Numeric.MaxEnt

The maximum entropy method, or MAXENT, is variational approach for computing probability distributions given a list of moment, or expected value, constraints.

Here are some links for background info. A good overview of applications: http://cmm.cit.nih.gov/maxent/letsgo.html On the idea of maximum entropy in general: http://en.wikipedia.org/wiki/Principle_of_maximum_entropy

Use this package to compute discrete maximum entropy distributions over a list of values and list of constraints.

Here is a the example from Probability the Logic of Science

>>> maxent ([1,2,3], [average 1.5])
Right [0.61, 0.26, 0.11]

The classic dice example

>>> maxent ([1,2,3,4,5,6], [average 4.5])
Right [.05, .07, 0.11, 0.16, 0.23, 0.34]

One can use different constraints besides the average value there.

As for why you want to maximize the entropy to find the probability constraint, I will say this for now. In the case of the average constraint it is a kin to choosing a integer partition with the most interger compositions. I doubt that makes any sense, but I will try to explain more with a blog post soon.

Documentation

type Constraint a = (ExpectationFunction a, a)

Constraint type. A function and the constant it equals.

Think of it as the pair (f, c) in the constraint

     Σ pₐ f(xₐ) = c

such that we are summing over all values .

For example, for a variance constraint the f would be (\x -> x*x) and c would be the variance.

type ExpectationFunction a = a -> a

A function that takes an index and value and returns a value. See average and variance for examples.

constraint :: RealFloat a => ExpectationFunction a -> a -> Constraint a

average :: RealFloat a => a -> Constraint a

variance :: RealFloat a => a -> Constraint a

Classic moment based

maxent

Arguments

:: (forall a. RealFloat a => ([a], [Constraint a]))

A pair of values that the distributions is over and the constraints

-> Either (Result, Statistics) [Double]

Either the a discription of what wrong or the probability distribution

Discrete maximum entropy solver where the constraints are all moment constraints.

General

type GeneralConstraint a = [a] -> [a] -> a

generalMaxent

Arguments

:: (forall a. RealFloat a => ([a], [(GeneralConstraint a, a)]))

A pair of values that the distributions is over and the constraints

-> Either (Result, Statistics) [Double]

Either the a discription of what wrong or the probability distribution

Most general solver This is the slowest but most flexible method. Although, I haven't tried using much...

Linear

data LinearConstraints a

Constructors

LC

Fields

matrix :: [[a]]

output :: [a]

Instances

Eq a => Eq (LinearConstraints a)

Show a => Show (LinearConstraints a)

linear

Arguments

:: Double

Tolerance for the numerical solver

-> (forall a. RealFloat a => LinearConstraints a)

The matrix A and column vector b

-> Either (Result, Statistics) [Double]

Either the a discription of what wrong or the probability distribution

This is for the linear case Ax = b x in this situation is the vector of probablities.

Consider the 1 dimensional circular convolution using hmatrix.

>>> import Numeric.LinearAlgebra
>>> fromLists [[0.68, 0.22, 0.1], [0.1, 0.68, 0.22], [0.22, 0.1, 0.68]] <> fromLists [[0.2], [0.5], [0.3]]
(3><1) [0.276, 0.426, 0.298]   

Now if we were given just the convolution and the output, we can use linear to infer the input.

>>> linear 3.0e-17 $ LC [[0.68, 0.22, 0.1], [0.1, 0.68, 0.22], [0.22, 0.1, 0.68]] [0.276, 0.426, 0.298]
Right [0.20000000000000004,0.4999999999999999,0.3]