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Pinchas A, Ben-Gal I, Painsky A. A Comparative Analysis of Discrete Entropy Estimators for Large-Alphabet Problems. ENTROPY (BASEL, SWITZERLAND) 2024; 26:369. [PMID: 38785618 PMCID: PMC11120205 DOI: 10.3390/e26050369] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2024] [Revised: 04/25/2024] [Accepted: 04/25/2024] [Indexed: 05/25/2024]
Abstract
This paper presents a comparative study of entropy estimation in a large-alphabet regime. A variety of entropy estimators have been proposed over the years, where each estimator is designed for a different setup with its own strengths and caveats. As a consequence, no estimator is known to be universally better than the others. This work addresses this gap by comparing twenty-one entropy estimators in the studied regime, starting with the simplest plug-in estimator and leading up to the most recent neural network-based and polynomial approximate estimators. Our findings show that the estimators' performance highly depends on the underlying distribution. Specifically, we distinguish between three types of distributions, ranging from uniform to degenerate distributions. For each class of distribution, we recommend the most suitable estimator. Further, we propose a sample-dependent approach, which again considers three classes of distribution, and report the top-performing estimators in each class. This approach provides a data-dependent framework for choosing the desired estimator in practical setups.
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Affiliation(s)
- Assaf Pinchas
- School of Electrical Engineering, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Irad Ben-Gal
- Industrial Engineering Department, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 6997801, Israel; (I.B.-G.); (A.P.)
| | - Amichai Painsky
- Industrial Engineering Department, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 6997801, Israel; (I.B.-G.); (A.P.)
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2
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De Gregorio J, Sánchez D, Toral R. Entropy Estimators for Markovian Sequences: A Comparative Analysis. ENTROPY (BASEL, SWITZERLAND) 2024; 26:79. [PMID: 38248204 PMCID: PMC11154276 DOI: 10.3390/e26010079] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Revised: 12/21/2023] [Accepted: 01/16/2024] [Indexed: 01/23/2024]
Abstract
Entropy estimation is a fundamental problem in information theory that has applications in various fields, including physics, biology, and computer science. Estimating the entropy of discrete sequences can be challenging due to limited data and the lack of unbiased estimators. Most existing entropy estimators are designed for sequences of independent events and their performances vary depending on the system being studied and the available data size. In this work, we compare different entropy estimators and their performance when applied to Markovian sequences. Specifically, we analyze both binary Markovian sequences and Markovian systems in the undersampled regime. We calculate the bias, standard deviation, and mean squared error for some of the most widely employed estimators. We discuss the limitations of entropy estimation as a function of the transition probabilities of the Markov processes and the sample size. Overall, this paper provides a comprehensive comparison of entropy estimators and their performance in estimating entropy for systems with memory, which can be useful for researchers and practitioners in various fields.
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Affiliation(s)
| | - David Sánchez
- Institute for Cross-Disciplinary Physics and Complex Systems IFISC (UIB-CSIC), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain; (J.D.G.); (R.T.)
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3
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Zhang Z. Several Basic Elements of Entropic Statistics. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1060. [PMID: 37510007 PMCID: PMC10377889 DOI: 10.3390/e25071060] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2023] [Revised: 07/11/2023] [Accepted: 07/12/2023] [Indexed: 07/30/2023]
Abstract
Inspired by the development in modern data science, a shift is increasingly visible in the foundation of statistical inference, away from a real space, where random variables reside, toward a nonmetrized and nonordinal alphabet, where more general random elements reside. While statistical inferences based on random variables are theoretically well supported in the rich literature of probability and statistics, inferences on alphabets, mostly by way of various entropies and their estimation, are less systematically supported in theory. Without the familiar notions of neighborhood, real or complex moments, tails, et cetera, associated with random variables, probability and statistics based on random elements on alphabets need more attention to foster a sound framework for rigorous development of entropy-based statistical exercises. In this article, several basic elements of entropic statistics are introduced and discussed, including notions of general entropies, entropic sample spaces, entropic distributions, entropic statistics, entropic multinomial distributions, entropic moments, and entropic basis, among other entropic objects. In particular, an entropic-moment-generating function is defined and it is shown to uniquely characterize the underlying distribution in entropic perspective, and, hence, all entropies. An entropic version of the Glivenko-Cantelli convergence theorem is also established.
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Affiliation(s)
- Zhiyi Zhang
- Department of Mathematics and Statistics, UNC Charlotte, Charlotte, NC 28223, USA
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4
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Wang T, Cook DJ, Fischer TR. The Indoor Predictability of Human Mobility: Estimating Mobility with Smart Home Sensors. IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTING 2023; 11:182-193. [PMID: 37457914 PMCID: PMC10348693 DOI: 10.1109/tetc.2022.3188939] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/18/2023]
Abstract
Analyzing human mobility patterns is valuable for understanding human behavior and providing location-anticipating services. In this work, we theoretically estimate the predictability of human movement for indoor settings, a problem that has not yet been tackled by the community. To validate the model, we utilize location data collected by ambient sensors in residential settings. The data support the model and allow us to contrast the predictability of various groups, including single-resident homes, homes with multiple residents, and homes with pets.
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Affiliation(s)
- Tinghui Wang
- School of EECS, Washington State University, Pullman, WA, USA
| | - Diane J Cook
- School of EECS, Washington State University, Pullman, WA, USA
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5
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Grassberger P. On Generalized Schürmann Entropy Estimators. ENTROPY 2022; 24:e24050680. [PMID: 35626564 PMCID: PMC9141067 DOI: 10.3390/e24050680] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/08/2022] [Revised: 04/26/2022] [Accepted: 05/09/2022] [Indexed: 02/04/2023]
Abstract
We present a new class of estimators of Shannon entropy for severely undersampled discrete distributions. It is based on a generalization of an estimator proposed by T. Schürmann, which itself is a generalization of an estimator proposed by myself.For a special set of parameters, they are completely free of bias and have a finite variance, something which is widely believed to be impossible. We present also detailed numerical tests, where we compare them with other recent estimators and with exact results, and point out a clash with Bayesian estimators for mutual information.
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Affiliation(s)
- Peter Grassberger
- Jülich Supercomputing Center, Jülich Research Center, D-52425 Jülich, Germany
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6
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Contreras Rodríguez L, Madarro-Capó EJ, Legón-Pérez CM, Rojas O, Sosa-Gómez G. Selecting an Effective Entropy Estimator for Short Sequences of Bits and Bytes with Maximum Entropy. ENTROPY (BASEL, SWITZERLAND) 2021; 23:561. [PMID: 33946438 PMCID: PMC8147137 DOI: 10.3390/e23050561] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/05/2021] [Revised: 04/26/2021] [Accepted: 04/28/2021] [Indexed: 11/22/2022]
Abstract
Entropy makes it possible to measure the uncertainty about an information source from the distribution of its output symbols. It is known that the maximum Shannon's entropy of a discrete source of information is reached when its symbols follow a Uniform distribution. In cryptography, these sources have great applications since they allow for the highest security standards to be reached. In this work, the most effective estimator is selected to estimate entropy in short samples of bytes and bits with maximum entropy. For this, 18 estimators were compared. Results concerning the comparisons published in the literature between these estimators are discussed. The most suitable estimator is determined experimentally, based on its bias, the mean square error short samples of bytes and bits.
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Affiliation(s)
- Lianet Contreras Rodríguez
- Facultad de Matemática y Computación, Instituto de Criptografía, Universidad de la Habana, Habana 10400, Cuba; (L.C.R.); (E.J.M.-C.); (C.M.L.-P.)
| | - Evaristo José Madarro-Capó
- Facultad de Matemática y Computación, Instituto de Criptografía, Universidad de la Habana, Habana 10400, Cuba; (L.C.R.); (E.J.M.-C.); (C.M.L.-P.)
| | - Carlos Miguel Legón-Pérez
- Facultad de Matemática y Computación, Instituto de Criptografía, Universidad de la Habana, Habana 10400, Cuba; (L.C.R.); (E.J.M.-C.); (C.M.L.-P.)
| | - Omar Rojas
- Facultad de Ciencias Económicas y Empresariales, Universidad Panamericana, Álvaro del Portillo 49, Zapopan, Jalisco 45010, Mexico;
| | - Guillermo Sosa-Gómez
- Facultad de Ciencias Económicas y Empresariales, Universidad Panamericana, Álvaro del Portillo 49, Zapopan, Jalisco 45010, Mexico;
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7
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Willis AD, Martin BD. Estimating diversity in networked ecological communities. Biostatistics 2020; 23:207-222. [PMID: 32432696 PMCID: PMC8759443 DOI: 10.1093/biostatistics/kxaa015] [Citation(s) in RCA: 58] [Impact Index Per Article: 14.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2019] [Revised: 03/04/2020] [Accepted: 03/08/2020] [Indexed: 01/09/2023] Open
Abstract
Comparing ecological communities across environmental gradients can be challenging, especially when the number of different taxonomic groups in the communities is large. In this setting, community-level summaries called diversity indices are widely used to detect changes in the community ecology. However, estimation of diversity indices has received relatively little attention from the statistical community. The most common estimates of diversity are the maximum likelihood estimates of the parameters of a multinomial model, even though the multinomial model implies strict assumptions about the sampling mechanism. In particular, the multinomial model prohibits ecological networks, where taxa positively and negatively co-occur. In this article, we leverage models from the compositional data literature that explicitly account for co-occurrence networks and use them to estimate diversity. Instead of proposing new diversity indices, we estimate popular diversity indices under these models. While the methodology is general, we illustrate the approach for the estimation of the Shannon, Simpson, Bray–Curtis, and Euclidean diversity indices. We contrast our method to multinomial, low-rank, and nonparametric methods for estimating diversity indices. Under simulation, we find that the greatest gains of the method are in strongly networked communities with many taxa. Therefore, to illustrate the method, we analyze the microbiome of seafloor basalts based on a 16S amplicon sequencing dataset with 1425 taxa and 12 communities.
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Affiliation(s)
- Amy D Willis
- Department of Biostatistics and Department of Statistics, University of Washington, Health Sciences Building, 1959 NE Pacific St, Seattle WA 98195, USA
| | - Bryan D Martin
- Department of Biostatistics and Department of Statistics, University of Washington, Health Sciences Building, 1959 NE Pacific St, Seattle WA 98195, USA
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8
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9
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Ferrari A. Modeling Information Content Via Dirichlet-Multinomial Regression Analysis. MULTIVARIATE BEHAVIORAL RESEARCH 2017; 52:259-270. [PMID: 28207283 DOI: 10.1080/00273171.2017.1279957] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Shannon entropy is being increasingly used in biomedical research as an index of complexity and information content in sequences of symbols, e.g. languages, amino acid sequences, DNA methylation patterns and animal vocalizations. Yet, distributional properties of information entropy as a random variable have seldom been the object of study, leading to researchers mainly using linear models or simulation-based analytical approach to assess differences in information content, when entropy is measured repeatedly in different experimental conditions. Here a method to perform inference on entropy in such conditions is proposed. Building on results coming from studies in the field of Bayesian entropy estimation, a symmetric Dirichlet-multinomial regression model, able to deal efficiently with the issue of mean entropy estimation, is formulated. Through a simulation study the model is shown to outperform linear modeling in a vast range of scenarios and to have promising statistical properties. As a practical example, the method is applied to a data set coming from a real experiment on animal communication.
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Affiliation(s)
- Alberto Ferrari
- a Department of Brain and Behavioural Sciences , University of Pavia
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10
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Jiao J, Venkat K, Han Y, Weissman T. Minimax Estimation of Functionals of Discrete Distributions. IEEE TRANSACTIONS ON INFORMATION THEORY 2015; 61:2835-2885. [PMID: 29375152 PMCID: PMC5786426 DOI: 10.1109/tit.2015.2412945] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size S is unknown and may be comparable with or even much larger than the number of observations n. We treat the respective regions where the functional is nonsmooth and smooth separately. In the nonsmooth regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the smooth regime, we apply a bias-corrected version of the maximum likelihood estimator (MLE). We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: 1) the entropy [Formula: see text] and 2) [Formula: see text], α > 0. We obtain the minimax L2 rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity n ≍ S/ln S for entropy estimation. We also demonstrate that the sample complexity for estimating Fα (P), 0 < α < 1, is n ≍ S1/α /ln S, which can be achieved by our estimator but not the MLE. For 1 < α < 3/2, we show the minimax L2 rate for estimating Fα (P) is (n ln n)-2(α-1) for infinite support size, while the maximum L2 rate for the MLE is n-2(α-1). For all the above cases, the behavior of the minimax rate-optimal estimators with n samples is essentially that of the MLE (plug-in rule) with n ln n samples, which we term "effective sample size enlargement." We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow-Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable.
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Affiliation(s)
- Jiantao Jiao
- Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA
| | - Kartik Venkat
- Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA
| | - Yanjun Han
- Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
| | - Tsachy Weissman
- Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA
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11
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Estimating Functions of Distributions Defined over Spaces of Unknown Size. ENTROPY 2013. [DOI: 10.3390/e15114668] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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12
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Chao A, Wang YT, Jost L. Entropy and the species accumulation curve: a novel entropy estimator via discovery rates of new species. Methods Ecol Evol 2013. [DOI: 10.1111/2041-210x.12108] [Citation(s) in RCA: 90] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Anne Chao
- Institute of Statistics; National Tsing Hua University; Hsin-Chu Taiwan 30043
| | - Y. T. Wang
- Institute of Statistics; National Tsing Hua University; Hsin-Chu Taiwan 30043
| | - Lou Jost
- EcoMinga Foundation; Via a Runtun Baños Tungurahua Ecuador
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13
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Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data. ENTROPY 2013. [DOI: 10.3390/e15051738] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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14
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Rempala GA, Seweryn M. Methods for diversity and overlap analysis in T-cell receptor populations. J Math Biol 2012; 67:1339-68. [PMID: 23007599 DOI: 10.1007/s00285-012-0589-7] [Citation(s) in RCA: 91] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2012] [Revised: 08/29/2012] [Indexed: 12/13/2022]
Abstract
The paper presents some novel approaches to the empirical analysis of diversity and similarity (overlap) in biological or ecological systems. The analysis is motivated by the molecular studies of highly diverse mammalian T-cell receptor (TCR) populations, and is related to the classical statistical problem of analyzing two-way contingency tables with missing cells and low cell counts. The new measures of diversity and overlap are proposed, based on the information-theoretic as well as geometric considerations, with the capacity to naturally up-weight or down-weight the rare and abundant population species. The consistent estimates are derived by applying the Good-Turing sample-coverage correction. In particular, novel consistent estimates of the Shannon entropy function and the Morisita-Horn index are provided. Data from TCR populations in mice are used to illustrate the empirical performance of the proposed methods vis a vis the existing alternatives.
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Affiliation(s)
- Grzegorz A Rempala
- Department of Biostatistics and Cancer Research Center, Georgia Health Sciences University, Augusta, GA, 30912, USA,
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15
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Abstract
A new nonparametric estimator of Shannon's entropy on a countable alphabet is proposed and analyzed against the well-known plug-in estimator. The proposed estimator is developed based on Turing's formula, which recovers distributional characteristics on the subset of the alphabet not covered by a size-n sample. The fundamental switch in perspective brings about substantial gain in estimation accuracy for every distribution with finite entropy. In general, a uniform variance upper bound is established for the entire class of distributions with finite entropy that decays at a rate of O(ln(n)/n) compared to O([ln(n)]2/n) for the plug-in. In a wide range of subclasses, the variance of the proposed estimator converges at a rate of O(1/n), and this rate of convergence carries over to the convergence rates in mean squared errors in many subclasses. Specifically, for any finite alphabet, the proposed estimator has a bias decaying exponentially in n. Several new bias-adjusted estimators are also discussed.
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Affiliation(s)
- Zhiyi Zhang
- Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, U.S.A
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16
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Pita-Almenar JD, Ranganathan GN, Koester HJ. Impact of cortical plasticity on information signaled by populations of neurons in the cerebral cortex. J Neurophysiol 2011; 106:1118-24. [PMID: 21653720 DOI: 10.1152/jn.01001.2010] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
The performance of neural codes to represent attributes of sensory signals has been evaluated in the vertebrate peripheral and central nervous system. Here, we determine how information signaled by populations of neurons is modified by plasticity. Suprathreshold neuronal responses from a large number of neurons were recorded in the juvenile mouse barrel cortex using dithered random-access scanning. Pairing of one input with another resulted in a long-lasting, input-specific modification of the cortical responses. Mutual information analysis indicated that cortical plasticity efficiently changed information signaled by populations of neurons. The contribution of neural correlations to the change in mutual information was negative. The largest factor limiting fidelity of mutual information after pairing was a low reliability of the modified cortical responses.
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Affiliation(s)
- Juan Diego Pita-Almenar
- Center for Learning and Memory, Section of Neurobiology, The University of Texas at Austin, 1 Univ. Sta., C7000, Austin, TX 78712, USA.
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17
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Rempala GA, Seweryn M, Ignatowicz L. Model for comparative analysis of antigen receptor repertoires. J Theor Biol 2010; 269:1-15. [PMID: 20955715 DOI: 10.1016/j.jtbi.2010.10.001] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2010] [Revised: 09/09/2010] [Accepted: 10/04/2010] [Indexed: 11/30/2022]
Abstract
In modern molecular biology one of the standard ways of analyzing a vertebrate immune system is to sequence and compare the counts of specific antigen receptor clones (either immunoglobulins or T-cell receptors) derived from various tissues under different experimental or clinical conditions. The resulting statistical challenges are difficult and do not fit readily into the standard statistical framework of contingency tables primarily due to the serious under-sampling of the receptor populations. This under-sampling is caused, on one hand, by the extreme diversity of antigen receptor repertoires maintained by the immune system and, on the other, by the high cost and labor intensity of the receptor data collection process. In most of the recent immunological literature the differences across antigen receptor populations are examined via non-parametric statistical measures of the species overlap and diversity borrowed from ecological studies. While this approach is robust in a wide range of situations, it seems to provide little insight into the underlying clonal size distribution and the overall mechanism differentiating the receptor populations. As a possible alternative, the current paper presents a parametric method that adjusts for the data under-sampling as well as provides a unifying approach to a simultaneous comparison of multiple receptor groups by means of the modern statistical tools of unsupervised learning. The parametric model is based on a flexible multivariate Poisson-lognormal distribution and is seen to be a natural generalization of the univariate Poisson-lognormal models used in the ecological studies of biodiversity patterns. The procedure for evaluating a model's fit is described along with the public domain software developed to perform the necessary diagnostics. The model-driven analysis is seen to compare favorably vis a vis traditional methods when applied to the data from T-cell receptors in transgenic mice populations.
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Affiliation(s)
- Grzegorz A Rempala
- Department of Biostatistics and the Cancer Center, Medical College of Georgia, Augusta, GA 30912, USA.
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18
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Gastpar MC, Gill PR, Huth AG, Theunissen FE. Anthropic Correction of Information Estimates and Its Application to Neural Coding. IEEE TRANSACTIONS ON INFORMATION THEORY 2010; 56:890-900. [PMID: 26900172 PMCID: PMC4758524 DOI: 10.1109/tit.2009.2037053] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Information theory has been used as an organizing principle in neuroscience for several decades. Estimates of the mutual information (MI) between signals acquired in neurophysiological experiments are believed to yield insights into the structure of the underlying information processing architectures. With the pervasive availability of recordings from many neurons, several information and redundancy measures have been proposed in the recent literature. A typical scenario is that only a small number of stimuli can be tested, while ample response data may be available for each of the tested stimuli. The resulting asymmetric information estimation problem is considered. It is shown that the direct plug-in information estimate has a negative bias. An anthropic correction is introduced that has a positive bias. These two complementary estimators and their combinations are natural candidates for information estimation in neuroscience. Tail and variance bounds are given for both estimates. The proposed information estimates are applied to the analysis of neural discrimination and redundancy in the avian auditory system.
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Affiliation(s)
- Michael C. Gastpar
- Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720 USA
- Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS), Delft University of Technology, Delft, The Netherlands ()
| | - Patrick R. Gill
- School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 USA ()
| | - Alexander G. Huth
- Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720 USA ()
| | - Frédéric E. Theunissen
- Helen Wills Neuroscience Institute and the Department of Psychology, University of California, Berkeley, CA 94720 USA ()
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19
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Abstract
Information estimates such as the direct method of Strong, Koberle, de Ruyter van Steveninck, and Bialek (1998) sidestep the difficult problem of estimating the joint distribution of response and stimulus by instead estimating the difference between the marginal and conditional entropies of the response. While this is an effective estimation strategy, it tempts the practitioner to ignore the role of the stimulus and the meaning of mutual information. We show here that as the number of trials increases indefinitely, the direct (or plug-in) estimate of marginal entropy converges (with probability 1) to the entropy of the time-averaged conditional distribution of the response, and the direct estimate of the conditional entropy converges to the time-averaged entropy of the conditional distribution of the response. Under joint stationarity and ergodicity of the response and stimulus, the difference of these quantities converges to the mutual information. When the stimulus is deterministic or nonstationary the direct estimate of information no longer estimates mutual information, which is no longer meaningful, but it remains a measure of variability of the response distribution across time.
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Affiliation(s)
- Vincent Q Vu
- Department of Statistics, University of California, Berkeley, CA 94720, USA.
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