.” The American Statistician 42 (4): 278–80.
Zhang, Zhiyi, and Michael Grabchak. 2014. Nonparametric Estimation of Küllback-Leibler Divergence.” Neural Computation 26 (11): 2570–93.

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