Information Bottleneck versus Maximum Likelihood Felix Polyakov

Overview of the talk  Brief review of the Information Bottleneck Maximum Likelihood Information Bottleneck and Maximum Likelihood Example from Image Segmentation

A Simple Example...

Simple Example

A new compact representation The document clusters preserve the relevant information between the documents and words

Feature Selection? NO ASSUMPTIONS about the source of the data Extracting relevant structure from data –functions of the data (statistics) that preserve information Information about what? Need a principle that is both general and precise.

DocumentsWords

The information bottleneck or relevance through distortion We would like the relevant partitioning T to compress X as much as possible, and to capture as much information about Y as possible X Y N. Tishby, F. Pereira, and W. Bialek

Goal: find q(T | X) –note Markovian independence relationT  X  Y

Variational problem Iterative algorithm

Overview of the talk Short review of the Information Bottleneck  Maximum Likelihood Information Bottleneck and Maximum Likelihood Example from Image Segmentation

A coin is known to be biased The coin is tossed three times – two heads and one tail Use ML to estimate the probability of throwing a head  Try P = 0.2  Try P = 0.6 Probability of a head Likelihood of the Data L(O) = 0.2 * 0.2 * 0.8 =  Try P = 0.4 L(O) = 0.4 * 0.4 * 0.6 = L(O) = 0.6 * 0.6 * 0.4 =  Try P = 0.8 L(O) = 0.8 * 0.8 * 0.2 = A simple example... Model: −p(head) = P −p(tail) = 1 - P

A bit more complicated example… : Mixture Model Three baskets with white (O = 1), grey (O = 2), and black (O = 3) balls B1B1 B2B2 B3B3 15 balls were drawn as follows: 1.Choose a basket according to p(i) =  b i 2.Draw the ball j from basket i with probability Use ML to estimate  given the observations: sequence of balls’ colors

Likelihood of observations Log Likelihood of observations Maximal Likelihood of observations

Likelihood of the observed data x – hidden random variables [e.g. basket] y – observed random variables [e.g. color]  - model parameters [e.g. they define p(y|x)]  0 – current estimate of model parameters

1.Expectation −Compute −Get 2.Maximization − Expectation-maximization algorithm (I) EM algorithm converges to local maxima

Log-likelihood is non-decreasing, examples

EM – another approach  Goal: Jensen’s inequality for concave function

1.Expectation 2.Maximization Expectation-maximization algorithm (II) (I) and (II) are equivalent

Scheme of the approach

Overview of the talk Short review of the Information Bottleneck Maximum Likelihood  Information Bottleneck and Maximum Likelihood for a toy problem Example from Image Segmentation

Words - Y Documents - X Topics - t t ~  (t) x ~  (x) y|t ~  (y|t)

Model parameters Example x i = 9 –t(9) = 2 –sample from  (y|2)  get y i = “Drug” −set n(9, “Drug”) = n(9, “Drug”) + 1 Sampling algorithm For i = 1:N –choose x i by sampling from  (x) –choose y i by sampling from  (y|t(x i )) –increase n(x i, y i ) by one

X t(X)  (y|t=1)  (y|t=2)  (y|t=3)

Toy problem: which parameters maximize the likelihood? = topics X = documents Y = words X Y t(x)  (y|t(x))

EM approach E-step M-step Normalization factor

IB approach Normalization factor

ML IB,, r is a scaling constant

X is uniformly distributed r = |X|  The EM algorithm is equivalent to the IB iterative algorithm,, IBML IB ML mapping Iterative IB EM

X is uniformly distributed  = n(x) IB ML IB ML mapping  All the fixed points of the likelihood L are mapped to all the fixed points of the IB-functional L = I(T;X) -  I(T;Y)  At the fixed points –log L  L + const

X is uniformly distributed  = n(x)  -(1/r) F -  H(Y) = L  -F  L + const  Every algorithm increases F, iff it decreases L

Deterministic case N  (or   ) EM: MLIB IB:

N  (or   ) –Do not speak about uniformity of X here  All the fixed points of L are mapped to all the fixed points of L  -F  L + const  Every algorithm which finds a fixed point of L, induces a fixed point of L and vice versa  In case of several different f.p., the solution that maximized L is mapped to the solution that minimizes L.

Example  (x) x 2/3Yellow submarine 1/3Red bull N=   =  t EMIB  (t) q(t) 11/22/3 21/21/3 This does not mean that q(t) =  (t)

 When N , every algorithm increases F iff it decrease L with   How large must N (or  ) be? How is it related to the “amount of uniformity” in n(x)?

Simulations for iIB

Simulations for EM

Simulations 200 runs = 100 (small N) (large N)  58 runs IIB converged to a smaller value of (-F) than EM  46 runs EM converged to (-F) related to a smaller value of L

Quality estimation for EM solution The quality of IB solution is measured through the theoretic upper bound Using mapping, one can adopt this measure for the ML esimation problem, for large enough N IB ML

Summary: IB versus ML ML and IB approaches are equivalent under certain conditions Models comparison –The mixture model assumes that Y is independent of X given T(X): X  T  Y –In the IB framework, T is defined through the IB Markovian independence relation: T  X  Y Can adapt the quality estimation measure from IB to the ML estimation problem, for large N

Overview of the talk Brief review of the Information Bottleneck Maximum Likelihood Information Bottleneck and Maximum Likelihood  Example from Image Segmentation ( L. Hermes et. al. )

The clustering model Pixels o i, i = 1, …, n Deterministic clusters c,, = 1, …, k Boolean assignment matrix M  M = {0, 1} n x k,  M i  Observations

oioi q r Observations

Likelihood Discretization of the color space into intervals I j Set Data likelihood

Relation to the IB

Log-Likelihood IB functional Assume that n i = const, set = n i  then L = -log L

Images generated from the learned statistics

References N. Tishby, F. Pereira, and W. Bialek. The Information Bottleneck method. Noam Slonim YairWeiss. Maximum Likelihood and the Information Bottleneck’ R. M. Neal and G. E. Hinton. A view of the EM algorithm that justifies incremental, sparse, and other variants. J. Goldberger. Lecture notes L. Hermes, T. Zoller, and J. M. Buhmann. Parametric distributional clustering for image segmentation. The end