The document discusses the principle of maximum entropy. It explains that maximum entropy is an approach for making probability assignments where the assigned probability distribution should have the largest entropy or uncertainty possible, subject to whatever information is known. It describes applications of maximum entropy modeling such as part-of-speech tagging and logistic regression. Maximum entropy and maximum likelihood methods are related as they both aim to make distributions as uniform as possible based on available information.

1. Principle of Maximum Entropy
Jiawang Liu
liujiawang@baidu.com
2012.6

2. Outline
 What is Entropy
 Principle of Maximum Entropy
 Relation
to Maximum Likelihood
 MaxEnt methods and Bayesian
 Applications
 NLP(POS tagging)
 Logistic regression
 Q&A

3. What is Entropy
In information theory, entropy is the measure of the
amount of information that is missing before reception
and is sometimes referred to as Shannon entropy.
 Uncertainty

4. Principle of Maximum Entropy
Subject to precisely stated prior data, which must be a
proposition that expresses testable information, the
probability distribution which best represents the
current state of knowledge is the one with largest
information theoretical entropy.
Why maximum entropy?
 Minimize commitment
 Model all that is known and assume nothing about what is unknown

5. Principle of Maximum Entropy
Overview
 Should guarantee the uniqueness and consistency of
probability assignments obtained by different methods
 Makes explicit our freedom in using different forms of
prior data
 Admits the most ignorance beyond the stated prior data

6. Principle of Maximum Entropy
Testable information
 The principle of maximum entropy is useful explicitly
only when applied to testable information
 A piece of information is testable if it can be determined
whether a give distribution is consistent with it.
 An example:
The expectation of the variable x is 2.87
and
p2 + p3 > 0.6

7. Principle of Maximum Entropy
General solution
 Entropy maximization with no testable information
 Given testable information
 Seek the probability distribution which maximizes information
entropy, subject to the constraints of the information.
 A constrained optimization problem. It can be typically solved
using the method of Lagrange Multipliers.

8. Principle of Maximum Entropy
General solution
 Question
 Seek the probability distribution which maximizes information
entropy, subject to some linear constraints.
 Mathematical problem
 Optimization Problem
 non-linear programming with linear constraints
 Idea
non-linear non-linear get result
programming programming
with linear with no
constraints constraints
• Lagrange • Let
• partial
multipliers derivative
differential
equals to 0

9. Principle of Maximum Entropy
General solution
 Constraints
 Some testable information I about a quantity x taking values in
{x1, x2,..., xn}. Express this information as m constraints on the
expectations of the functions fk, that is, we require our
probability distribution to satisfy
 Furthermore, the probabilities must sum to one, giving the
constraint
 Objective function

10. Principle of Maximum Entropy
General solution
 The probability distribution with maximum information
entropy subject to these constraints is
 The normalization constant is determined by
 The λk parameters are Lagrange multipliers whose
particular values are determined by the constraints
according to
 These m simultaneous equations do not generally possess a closed form
solution, and are usually solved by numerical methods.

11. Principle of Maximum Entropy
Training Model
 Generalized Iterative Scaling (GIS) (Darroch and
Ratcliff, 1972)
 Improved Iterative Scaling (IIS) (Della Pietra et al.,
1995)

12. Principle of Maximum Entropy
Training Model
 Generalized Iterative Scaling (GIS) (Darroch and
Ratcliff, 1972)
 Compute dj, j=1, …, k+1
 Initialize (any values, e.g., 0)
 Repeat until converge
• For each j
– Compute
– Compute
– Update

13. Principle of Maximum Entropy
Training Model
 Generalized Iterative Scaling (GIS) (Darroch and
Ratcliff, 1972)
 The running time of each iteration is O(NPA):
• N: the training set size
• P: the number of classes
• A: the average number of features that are active for a given
event (a, b).

14. Principle of Maximum Entropy
Relation to Maximum Likelihood
 Likelihood function
 P(x) is the distribution of estimation
 is the empirical distribution
 Log-Likelihood function

15. Principle of Maximum Entropy
Relation to Maximum Likelihood
 Theorem
 The model p*C with maximum entropy is the model in the
parametric family p(y|x) that maximizes the likelihood of the
training sample.
 Coincidence?
 Entropy – the measure of uncertainty
 Likelihood – the degree of identical to knowledge
 Maximum entropy - assume nothing about what is unknown
 Maximum likelihood – impartially understand the knowledge
Knowledge = complementary set of uncertainty

16. Principle of Maximum Entropy
MaxEnt methods and Bayesian
 Bayesian methods
p(H|DI) = p(H|I)p(D|HI) / p(D|I)
 H stands for some hypothesis whose truth we want to judge
 D for a set of data
 I for prior information
 Difference
 A single application of Bayes’ theorem gives us only a
probability, not a probability distribution
 MaxEnt gives us necessarily a probability distribution, not just a
probability.

17. Principle of Maximum Entropy
MaxEnt methods and Bayesian
 Difference (continue)
 Bayes’ theorem cannot determine the numerical value of any
probability directly from our information. To apply it one must first
use some other principle to translae information into numerical
values for p(H|I), p(D|HI), p(D|I)
 MaxEnt does not require for input the numerical values of any
probabilities on the hypothesis space.
 In common
 The updating of a state of knowledge
 Bayes’ rule and MaxEnt are completely compatible and can be
seen as special cases of the method of MaxEnt. (Giffin et al.
2007)

18. Applications
Maximum Entropy Model
 NLP: POS Tagging, Parsing, PP attachment, Text
Classification, LM, …
 POS Tagging
 Features
 Model

19. Applications
Maximum Entropy Model
 POS Tagging
 Tagging with MaxEnt Model
The conditional probability of a tag sequence t1,…, tn is
given a sentence w1,…, wn and contexts C1,…, Cn
 Model Estimation
• The model should reflect the data
– use the data to constrain the model
• What form should the constraints take?
– constrain the expected value of each feature

20. Applications
Maximum Entropy Model
 POS Tagging
 The Constraints
• Expected value of each feature must satisfy some constraint Ki
• A natural choice for Ki is the average empirical count
• derived from the training data (C1, t1), (C2, t2)…, (Cn, tn)

21. Applications
Maximum Entropy Model
 POS Tagging
 MaxEnt Model
• The constraints do not uniquely identify a model
• The maximum entropy model is the most uniform model
– makes no assumptions in addition to what we know from the data
• Set the weights to give the MaxEnt model satisfying the constraints
– use Generalised Iterative Scaling (GIS)
 Smoothing
• empirical counts for low frequency features can be unreliable
• Common smoothing technique is to ignore low frequency features
• Use a prior distribution on the parameters

22. Applications
Maximum Entropy Model
 Logistic regression
 Classification
• Linear regression for classification
• The problems of linear regression for classification

23. Applications
Maximum Entropy Model
 Logistic regression
 Hypothesis representation
• What function is used to represent our hypothesis in classification
• We want our classifier to output values between 0 and 1
• When using linear regression we did hθ(x) = (θT x)
• For classification hypothesis representation we do
hθ(x) = g((θT x))
Where we define g(z), z is a real number
g(z) = 1/(1 + e-z)
This is the sigmoid function, or the logistic function

24. Applications
Maximum Entropy Model
 Logistic regression
 Cost function for logistic regression
• Hypothesis representation
• Linear regression uses the following function to determine θ
• Define cost(hθ(xi), y) = 1/2(hθ(xi) - yi)2
• Redefine J(Θ)
• J(Θ) does not work for logistic regression, since it’s a non-convex function

25. Applications
Maximum Entropy Model
 Logistic regression
 Cost function for logistic regression
• A convex logistic regression cost function

26. Applications
Maximum Entropy Model
 Logistic regression
 Simplified cost function
• For binary classification problems y is always 0 or 1
• So we can write cost function is
cost(hθ, (x),y) = -ylog( hθ(x) ) - (1-y)log( 1- hθ(x) )
• So, in summary, our cost function for the θ parameters can be defined as
• Find parameters θ which minimize J(θ)

27. Applications
Maximum Entropy Model
 Logistic regression
 How to minimize the logistic regression cost function
 Use gradient descent to minimize J(θ)

28. Applications
Maximum Entropy Model
 Logistic regression
 Advanced optimization
• Good for large machine learning problems (e.g. huge feature set)
• What is gradient descent actually doing?
– compute J(θ) and the derivatives
– plug these values into gradient descent
• Alternatively, instead of gradient descent to minimize the cost function we
could use
– Conjugate gradient
– BFGS (Broyden-Fletcher-Goldfarb-Shanno)
– L-BFGS (Limited memory - BFGS)

29. Applications
Maximum Entropy Model
 Logistic regression
 Why do we chose this function when other cost functions exist?
• This cost function can be derived from statistics using the principle
of maximum likelihood estimation
– Note this does mean there's an underlying Gaussian assumption
relating to the distribution of features
• Also has the nice property that it's convex

30. Q&A
Thanks!

31. Reference
 Jaynes, E. T., 1988, 'The Relation of Bayesian and Maximum Entropy Methods',
in Maximum-Entropy and Bayesian Methods in Science and Engineering (Vol. 1),
Kluwer Academic Publishers, p. 25-26.
 https://www.coursera.org/course/ml
 The elements of statistical learning, 4.4.
 Kitamura, Y., 2006, Empirical Likelihood Methods in Econometrics: Theory and
Practice, Cowles Foundation Discussion Papers 1569, Cowles Foundation, Yale
University
 http://en.wikipedia.org/wiki/Principle_of_maximum_entropy
 Lazar, N., 2003, "Bayesian Empirical Likelihood", Biometrika, 90, 319-326.
 Giffin, A. and Caticha, A., 2007, Updating Probabilities with Data and Moments
 Guiasu, S. and Shenitzer, A., 1985, 'The principle of maximum entropy', The
Mathematical Intelligencer, 7(1), 42-48.
 Harremoës P. and Topsøe F., 2001, Maximum Entropy Fundamentals, Entropy,
3(3), 191-226.
 http://en.wikipedia.org/wiki/Logistic_regression