Distribution of Mutual Information
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Connection Between Dirichlet Distributions and a Scale-Invariant Probabilistic Model Based on Leibniz-Like Pyramids
Home Search Collections Journals About Contact us My IOPscience Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids This content has been downloaded from IOPscience. Please scroll down to see the full text. View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 152.84.125.254 This content was downloaded on 22/12/2014 at 23:17 Please note that terms and conditions apply. Journal of Statistical Mechanics: Theory and Experiment Connection between Dirichlet distributions and a scale-invariant probabilistic model based on J. Stat. Mech. Leibniz-like pyramids A Rodr´ıguez1 and C Tsallis2,3 1 Departamento de Matem´atica Aplicada a la Ingenier´ıa Aeroespacial, Universidad Polit´ecnica de Madrid, Pza. Cardenal Cisneros s/n, 28040 Madrid, Spain 2 Centro Brasileiro de Pesquisas F´ısicas and Instituto Nacional de Ciˆencia e Tecnologia de Sistemas Complexos, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil (2014) P12027 3 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA E-mail: [email protected] and [email protected] Received 4 November 2014 Accepted for publication 30 November 2014 Published 22 December 2014 Online at stacks.iop.org/JSTAT/2014/P12027 doi:10.1088/1742-5468/2014/12/P12027 Abstract. We show that the N limiting probability distributions of a recently introduced family of d→∞-dimensional scale-invariant probabilistic models based on Leibniz-like (d + 1)-dimensional hyperpyramids (Rodr´ıguez and Tsallis 2012 J. Math. Phys. 53 023302) are given by Dirichlet distributions for d = 1, 2, ... -
Lecture 3: Entropy, Relative Entropy, and Mutual Information 1 Notation 2
EE376A/STATS376A Information Theory Lecture 3 - 01/16/2018 Lecture 3: Entropy, Relative Entropy, and Mutual Information Lecturer: Tsachy Weissman Scribe: Yicheng An, Melody Guan, Jacob Rebec, John Sholar In this lecture, we will introduce certain key measures of information, that play crucial roles in theoretical and operational characterizations throughout the course. These include the entropy, the mutual information, and the relative entropy. We will also exhibit some key properties exhibited by these information measures. 1 Notation A quick summary of the notation 1. Discrete Random Variable: U 2. Alphabet: U = fu1; u2; :::; uM g (An alphabet of size M) 3. Specific Value: u; u1; etc. For discrete random variables, we will write (interchangeably) P (U = u), PU (u) or most often just, p(u) Similarly, for a pair of random variables X; Y we write P (X = x j Y = y), PXjY (x j y) or p(x j y) 2 Entropy Definition 1. \Surprise" Function: 1 S(u) log (1) , p(u) A lower probability of u translates to a greater \surprise" that it occurs. Note here that we use log to mean log2 by default, rather than the natural log ln, as is typical in some other contexts. This is true throughout these notes: log is assumed to be log2 unless otherwise indicated. Definition 2. Entropy: Let U a discrete random variable taking values in alphabet U. The entropy of U is given by: 1 X H(U) [S(U)] = log = − log (p(U)) = − p(u) log p(u) (2) , E E p(U) E u Where U represents all u values possible to the variable. -
Lecture 24: Dirichlet Distribution and Dirichlet Process
Stat260: Bayesian Modeling and Inference Lecture Date: April 28, 2010 Lecture 24: Dirichlet distribution and Dirichlet Process Lecturer: Michael I. Jordan Scribe: Vivek Ramamurthy 1 Introduction In the last couple of lectures, in our study of Bayesian nonparametric approaches, we considered the Chinese Restaurant Process, Bayesian mixture models, stick breaking, and the Dirichlet process. Today, we will try to gain some insight into the connection between the Dirichlet process and the Dirichlet distribution. 2 The Dirichlet distribution and P´olya urn First, we note an important relation between the Dirichlet distribution and the Gamma distribution, which is used to generate random vectors which are Dirichlet distributed. If, for i ∈ {1, 2, · · · , K}, Zi ∼ Gamma(αi, β) independently, then K K S = Zi ∼ Gamma αi, β i=1 i=1 ! X X and V = (V1, · · · ,VK ) = (Z1/S, · · · ,ZK /S) ∼ Dir(α1, · · · , αK ) Now, consider the following P´olya urn model. Suppose that • Xi - color of the ith draw • X - space of colors (discrete) • α(k) - number of balls of color k initially in urn. We then have that α(k) + j<i δXj (k) p(Xi = k|X1, · · · , Xi−1) = α(XP) + i − 1 where δXj (k) = 1 if Xj = k and 0 otherwise, and α(X ) = k α(k). It may then be shown that P n α(x1) α(xi) + j<i δXj (xi) p(X1 = x1, X2 = x2, · · · , X = x ) = n n α(X ) α(X ) + i − 1 i=2 P Y α(1)[α(1) + 1] · · · [α(1) + m1 − 1]α(2)[α(2) + 1] · · · [α(2) + m2 − 1] · · · α(C)[α(C) + 1] · · · [α(C) + m − 1] = C α(X )[α(X ) + 1] · · · [α(X ) + n − 1] n where 1, 2, · · · , C are the distinct colors that appear in x1, · · · ,xn and mk = i=1 1{Xi = k}. -
The Exciting Guide to Probability Distributions – Part 2
The Exciting Guide To Probability Distributions – Part 2 Jamie Frost – v1.1 Contents Part 2 A revisit of the multinomial distribution The Dirichlet Distribution The Beta Distribution Conjugate Priors The Gamma Distribution We saw in the last part that the multinomial distribution was over counts of outcomes, given the probability of each outcome and the total number of outcomes. xi f(x1, ... , xk | n, p1, ... , pk)= [n! / ∏xi!] ∏pi The count of The probability of each outcome. each outcome. That’s all smashing, but suppose we wanted to know the reverse, i.e. the probability that the distribution underlying our random variable has outcome probabilities of p1, ... , pk, given that we observed each outcome x1, ... , xk times. In other words, we are considering all the possible probability distributions (p1, ... , pk) that could have generated these counts, rather than all the possible counts given a fixed distribution. Initial attempt at a probability mass function: Just swap the domain and the parameters: The RHS is exactly the same. xi f(p1, ... , pk | n, x1, ... , xk )= [n! / ∏xi!] ∏pi Notational convention is that we define the support as a vector x, so let’s relabel p as x, and the counts x as α... αi f(x1, ... , xk | n, α1, ... , αk )= [n! / ∏ αi!] ∏xi We can define n just as the sum of the counts: αi f(x1, ... , xk | α1, ... , αk )= [(∑αi)! / ∏ αi!] ∏xi But wait, we’re not quite there yet. We know that probabilities have to sum to 1, so we need to restrict the domain we can draw from: s.t. -
Stat 5101 Lecture Slides: Deck 8 Dirichlet Distribution
Stat 5101 Lecture Slides: Deck 8 Dirichlet Distribution Charles J. Geyer School of Statistics University of Minnesota This work is licensed under a Creative Commons Attribution- ShareAlike 4.0 International License (http://creativecommons.org/ licenses/by-sa/4.0/). 1 The Dirichlet Distribution The Dirichlet Distribution is to the beta distribution as the multi- nomial distribution is to the binomial distribution. We get it by the same process that we got to the beta distribu- tion (slides 128{137, deck 3), only multivariate. Recall the basic theorem about gamma and beta (same slides referenced above). 2 The Dirichlet Distribution (cont.) Theorem 1. Suppose X and Y are independent gamma random variables X ∼ Gam(α1; λ) Y ∼ Gam(α2; λ) then U = X + Y V = X=(X + Y ) are independent random variables and U ∼ Gam(α1 + α2; λ) V ∼ Beta(α1; α2) 3 The Dirichlet Distribution (cont.) Corollary 1. Suppose X1;X2;:::; are are independent gamma random variables with the same shape parameters Xi ∼ Gam(αi; λ) then the following random variables X1 ∼ Beta(α1; α2) X1 + X2 X1 + X2 ∼ Beta(α1 + α2; α3) X1 + X2 + X3 . X1 + ··· + Xd−1 ∼ Beta(α1 + ··· + αd−1; αd) X1 + ··· + Xd are independent and have the asserted distributions. 4 The Dirichlet Distribution (cont.) From the first assertion of the theorem we know X1 + ··· + Xk−1 ∼ Gam(α1 + ··· + αk−1; λ) and is independent of Xk. Thus the second assertion of the theorem says X1 + ··· + Xk−1 ∼ Beta(α1 + ··· + αk−1; αk)(∗) X1 + ··· + Xk and (∗) is independent of X1 + ··· + Xk. That proves the corollary. 5 The Dirichlet Distribution (cont.) Theorem 2. -
Parameter Specification of the Beta Distribution and Its Dirichlet
%HWD'LVWULEXWLRQVDQG,WV$SSOLFDWLRQV 3DUDPHWHU6SHFLILFDWLRQRIWKH%HWD 'LVWULEXWLRQDQGLWV'LULFKOHW([WHQVLRQV 8WLOL]LQJ4XDQWLOHV -5HQpYDQ'RUSDQG7KRPDV$0D]]XFKL (Submitted January 2003, Revised March 2003) I. INTRODUCTION.................................................................................................... 1 II. SPECIFICATION OF PRIOR BETA PARAMETERS..............................................5 A. Basic Properties of the Beta Distribution...............................................................6 B. Solving for the Beta Prior Parameters...................................................................8 C. Design of a Numerical Procedure........................................................................12 III. SPECIFICATION OF PRIOR DIRICHLET PARAMETERS................................. 17 A. Basic Properties of the Dirichlet Distribution...................................................... 18 B. Solving for the Dirichlet prior parameters...........................................................20 IV. SPECIFICATION OF ORDERED DIRICHLET PARAMETERS...........................22 A. Properties of the Ordered Dirichlet Distribution................................................. 23 B. Solving for the Ordered Dirichlet Prior Parameters............................................ 25 C. Transforming the Ordered Dirichlet Distribution and Numerical Stability ......... 27 V. CONCLUSIONS........................................................................................................ 31 APPENDIX................................................................................................................... -
A Statistical Framework for Neuroimaging Data Analysis Based on Mutual Information Estimated Via a Gaussian Copula
bioRxiv preprint doi: https://doi.org/10.1101/043745; this version posted October 25, 2016. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. A statistical framework for neuroimaging data analysis based on mutual information estimated via a Gaussian copula Robin A. A. Ince1*, Bruno L. Giordano1, Christoph Kayser1, Guillaume A. Rousselet1, Joachim Gross1 and Philippe G. Schyns1 1 Institute of Neuroscience and Psychology, University of Glasgow, 58 Hillhead Street, Glasgow, G12 8QB, UK * Corresponding author: [email protected] +44 7939 203 596 Abstract We begin by reviewing the statistical framework of information theory as applicable to neuroimaging data analysis. A major factor hindering wider adoption of this framework in neuroimaging is the difficulty of estimating information theoretic quantities in practice. We present a novel estimation technique that combines the statistical theory of copulas with the closed form solution for the entropy of Gaussian variables. This results in a general, computationally efficient, flexible, and robust multivariate statistical framework that provides effect sizes on a common meaningful scale, allows for unified treatment of discrete, continuous, uni- and multi-dimensional variables, and enables direct comparisons of representations from behavioral and brain responses across any recording modality. We validate the use of this estimate as a statistical test within a neuroimaging context, considering both discrete stimulus classes and continuous stimulus features. We also present examples of analyses facilitated by these developments, including application of multivariate analyses to MEG planar magnetic field gradients, and pairwise temporal interactions in evoked EEG responses. -
Generalized Dirichlet Distributions on the Ball and Moments
Generalized Dirichlet distributions on the ball and moments F. Barthe,∗ F. Gamboa,† L. Lozada-Chang‡and A. Rouault§ September 3, 2018 Abstract The geometry of unit N-dimensional ℓp balls (denoted here by BN,p) has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest, such questions have applications in several probabilistic and geometric contexts [BGMN05]. In this paper, our aim is to revisit some known results of this flavour with a new point of view. Roughly speaking, we will endow BN,p with some kind of Dirichlet distribution that generalizes the uniform one and will follow the method developed in [Ski67], [CKS93] in the context of the randomized moment space. The main idea is to build a suitable coordinate change involving independent random variables. Moreover, we will shed light on connections between the randomized balls and the randomized moment space. n Keywords: Moment spaces, ℓp -balls, canonical moments, Dirichlet dis- tributions, uniform sampling. AMS classification: 30E05, 52A20, 60D05, 62H10, 60F10. arXiv:1002.1544v2 [math.PR] 21 Oct 2010 1 Introduction The starting point of our work is the study of the asymptotic behaviour of the moment spaces: 1 [0,1] j MN = t µ(dt) : µ ∈M1([0, 1]) , (1) (Z0 1≤j≤N ) ∗Equipe de Statistique et Probabilit´es, Institut de Math´ematiques de Toulouse (IMT) CNRS UMR 5219, Universit´ePaul-Sabatier, 31062 Toulouse cedex 9, France. e-mail: [email protected] †IMT e-mail: [email protected] ‡Facultad de Matem´atica y Computaci´on, Universidad de la Habana, San L´azaro y L, Vedado 10400 C.Habana, Cuba. -
Information Theory 1 Entropy 2 Mutual Information
CS769 Spring 2010 Advanced Natural Language Processing Information Theory Lecturer: Xiaojin Zhu [email protected] In this lecture we will learn about entropy, mutual information, KL-divergence, etc., which are useful concepts for information processing systems. 1 Entropy Entropy of a discrete distribution p(x) over the event space X is X H(p) = − p(x) log p(x). (1) x∈X When the log has base 2, entropy has unit bits. Properties: H(p) ≥ 0, with equality only if p is deterministic (use the fact 0 log 0 = 0). Entropy is the average number of 0/1 questions needed to describe an outcome from p(x) (the Twenty Questions game). Entropy is a concave function of p. 1 1 1 1 7 For example, let X = {x1, x2, x3, x4} and p(x1) = 2 , p(x2) = 4 , p(x3) = 8 , p(x4) = 8 . H(p) = 4 bits. This definition naturally extends to joint distributions. Assuming (x, y) ∼ p(x, y), X X H(p) = − p(x, y) log p(x, y). (2) x∈X y∈Y We sometimes write H(X) instead of H(p) with the understanding that p is the underlying distribution. The conditional entropy H(Y |X) is the amount of information needed to determine Y , if the other party knows X. X X X H(Y |X) = p(x)H(Y |X = x) = − p(x, y) log p(y|x). (3) x∈X x∈X y∈Y From above, we can derive the chain rule for entropy: H(X1:n) = H(X1) + H(X2|X1) + .. -
Spatially Constrained Student's T-Distribution Based Mixture Model
J Math Imaging Vis (2018) 60:355–381 https://doi.org/10.1007/s10851-017-0759-8 Spatially Constrained Student’s t-Distribution Based Mixture Model for Robust Image Segmentation Abhirup Banerjee1 · Pradipta Maji1 Received: 8 January 2016 / Accepted: 5 September 2017 / Published online: 21 September 2017 © Springer Science+Business Media, LLC 2017 Abstract The finite Gaussian mixture model is one of the Keywords Segmentation · Student’s t-distribution · most popular frameworks to model classes for probabilistic Expectation–maximization · Hidden Markov random field model-based image segmentation. However, the tails of the Gaussian distribution are often shorter than that required to model an image class. Also, the estimates of the class param- 1 Introduction eters in this model are affected by the pixels that are atypical of the components of the fitted Gaussian mixture model. In In image processing, segmentation refers to the process this regard, the paper presents a novel way to model the image of partitioning an image space into some non-overlapping as a mixture of finite number of Student’s t-distributions for meaningful homogeneous regions. It is an indispensable step image segmentation problem. The Student’s t-distribution for many image processing problems, particularly for med- provides a longer tailed alternative to the Gaussian distri- ical images. Segmentation of brain images into three main bution and gives reduced weight to the outlier observations tissue classes, namely white matter (WM), gray matter (GM), during the parameter estimation step in finite mixture model. and cerebro-spinal fluid (CSF), is important for many diag- Incorporating the merits of Student’s t-distribution into the nostic studies. -
Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics
axioms Article Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics David J. Galas 1,*, Gregory Dewey 2, James Kunert-Graf 1 and Nikita A. Sakhanenko 1 1 Pacific Northwest Research Institute, 720 Broadway, Seattle, WA 98122, USA; [email protected] (J.K.-G.); [email protected] (N.A.S.) 2 Albany College of Pharmacy and Health Sciences, 106 New Scotland Avenue, Albany, NY 12208, USA; [email protected] * Correspondence: [email protected] Academic Editors: Abbe Mowshowitz and Matthias Dehmer Received: 31 January 2017; Accepted: 27 March 2017; Published: 1 April 2017 Abstract: Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. -
Fundamentals of Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Independent Vector Analysis (IVA)
Advanced Signal Processing Group Fundamentals of Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Independent Vector Analysis (IVA) Dr Mohsen Naqvi School of Electrical and Electronic Engineering, Newcastle University [email protected] UDRC Summer School Advanced Signal Processing Group Acknowledgement and Recommended text: Hyvarinen et al., Independent Component Analysis, Wiley, 2001 UDRC Summer School 2 Advanced Signal Processing Group Source Separation and Component Analysis Mixing Process Un-mixing Process 푥 푠1 1 푦1 H W Independent? 푠푛 푥푚 푦푛 Unknown Known Optimize Mixing Model: x = Hs Un-mixing Model: y = Wx = WHs =PDs UDRC Summer School 3 Advanced Signal Processing Group Principal Component Analysis (PCA) Orthogonal projection of data onto lower-dimension linear space: i. maximizes variance of projected data ii. minimizes mean squared distance between data point and projections UDRC Summer School 4 Advanced Signal Processing Group Principal Component Analysis (PCA) Example: 2-D Gaussian Data UDRC Summer School 5 Advanced Signal Processing Group Principal Component Analysis (PCA) Example: 1st PC in the direction of the largest variance UDRC Summer School 6 Advanced Signal Processing Group Principal Component Analysis (PCA) Example: 2nd PC in the direction of the second largest variance and each PC is orthogonal to other components. UDRC Summer School 7 Advanced Signal Processing Group Principal Component Analysis (PCA) • In PCA the redundancy is measured by correlation between data elements • Using