## Estimating Entropies and Informations

*11 Apr 2022 10:14*

The central mathematical objects in information theory are the entropies of random variables. These ("Shannon") entropies are properties of the probability distributions of the variables, rather than of particular realizations. (This is unlike the Boltzmann entropy of statistical mechanics, which is an objective property of the macroscopic state, at least once we fix our partition of microscopic states into macroscopic states. Confusing the two entropies is common but bad.) The question which concerns me here is how to estimate the entropy of the distribution, given a sample of realizations. The most obvious approach, if one knows the form of the distribution but not the parameters, is to estimate the parameters and then plug in. But it feels like one should be able to do this more non-parametrically. The obvious non-parametric estimator is of course just the entropy of the empirical distribution, sometimes called the "empirical entropy". However, the empirical distribution isn't always the best estimate of the true distribution (one might perfer, e.g., some kind of kernel density estimate). For that matter, we often don't really care about the distribution, just its entropy, so some more direct estimator would be nice.

What would be really nice would be to not just have point estimates but also
confidence intervals. Non-parametrically, my guess is that the only feasible
way to do this is bootstrapping. [**Update, 2022**: That guess was
wrong! Moon and Hero (2014) and Kandasamy et al. (2015) both give
non-parametric estimators with (asymptotic) confidence intervals that don't
require bootstrapping. I particularly like the Kandasamy et al. approach, but
I admit to a certain amount of home-team bias there.]

For finite alphabets, one approach would be to use something like variable length Markov chains, or causal state reconstruction, to reconstruct a get machine capable of generating the sequence. From the machine, it is easy to calculate the entropy of words or blocks of any finite length, and even the entropy rate. My experience with using CSSR is that the entropy rate estimates can get very good even when the over-all reconstruction of the structure is very poor, but I don't have any real theory on that. I suspect CSSR converges on the true entropy rate faster than do variable length Markov chains, because the former has greater expressive power, but again I don't know that for sure.

Using gzip is a bad idea (for this purpose; it works fine for data compression).

(Thanks to Sankaran Ramakrishnan for pointing out a think-o in an earlier version.)

See also:
Bootstrapping Entropy Estimates;
Complexity, Entropy and the Physics of `gzip`;
Directed Information and Transfer Entropy (where I stash references on estimating those quantities)

- Recommended (somewhat miscellaneous):
- D. J. Albers, George Hripcsak, "Estimation of time-delayed mutual information and bias for irregularly and sparsely sampled time-series", arxiv:1110.1615
- Jose M. Amigo, Janusz Szczepanski, Elek Wajnryb and Maria
V. Sanchez-Vives, "Estimating the Entropy Rate of Spike Trains via Lempel-Ziv
Complexity",
Neural
Computation
**16**(2004): 717--736 [Normally, I have strong views on using Lempel-Ziv to measure entropy rates, but here they are using the 1976 Lempel-Ziv definitions, not the 1978 ones. The difference is subtle, but important; 1978 leads to gzip and practical compression algorithms, but very bad entropy estimates; 1976 leads, as they show numerically, to reasonable entropy rate estimates, at least for some processes. Thanks to Dr. Szczepanski for correspondence about this paper.] - Thomas B. Berrett, Richard J. Samworth, Ming Yuan, "Efficient multivariate entropy estimation via k-nearest neighbour distances", Annals of Statistics (forthcoming), arxiv:1606.00304
- J.-R. Chazottes and D. Gabrielli, "Large deviations for empirical
entropies of Gibbsian sources", Nonlinearity
**18**(2005): 2545--2563 , math.PR/0406083 [This is a very cool result which shows that block entropies, and entropy rates estimated from those blocks, obey the large deviation principle even as one lets the length of the blocks grow with the amount of data, provided the block-length doesn't grow too quickly (only logarithmically). I wish I could write papers like this.] - Simon DeDeo, Robert X. D. Hawkins, Sara Klingenstein, Tim Hitchcock. "Bootstrap Methods for the Empirical Study of Decision-Making and Information Flows in Social Systems", arxiv:1302.0907
- John W. Fisher III, Alexander T. Ihler and Paula A. Viola, "Learning Informative Statistics: A Nonparametric Approach", pp. 900--906 in NIPS 12 (1999) [PDF reprint. I'd call this more of a semi-parametric approach than a fully non-parametric one; they assume a parametric form for the dependence structure, but are agnostic about the distributions of innovations, and so try to maximize non-parametrically estimated mutual informations.]
- Yongmiao Hong and Halbert White, "Asymptotic Distribution Theory
for Nonparametric Entropy Measures of Serial Dependence",
Econometrica
**73**(2005): 837--901 [JSTOR; PDF reprint via Prof. White] - Kirthevasan Kandasamy, Akshay Krishnamurthy, Barnabas Poczos, Larry Wasserman, James M. Robins, "Nonparametric von Mises Estimators for Entropies, Divergences and Mutual Informations", NIPS 2015, arxiv:1411.4342
- Matthew B. Kennel, Jonathon Shlens, Henry D. I. Abarbanel and
E. J. Chichilnisky, "Estimating Entropy Rates with Bayesian Confidence
Intervals", Neural
Computation
**17**(2005): 1531--1576 - Ioannis
Kontoyiannis, P. H. Algoet, Yu. M. Suhov and A. J. Wyner, "Nonparametric
Entropy Estimation for Stationary Processes and Random Fields, with
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**44**(1998): 1319--1327 - Alexander Kraskov, Harald Stögbauer and Peter Grassberger,
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**69**(2004): 066138, cond-mat/0305641 - Akshay Krishnamurthy, Kirthevasan Kandasamy, Barnabas Poczos, Larry Wasserman, "Nonparametric Estimation of Renyi Divergence and Friends", arxiv:1402.2966
- Kevin R. Moon, Alfred O. Hero III, "Multivariate f-Divergence Estimation With Confidence", arxiv:1411.2045
- Ilya Nemenman, "Inference of entropies of discrete random variables with unknown cardinalities," physics/0207009
- Dávid Pál, Barnabás Póczos, Csaba Szepesvári, "Estimation of Rényi Entropy and Mutual Information Based on Generalized Nearest-Neighbor Graphs", arxiv:1003.1954
- Liam Paninski, "Estimation of Entropy and Mutual Information",
Neural Computation
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symbol sequences," Chaos
**6**(1996): 414--427, cond-mat/0203436 - Jonathon Shlens, Matthew B. Kennel, Henry D. I. Abarbanel, E. J. Chichilnisky, "Estimating Information Rates with Confidence Intervals in Neural Spike Trains", Neural Computation
**19**(2007): 1683--1719 - Jonathan D. Victor, "Asymptotic Bias in Information Estimates and
the Exponential (Bell) Polynomials", Neural
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**12**(2000): 2797--2804 [Calculates the bias in the empirical entropy, as an estimator of the true entropy, under IID sampling of a discrete space. Interestingly, the first-order (1/n) term in the bias does not depend on the actual distribution, though the higher-order terms do.] - Vincent Q. Vu, Bin Yu, Robert E. Kass, "Information In The Non-Stationary Case", arxiv:0806.3978
- Benjamin Weiss, Single Orbit Dynamics [Discusses procedures for non-parametrically estimating entropy of suitably ergodic sources, using just one realization of the process.]

- Modesty forbids me to recommend:
- Octavio César Mesner and CRS, "Conditional Mutual Information Estimation for Mixed Discrete and Continuous Variables with Nearest Neighbors", IEEE Transactions on Information Theory forthcoming, arxiv:1912.03387

- To read:
- Evan Archer, Il Memming Park, Jonathan Pillow, "Bayesian Entropy Estimation for Countable Discrete Distributions", arxiv:1302.0328
- Andrew D. Back, Daniel Angus and Janet Wiles, "Determining the Number of Samples Required to Estimate Entropy in Natural Sequences", IEEE Transactions on Information Theory
**65**(2019): 4345--4352 - M. S. Baptista, E. J. Ngamga, Paulo R. F. Pinto, Margarida Brito, J. Kurths, "Kolmogorov-Sinai entropy from recurrence times", arxiv:0908.3401
- Andre C. Barato, David Hartich and Udo Seifert, "Rate of Mutual Information Between Coarse-Grained Non-Markovian Variables", Journal of Statistical Physics
**153**(2013): 460--478 - D. Benedetto, E. Caglioti, G. Cristadoro, M. Degli Esposti, "Relative entropy via non-sequential recursive pair substitutions", arxiv:1007.3384 [The first two authors were also the lead authors of the epic-fail "Language Trees and Zipping" paper, arxiv:cond-mat/0108530, but perhaps they've improved.]
- Visar Berisha, Alfred O. Hero, "Empirical non-parametric estimation of the Fisher Information",
IEEE Signal Processing
Letters
**22**(2015): 988--992, arxiv:1408.1182 - John A. Berkowitz and Tatyana O. Sharpee, "Quantifying Information Conveyed by Large Neuronal Populations",
Neural Computation
**31**(2019): 1015--1047 - Juan A. Bonachela, Haye Hinrichsen, Miguel A. Munoz, "Entropy estimates of small data sets", arxiv:0804.4561
- Salim Bouzebda and Issam Elhattab, "Uniform-in-bandwidth consistency for kernel-type estimators of Shannon's entropy", Electronic Journal of Statistics
**5**(2011): 440--459 - Clive G. Bowsher, Margaritis Voliotis, "Mutual Information and Conditional Mean Prediction Error", arxiv:1407.7165
- H. Cai, S. R. Kulkarni and S. Verdu, "Universal Entropy Estimation
via Block Sorting", IEEE Transactions on Information Theory
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**71**(2005): 066208 - Ishanu Chattopadhyay, Hod Lipson, "Computing Entropy Rate Of Symbol Sources & A Distribution-free Limit Theorem", arxiv:1401.0711
- J.-R. Chazottes and E. Uglade, "Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources", math.DS/0401093
- Gabriela Ciuperca and Valerie Girardin, "Estimation of the
Entropy Rate of a Countable Markov Chain", Communications in Statistics: Theory and Methods
**36**(2007): 2543--2557 - G. Ciuperca, V. Girardin and L. Lhote, "Computation and Estimation of Generalized Entropy Rates for Denumerable Markov Chains", IEEE Transactions on Information Theory
**57**(2011): 4026--4034 [The estimation is just plugging in the MLE of the parameters, for finitely-parametrized chains, but they claim to show that works well] - J.-R. Chazottes, C. Maldonado, "Concentration bounds for entropy estimation of one-dimensional Gibbs measures", arxiv:1102.1816
- Tommy W. S. Chow and D. Huang, "Estimating Optimal Feature Subsets
Using Efficient Estimation of High-Dimensional Mutual Information", IEEE
Transactions on Neural Networks
**16**(2005): 213--224 - Peter Clifford and Ioana Ada Cosma, "A simple sketching algorithm for entropy estimation", AISTATS 2013 196--206, arxiv:0908.3961
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- J. M. Finn, J. D. Goettee, Z. Toroczkai, M. Anghel and B. P. Wood,
"Estimation of entropies and dimensions by nonlinear symbolic time series
analysis", Chaos
**13**(2003): 444--456 - Shuyang Gao, Greg Ver Steeg and Aram Galstyan
- "Estimating Mutual Information by Local Gaussian Approximation", UAI 2015
- "Efficient Estimation of Mutual Information for Strongly Dependent Variables", arxiv:1411.2003

- Yun Gao, Ioannis Kontoyiannis, Elie Bienenstock
- "From the entropy to the statistical structure of spike trains", arxiv:0710.4117
- "Estimating the entropy of binary time series: Methodology, some theory and a simulation study", arxiv:0802.4363

- M. N. Goria, N. N. Leonenko, V. V. Mergel and Pl L. Novi
Inverardi, "A new class of random vector entropy estimators and its applications in testing statistical hypotheses", Journal of Nonparametric Statistics
**17**(2005): 277--297 - Peter Grassberger, "Data Compression and Entropy Estimates by Non-sequential Recursive Pair Substitution," physics/0207023 [On Jimenez-Montano, Ebeling and Poeschel]
- Jean Hausser and Korbinian Strimmer, "Entropy Inference and the James-Stein Estimator, with Application to Nonlinear Gene Association Networks",
Journal of Machine Learning Research
**10**(2009): 1469--1484 - Detlef Holstein and Holger Kantz, "Optimal Markov approximations and generalized embeddings", Physical Review E
**79**(2009): 056202 - Wentao Huang and Kechen Zhang, "Information-Theoretic Bounds and Approximations in Neural Population Coding", Neural Computation
**30**(2018): 885--944 - Marcus Hutter, Marco Zaffalon, "Distribution of Mutual Information from Complete and Incomplete Data", Computational Statistics and
Data Analysis
**48**(2005): 633--657, arxiv:cs/0403025 - Siddharth Jain, Rakesh Kumar Bansal, "On Large Deviation Property of Recurrence Times", ISIT 2013, arxiv:1303.1093
- Jiantao Jiao, Kartik Venkat, Tsachy Weissman
- "Order-Optimal Estimation of Functionals of Discrete Distributions", arxiv:1406.6956
- "Maximum Likelihood Estimation of Functionals of Discrete Distributions", arxiv:1406.6959

- Miguel Angel Jimenez-Montano, Werner Ebeling, and Thorsten Poeschel, "SYNTAX: A computer program to compress a sequence and to estimate its information content," cond-mat/0204134
- David Källberg, Nikolaj Leonenko, Oleg Seleznjev, "Statistical Inference for Rényi Entropy Functionals", arxiv:1103.4977
- Alexei Kaltchenko, "Algorithms for Estimating Information Distance with Applications to Bioinformatics and Linguistics", cs.CC/0404039
- Shiraj Khan, Sharba Bandyopadhyay, Auroop R. Ganguly, Sunil Saigal,
David J. Erickson, III, Vladimir Protopopescu, and George Ostrouchov, "Relative
performance of mutual information estimation methods for quantifying the
dependence among short and noisy data",
Physical Review
E
**76**(2007): 026209 - John M. Konstantinides and Ioannis Andreadis, "Optimal Code Length Estimates From Dependent Samples With Bounds on the Estimation Error", IEEE communications Letters
**20**(2016): 2358--2361 - Nikolai Leonenko, Luc Pronzato and Vippal Savani, "A class
of Rényi information estimators for multidimensional densities",
Annals of Statistics
**36**(2008): 2153--2182, arxiv:0810.5302 - Annick Lesne, Jean-Luc Blanc and Laurent Pezard, "Entropy
estimation of very short symbolic sequences", Physical Review E
**79**(2009): 046208 - Christophe Letellier, "Estimating the Shannon Entropy: Recurrence
Plots versus Symbolic Dynamics", Physical Review
Letters
**96**(2006): 254102 - Johan Lim, "Estimation of the Entropy Functional from Dependent
Samples", Communications in
Statistics: Theory and Methods
**36**(2007): 1577--1589 - Tiger W. Lin and George N. Reeke, "A Continuous Entropy Rate
Estimator for Spike Trains Using a K-Means-Based Context
Tree", Neural
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**22**(2010): 998--1024 - Han Liu, Larry Wasserman and John D. Lafferty, "Exponential Concentration for Mutual Information Estimation with Application to Forests", NIPS 2012
- Jay Mardia, Jiantao Jiao, Ervin Tánczos, Robert D. Nowak, Tsachy Weissman, "Concentration Inequalities for the Empirical Distribution", arxiv:1809.06522
- Ilya Nemenman, William Bialek and Rob de Ruyter van Steveninck, "Entropy and information in neural spike trains: Progress on the sampling problem", 0306063
- XuanLong Nguyen, Martin J. Wainwright, Michael I. Jordan, "Estimating divergence functionals and the likelihood ratio by convex risk minimization", arxiv:0809.0853
- Yung-Kyun Noh, Masashi Sugiyama, Song Liu, Marthinus C. du Plessis,
"Bias Reduction and Metric Learning for Nearest-Neighbor Estimation of Kullback-Leibler Divergence", Neural Computation
**30**(2018): 1930--1960 - Morteza Noshad, Yu Zeng, Alfred O. Hero III, "Scalable Mutual Information Estimation using Dependence Graphs", arxiv:1801.09125
- Sebastian Nowozin, "Improved Information Gain Estimates for Decision Tree Induction", arxiv:1206.4620
- Leandro Pardo, Statistical Inference Based on Divergence Measures
- Liam Paninski, "Estimating Entropy on
*m*Bins Given Fewer Than*m*Samples", IEEE Transactions on Information Theory**50**(2004): 2200--2203 - Angeliki Papana and Dimitris Kugiumtzis, "Evaluation of Mutual Information Estimators for Time Series", arxiv:0904.4753
- Paulo R. F. Pinto, M. S. Baptista, Isabel S. Labouriau, "Density of first Poincaré returns, periodic orbits, and Kolmogorov-Sinai entropy", arxiv:0908.4575
- Barnabas Poczos), Zoubin Ghahramani, Jeff Schneider, "Copula-based Kernel Dependency Measures", arxiv:1206.4682
- G. Pola, R. S. Petersen, A. Thiele, M. P. Young and S. Panzeri,
"Data-Robust Tight Lower Bounds to the Information Carried by Spike Times of a
Neuronal Population", Neural
Computation
**17**(2005): 1962--2005 - Paul K. Rubenstein, Olivier Bousquet, Josip Djolonga, Carlos Riquelme, Ilya Tolstikhin, "Practical and Consistent Estimation of f-Divergences", arxiv:1905.11112
- Avraham Ruderman, Mark Reid, Dario Garcia-Garcia, James Petterson, "Tighter Variational Representations of f-Divergences via Restriction to Probability Measures", arxiv:1206.4664
- Luis G. Sanchez Giraldo, Murali Rao, Jose C. Principe, "Measures of Entropy from Data Using Infinitely Divisible Kernels", arxiv:1211.2459
- Hailin Sang Yongli Sang Fangjun Xu, "Kernel Entropy Estimation for Linear Processes", Journal of Time Series Analysis
**39**(2018): 563--591 - Thomas Schürmann
- "Bias Analysis in Entropy Estimates", cond-mat/0403192
- "Scaling behaviour of entropy estimates," cond-mat/0203409
- "A Note on Entropy Estimation",
Neural Computation
**27**(2015): 2097--2106

- J. F. Silva and S. Narayanan, "Complexity-Regularized Tree-Structured Partition for Mutual Information Estimation", IEEE Transactions on Information Theory
**58**(2012): 1940--1952 - Kumar Sricharan, "Ensemble Estimators for Multivariate Entropy Estimation", IEEE Transactions on Information
Theory
**59**(2013): 4374--4388 - Kumar Sricharan, Alfred O. Hero III, "Ensemble estimators for multivariate entropy estimation", arxiv:1203.5829
- Kumar Sricharan, Raviv Raich, Alfred O. Hero III
- "Empirical estimation of entropy functionals with confidence", arxiv:1012.4188
- "Estimation of Nonlinear Functionals of Densities With Confidence", IEEE Transactions on Information Theory
**58**(2012): 4135--4159

- Taiji Suzuki, Masashi Sugiyama and Toshiyuki Tanaka, "Mutual information approximation via maximum likelihood estimation of desnity ratio", ISIT 2009 [PDF preprint via Prof. Sugiyama]
- Evgeniy Timofeev, Alexei Kaltchenko, "Nearest-neighbor Entropy Estimators with Weak Metrics", arxiv:1205.5856
- Nicholas Watters and George N. Reeke, "Neuronal Spike Train Entropy Estimation by History Clustering", Neural Computation
**26**(2014): 1840--1872 - Yihong Wu, Pengkun Yang, "Minimax rates of entropy estimation on large alphabets via best polynomial approximation", arxiv:1407.0381
- Alexander L Young, David B Dunson, "Efficient Entropy Estimation for Stationary Time Series", arxiv:1904.05850
- Xianli Zeng, Yingcun Xia, and Howell Tong, "Jackknife approach to the estimation of mutual information", Proceedings of the National Academy of Sciences (USA)
**115**(2018): 9956--9961 - Zhiyi Zhang
- "Entropy Estimation in Turing's Perspective", Neural Computation
**24**(2012): 1368--1389 - "A Normal Law for the Plug-in Estimator of Entropy",
IEEE Transactions on
Information Theory
**58**(2012): 2745--2747

- "Entropy Estimation in Turing's Perspective", Neural Computation
- Zhiyi Zhang and Michael Grabchak, "Nonparametric Estimation
of Küllback-Leibler Divergence", Neural Computation
**26**(2014): 2570--2593 [Kullback did not spell his name with an umlaut]