Not: what you hope to get from the newspaper. 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. If you wish to estimate such a quantity empirically, from your experiment, that’s adifferent problem.

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

## Shannon Information

Vanilla information, thanks be to Claude Shannon. You have are given a discrete random process of specified parameters. How much can you compress it down to a more parsimonious process? (leaving coding theory aside for the moment.)

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

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

Over at the Functional equations page I note that Tom Leinster has a 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, even if usually in sentences like “unlike the K-L divergences”. Or you could call it the “relative entropy”, but that sounds like something to do with my uncle after the seventh round of Christmas drinks.

It is defined between the probability mass functions of two discrete random variables, \(P,Q\), where those probability mass functions are given \(p(x)\( and\)q(x)) respectively.

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

## 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

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

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 emphasis different characteristics:

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

## 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?

### Alternative formulations and 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.

### To Read

John Baez’s A Characterisation of Entropy etc http://johncarlosbaez.wordpress.com/category/information-and-entropy/

Daniel Ellerman’s History of the Logical Entropy Formula and From Partition Logic to Information Theory, which he has now written up as Elle17.

## Refs

- KeOb94: M. S. Keane, George L. O’Brien (1994) A Bernoulli Factory.
*ACM Trans. Model. Comput. Simul.*, 4(2), 213–219. DOI - Leon08: Nikolai Leonenko (2008) A class of Rényi information estimators for multidimensional densities.
*The Annals of Statistics*, 36(5), 2153–2182. DOI - Shan48: Claude E Shannon (1948) A mathematical theory of communication.
*The Bell Syst Tech J*, 27, 379–423. - Kell56: J L Kelly Jr (1956) A new interpretation of information rate.
*Bell System Technical Journal*, 35(3), 917–926. - Schü15: Thomas Schürmann (2015) A Note on Entropy Estimation.
*Neural Computation*, 27(10), 2097–2106. DOI - BiWo04: Stefan Bieniawski, 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 (Vol. 4, pp. 1230–1231). IEEE Computer Society
- SlTi00: Noam Slonim, Naftali Tishby (2000) Agglomerative information bottleneck.
*Advances in Neural Information Processing Systems*, 12, 617–623. - Chai77: Gregory J Chaitin (1977) Algorithmic information theory.
*IBM Journal of Research and Development*. - GáTV01: Péter Gács, JohnT. Tromp, Paul M.B. Vitányi (2001) Algorithmic statistics.
*IEEE Transactions on Information Theory*, 47(6), 2443–2463. DOI - ChLi68: C K Chow, C N Liu (1968) Approximating discrete probability distributions with dependence trees.
*IEEE Transactions on Information Theory*, 14, 462–467. DOI - SHRK06: Cosma Rohilla Shalizi, Robert Haslinger, Jean-Baptiste Rouquier, Kristina L Klinkner, Cristopher Moore (2006) Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems.
*Physical Review E*, 73(3). DOI - Stud16: Milan Studený (2016) Basic facts concerning supermodular functions.
*ArXiv:1612.06599 [Math, Stat]*. - Shib97: Ritei Shibata (1997) Bootstrap estimate of Kullback-Leibler information for model selection.
*Statistica Sinica*, 7, 375–394. - WoWT00: David H Wolpert, Kevin R Wheeler, Kagan Tumer (2000) Collective intelligence for control of distributed dynamical systems.
*EPL (Europhysics Letters)*, 49, 708. DOI - BiNT01: William Bialek, Ilya Nemenman, Naftali Tishby (2001) Complexity through nonextensivity.
*Physica A: Statistical and Theoretical Physics*, 302(1–4), 89–99. DOI - Lang90: Chris G. Langton (1990) Computation at the edge of chaos: Phase transitions and emergent computation.
*Physica D: Nonlinear Phenomena*, 42(1–3), 12–37. DOI - RaSa12: Maxim Raginsky, Igal Sason (2012) Concentration of Measure Inequalities in Information Theory, Communications and Coding.
*Foundations and Trends in Communications and Information Theory*. - TuWo04: Kagan Tumer, David H Wolpert (2004) Coordination in Large Collectives- Chapter 1.
- WoLa02: David H Wolpert, John W Lawson (2002) Designing agent collectives for systems with Markovian dynamics. (pp. 1066–1073). DOI
- LiPr10: Joseph T Lizier, Mikhail Prokopenko (2010) Differentiating information transfer and causal effect.
*The European Physical Journal B - Condensed Matter and Complex Systems*, 73(4), 605–615. DOI - Ragi11: M. Raginsky (2011) Directed information and Pearl’s causal calculus. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton) (pp. 958–965). DOI
- WeKP13: T. Weissman, Y. H. Kim, H. H. Permuter (2013) Directed Information, Causal Estimation, and Communication in Continuous Time.
*IEEE Transactions on Information Theory*, 59(3), 1271–1287. DOI - Lin91: Jianhua Lin (1991) Divergence measures based on the Shannon entropy.
*IEEE Transactions on Information Theory*, 37(1), 145–151. DOI - Gran63: Clive W J Granger (1963) Economic processes involving feedback.
*Information and Control*, 6(1), 28–48. DOI - GaVG15: Shuyang Gao, Greg Ver Steeg, Aram Galstyan (2015) Efficient Estimation of Mutual Information for Strongly Dependent Variables. In Journal of Machine Learning Research (pp. 277–286).
- CoTh06: Thomas M Cover, Joy A Thomas (2006)
*Elements of Information Theory*. Wiley-Interscience - NeSB01: Ilya Nemenman, Fariel Shafee, William Bialek (2001) Entropy and inference, revisited. In arXiv:physics/0108025.
- SKRB98: Steven P Strong, Roland Koberle, Rob R de Ruyter van Steveninck, William Bialek (1998) Entropy and Information in Neural Spike Trains.
*Phys. Rev. Lett.*, 80(1), 197–200. DOI - Gray91: Robert M Gray (1991)
*Entropy and Information Theory*. New York: Springer-Verlag - HaSt09: Jean Hausser, Korbinian Strimmer (2009) Entropy Inference and the James-Stein Estimator, with Application to Nonlinear Gene Association Networks.
*Journal of Machine Learning Research*, 10, 1469. - WoWo94: David R. Wolf, 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*. - WoWo94: David H. Wolpert, 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*. - KrSG04: Alexander Kraskov, Harald Stögbauer, Peter Grassberger (2004) Estimating mutual information.
*Physical Review E*, 69, 066138. DOI - Roul99: Mark S Roulston (1999) Estimating the errors on measured entropy and mutual information.
*Physica D: Nonlinear Phenomena*, 125(3–4), 285–294. DOI - Pani03: Liam Paninski (2003) Estimation of entropy and mutual information.
*Neural Computation*, 15(6), 1191–1253. DOI - EaNo98: John Earman, 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), 435–471. DOI - EaNo99: John Earman, John D Norton (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. DOI - Gras88: Peter Grassberger (1988) Finite sample corrections to entropy and dimension estimates.
*Physics Letters A*, 128(6–7), 369–373. DOI - PhTT14: Vu N. Phat, Nguyen T. Thanh, Hieu Trinh (2014) Full-Order observer design for nonlinear complex large-scale systems with unknown time-varying delayed interactions.
*Complexity*, n/a-n/a. DOI - WoWT99: David H Wolpert, Kevin R Wheeler, Kagan Tumer (1999) General principles of learning-based multi-agent systems. (pp. 77–83). DOI
- Jayn65: Edwin Thompson Jaynes (1965) Gibbs vs Boltzmann Entropies.
*American Journal of Physics*, 33, 391–398. DOI - BaBS09: Lionel Barnett, Adam B. Barrett, Anil K. Seth (2009) Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables.
*Physical Review Letters*, 103(23), 238701. DOI - Eich01: Michael Eichler (2001) Granger-causality graphs for multivariate time series.
*Granger-Causality Graphs for Multivariate Time Series*. - CaMR05: O Cappé, E Moulines, T Ryden (2005)
*Inference in hidden Markov models*. Springer Verlag - KKPW14: Kirthevasan Kandasamy, Akshay Krishnamurthy, Barnabas Poczos, Larry Wasserman, James M. Robins (2014) Influence Functions for Machine Learning: Nonparametric Estimators for Entropies, Divergences and Mutual Informations.
*ArXiv:1411.4342 [Stat]*. - Riss07: Jorma Rissanen (2007)
*Information and complexity in statistical modeling*. New York: Springer - TaTB07: Samuel F Taylor, Naftali Tishby, William Bialek (2007) Information and fitness.
*Arxiv Preprint ArXiv:0712.4382*. - Calu02: Cristian S Calude (2002)
*Information and Randomness : An Algorithmic Perspective*. Springer - ShCr02: Cosma Rohilla Shalizi, 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. DOI - Lesk12: Jure Leskovec (2012) Information Diffusion and External Influence in Networks.
*Eprint ArXiv:1206.1331*. - Amar01: Shunʼichi Amari (2001) Information geometry on hierarchy of probability distributions.
*IEEE Transactions on Information Theory*, 47, 1701–1711. DOI - HiUl85: Hironori Hirata, Robert E Ulanowicz (1985) Information theoretical analysis of the aggregation and hierarchical structure of ecological networks.
*Journal of Theoretical Biology*, 116(3), 321–341. DOI - Shal00a: Cosma Rohilla Shalizi (n.d.-a)
*Information Theory* - Akai73: Hirotogu Akaike (1973) Information Theory and an Extension of the Maximum Likelihood Principle. In Proceeding of the Second International Symposium on Information Theory (pp. 199–213). Budapest: Akademiai Kiado
- Pink56: Richard C. Pinkerton (1956) Information theory and melody.
*Scientific American*, 194(2), 77–86. DOI - Cohe62: Joel E. Cohen (1962) Information theory and music.
*Behavioral Science*, 7(2), 137–163. DOI - Jayn63: Edwin Thompson Jaynes (1963) Information Theory and Statistical Mechanics. In Statistical Physics (Vol. 3).
- CsSh04: Imre Csiszár, Paul C Shields (2004) Information theory and statistics: a tutorial.
*Foundations and Trends™ in Communications and Information Theory*, 1(4), 417–528. DOI - Dewa03: Roderick C Dewar (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, 631–641. DOI - TiPo11: Naftali Tishby, Daniel Polani (2011) Information theory of decisions and actions. In PERCEPTION-ACTION CYCLE (pp. 601–636). Springer
- Wolp06a: David H Wolpert (2006a) Information Theory–The Bridge Connecting Bounded Rational Game Theory and Statistical Physics. In Complex Engineered Systems (pp. 262–290). Springer Berlin Heidelberg
- ChDP11: G Chiribella, G M D’Ariano, P Perinotti (2011) Informational derivation of Quantum Theory.
*Physical Review A*, 84(1), 012311. DOI - SATB05: Noam Slonim, Gurinder S Atwal, Gašper Tkačik, William Bialek (2005) Information-based clustering.
*Proceedings of the National Academy of Sciences of the United States of America*, 102, 18297–18302. DOI - XuRa17: Aolin Xu, Maxim Raginsky (2017) Information-theoretic analysis of generalization capability of learning algorithms. In Advances In Neural Information Processing Systems.
- Parr64: William Parry (1964) Intrinsic Markov chains.
*Transactions of the American Mathematical Society*, 112(1), 55–66. DOI - CoGG89: Thomas M. Cover, Péter Gács, Robert M. Gray (1989) Kolmogorov’s Contributions to Information Theory and Algorithmic Complexity.
*The Annals of Probability*, 17(3), 840–865. DOI - VeVi04: N.K. Vereshchagin, Paul MB Vitányi (2004) Kolmogorov’s structure functions and model selection.
*IEEE Transactions on Information Theory*, 50(12), 3265–3290. DOI - PlNo00: Joshua B Plotkin, Martin A Nowak (2000) Language Evolution and Information Theory.
*Journal of Theoretical Biology*, 205, 147–159. DOI - PaVe08: D.P. Palomar, S. Verdu (2008) Lautum Information.
*IEEE Transactions on Information Theory*, 54(3), 964–975. DOI - ElFr05: Gal Elidan, Nir Friedman (2005) Learning Hidden Variable Networks: The Information Bottleneck Approach.
*Journal of Machine Learning Research*, 6, 81–127. - LiPZ08: Joseph T Lizier, Mikhail Prokopenko, Albert Y Zomaya (2008) Local information transfer as a spatiotemporal filter for complex systems.
*Physical Review E*, 77, 026110. DOI - Mart15: Katalin Marton (2015) Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance.
*ArXiv:1507.02803 [Math]*. - Elle17: David Ellerman (2017, May 22) Logical Information Theory: New Foundations for Information Theory.
- Vitá06: Paul M Vitányi (2006) Meaningful information.
*IEEE Transactions on Information Theory*, 52(10), 4617–4626. DOI - Schr00: Thomas Schreiber (2000) Measuring information transfer.
*Physical Review Letters*, 85(2), 461–464. - Croo07: Gavin E Crooks (2007) Measuring Thermodynamic Length.
*Physical Review Letters*, 99(10), 100602. DOI - Sing85: Nirvikar Singh (1985) Monitoring and Hierarchies: The Marginal Value of Information in a Principal-Agent Model.
*Journal of Political Economy*, 93(3), 599–609. - MoHe14: Kevin R. Moon, Alfred O. Hero III (2014) Multivariate f-Divergence Estimation With Confidence. In NIPS 2014.
- BaBS10: Adam B Barrett, Lionel Barnett, Anil K Seth (2010) Multivariate Granger causality and generalized variance.
*Phys. Rev. E*, 81(4), 041907. DOI - FMST01: Nir Friedman, Ori Mosenzon, Noam Slonim, Naftali Tishby (2001) Multivariate information bottleneck. In Proceedings of the Seventeenth conference on Uncertainty in artificial intelligence (pp. 152–161). San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.
- SlFT06: Noam Slonim, Nir Friedman, Naftali Tishby (2006) Multivariate information bottleneck.
*Neural Computation*, 18(8), 1739–1789. DOI - HaOp97: David Haussler, Manfred Opper (1997) Mutual information, metric entropy and cumulative relative entropy risk.
*The Annals of Statistics*, 25(6), 2451–2492. DOI - ZhGr14: Zhiyi Zhang, Michael Grabchak (2014) Nonparametric Estimation of Küllback-Leibler Divergence.
*Neural Computation*, 26(11), 2570–2593. DOI - RyRy10: Daniil Ryabko, Boris Ryabko (2010) Nonparametric Statistical Inference for Ergodic Processes.
*IEEE Transactions on Information Theory*, 56(3), 1430–1435. DOI - LiVa06: F Liese, I Vajda (2006) On Divergences and Informations in Statistics and Information Theory.
*IEEE Transactions on Information Theory*, 52(10), 4394–4412. DOI - KuLe51: S Kullback, R A Leibler (1951) On Information and Sufficiency.
*The Annals of Mathematical Statistics*, 22(1), 79–86. - StVe98: Milan Studený, Jiřina Vejnarová (1998) On multiinformation function as a tool for measuring stochastic dependence. In Learning in graphical models (pp. 261–297). Cambridge, Mass.: MIT Press
- Dahl96: R Dahlhaus (1996) On the Kullback-Leibler information divergence of locally stationary processes.
*Stochastic Processes and Their Applications*, 62(1), 139–168. DOI - Shal00b: Cosma Rohilla Shalizi (n.d.-b)
*Optimal Prediction* - KrGu09: Andreas Krause, Carlos Guestrin (2009) Optimal value of information in graphical models.
*J. Artif. Int. Res.*, 35(1), 557–591. - FrPo10: Peter I Frazier, Warren B Powell (2010) Paradoxes in Learning and the Marginal Value of Information.
*Decision Analysis*, 7(4), 378–403. DOI - BiNT06: William Bialek, Ilya Nemenman, Naftali Tishby (2006) Predictability, Complexity, and Learning.
*Neural Computation*, 13(11), 2409–2463. DOI - ABDG08: N. Ay, N. Bertschinger, R. Der, F. Güttler, E. Olbrich (2008) Predictive information and explorative behavior of autonomous robots.
*The European Physical Journal B - Condensed Matter and Complex Systems*, 63(3), 329–339. DOI - VeVi10: N.K. Vereshchagin, Paul MB Vitányi (2010) Rate Distortion and Denoising of Individual Data Using Kolmogorov Complexity.
*IEEE Transactions on Information Theory*, 56(7), 3438–3454. DOI - WeVe12: Claudio Weidmann, Martin Vetterli (2012) Rate Distortion Behavior of Sparse Sources.
*IEEE Transactions on Information Theory*, 58(8), 4969–4992. DOI - Odum88: Howard T Odum (1988) Self-Organization, Transformity, and Information.
*Science*, 242(4882), 1132–1139. - Spen02: Michael Spence (2002) Signaling in Retrospect and the Informational Structure of Markets.
*American Economic Review*, 92, 434–459. DOI - Seth06: James P Sethna (2006)
*Statistical mechanics: entropy, order parameters, and complexity*. Oxford University Press, USA - FeCr04: David P Feldman, James P Crutchfield (2004) Synchronizing to Periodicity: the Transient Information and Synchronization Time of Periodic Sequences.
*Advances in Complex Systems*, 7(03), 329–355. DOI - Mayn00: John Maynard Smith (2000) The Concept of Information in Biology.
*Philosophy of Science*, 67(2), 177–194. - Fris10: Karl Friston (2010) The free-energy principle: a unified brain theory?
*Nature Reviews Neuroscience*, 11(2), 127. DOI - TiPB00: Naftali Tishby, Fernando C Pereira, William Bialek (2000) The information bottleneck method.
*ArXiv:Physics/0004057*. - StGa15: Greg Ver Steeg, Aram Galstyan (2015) The Information Sieve.
*ArXiv:1507.02284 [Cs, Math, Stat]*. - Chai02: Gregory J Chaitin (2002) The intelligibility of the universe and the notions of simplicity, complexity and irreducibility.
- Shie98: P C Shields (1998) The interactions between ergodic theory and information theory.
*IEEE Transactions on Information Theory*, 44(6), 2079–2093. DOI - LeAW07: V. Lecomte, C. Appert-Rolland, F. van Wijland (2007) Thermodynamic Formalism for Systems with Markov Dynamics.
*Journal of Statistical Physics*, 127(1), 51–106. DOI - Smit08a: D Eric Smith (2008a) Thermodynamics of natural selection I: Energy flow and the limits on organization.
*Journal of Theoretical Biology*, 252, 185–197. DOI - Smit08b: D Eric Smith (2008b) Thermodynamics of natural selection II: Chemical Carnot cycles.
*Journal of Theoretical Biology*, 252, 198–212. DOI - Smit08c: D Eric Smith (2008c) Thermodynamics of natural selection III: Landauer’s principle in computation and chemistry.
*Journal of Theoretical Biology*, 252(2), 213–220. DOI - Kolm68: A N Kolmogorov (1968) Three approaches to the quantitative definition of information.
*International Journal of Computer Mathematics*, 2(1), 157–168. - Klir06: George J Klir (2006)
*Uncertainty and information*. Wiley Online Library - Wolp06b: David H Wolpert (2006b) What Information Theory says about Bounded Rational Best Response. In The Complex Networks of Economic Interactions (pp. 293–306). Springer
- ShMo03: Cosma Rohilla Shalizi, Cristopher Moore (2003) What Is a Macrostate? Subjective Observations and Objective Dynamics.
*Eprint ArXiv:Cond-Mat/0303625*.