Hugo W.L. ter Doest

# NAME

MaxEntropy - Perl5 module for Maximum Entropy Modeling and Feature Induction

# SYNOPSIS

  use Statistics::MaxEntropy;

# debugging messages; default 0
$Statistics::MaxEntropy::debug = 0; # maximum number of iterations for IIS; default 100$Statistics::MaxEntropy::NEWTON_max_it = 100;

# minimal distance between new and old x for Newton's method;
# default 0.001
$Statistics::MaxEntropy::NEWTON_min = 0.001; # maximum number of iterations for Newton's method; default 100$Statistics::MaxEntropy::KL_max_it = 100;

# minimal distance between new and old x; default 0.001
$Statistics::MaxEntropy::KL_min = 0.001; # the size of Monte Carlo samples; default 1000$Statistics::MaxEntropy::SAMPLE_size = 1000;

# creation of a new event space from an events file
$events = Statistics::MaxEntropy::new($vectype, $file); # Generalised Iterative Scaling, "corpus" means no sampling$events->scale("corpus", "gis");

# Improved Iterative Scaling, "mc" means Monte Carlo sampling
$events->scale("mc", "iis"); # Feature Induction algorithm, also see Statistics::Candidates POD$candidates = Statistics::Candidates->new($candidates_file);$events->fi("iis", $candidates,$nr_to_add, "mc");

# writing new events, candidates, and parameters files
$events->write($some_other_file);
$events->write_parameters($file);
$events->write_parameters_with_names($file);

# dump/undump the event space to/from a file
$events->dump($file);
$events->undump($file);

# DESCRIPTION

This module is an implementation of the Generalised and Improved Iterative Scaling (GIS, IIS) algorithms and the Feature Induction (FI) algorithm as defined in (Darroch and Ratcliff 1972) and (Della Pietra et al. 1997). The purpose of the scaling algorithms is to find the maximum entropy distribution given a set of events and (optionally) an initial distribution. Also a set of candidate features may be specified; then the FI algorithm may be applied to find and add the candidate feature(s) that give the largest gain' in terms of Kullback Leibler divergence when it is added to the current set of features.

Events are specified in terms of a set of feature functions (properties) f_1...f_k that map each event to {0,1} or to the natural numbers: an event is a string of bits. In addition, the frequency of each event is given. We assume the event space to have a probability distribution that can be described by

If you have a Perl earlier than 5.005, then you need Data::Dumper module by Gurusamy Sarathy. It can be obtained from CPAN just like this module.

## CONFIGURATION VARIABLES

$Statistics::MaxEntropy::debug If set to 1, lots of debug information, and intermediate results will be output. Default: 0 $Statistics::MaxEntropy::NEWTON_max_it

Sets the maximum number of iterations in Newton's method. Newton's method is applied to find the new parameters \alpha_i of the features f_i. Default: 100.

$Statistics::MaxEntropy::NEWTON_min Sets the minimum difference between x' and x in Newton's method (used for computing parameter updates in IIS); if either the maximum number of iterations is reached or the difference between x' and x is small enough, the iteration is stopped. Default: 0.001. Sometimes features have Infinity or -Infinity as a solution; these features are excluded from future iterations. $Statistics::MaxEntropy::KL_max_it

Sets the maximum number of iterations applied in the IIS algorithm. Default: 100.

$Statistics::MaxEntropy::KL_min Sets the minimum difference between KL divergences of two distributions in the IIS algorithm; if either the maximum number of iterations is reached or the difference between the divergences is enough, the iteration is stopped. Default: 0.001. $Statistics::MaxEntropy::SAMPLE_size

Determines the number of (unique) events a sample should contain. Only makes sense if for sampling "mc" is selected (see below). Its default is 1000.

## METHODS

new
 $vectype = "binary"; # or "integer"$events = Statistics::MaxEntropy::new($vectype,$events_file);

A new event space is created, and the events are read from $file. The events file is not required. The syntax of events files is described in "FILE SYNTAX". The $vectype parameter specifies how nonzero feature values should be interpreted as binary values or not.

write
 $events->write($file);

Writes the events to a file. Its syntax is described in "FILE SYNTAX".

scale
 $events->scale($sample, $scaler); If $scaler equals "gis", the Generalised Iterative Scaling algorithm (Darroch and Ratcliff 1972) is applied on the event space; $scaler equals "iis", the Improved Iterative Scaling Algorithm (Della Pietra et al. 1997) is used. If $sample is "corpus", there is no sampling done to re-estimate the parameters (the events previously read are considered a good sample); if it equals "mc" Monte Carlo (Metropolis-Hastings) sampling is performed to obtain a random sample; if $sample is "enum" the complete event space is enumerated. fi  fi($scaler, $candidates,$nr_to_add, $sampling); Calls the Feature Induction algorithm. The parameter $nr_to_add is for the number of candidates it should add. If this number is greater than the number of candidates, all candidates are added. Meaningfull values for $scaler are "gis" and "iis"; default is "gis" (see previous item). $sampling should be one of "corpus", "mc", "enum". $candidates should be in the Statistics::Candidates class: $candidates = Statistics::Candidates->new($file); write_parameters $events->write_parameters($file); write_parameters_with_names $events->write_parameters_with_names($file); dump $events->dump($file); $events is written to $file using Data::Dumper. undump $events = Statistics::MaxEntropy->undump($file); The contents of file $file is read and eval'ed into \$events.

# FILE SYNTAX

Lines that start with a # and empty lines are ignored.

Below we give the syntax of in and output files.

## EVENTS FILE (input/output)

Syntax of the event file (n features, and m events); the following holds for features:

• each line is an event;

• each column represents a feature function; the co-domain of a feature function is N;

• constant features (i.e. columns that are completely 0 or 1) are forbidden;

• 2 or more events should be specified (this is in fact a consequence of the previous requirement;

The frequency of each event precedes the feature columns. Features are indexed from right to left. Each f_ij and each freq_i is an integer:

    name_1 <tab> name_2 ... name_n-1 <tab> name_n <cr>
freq_1 <white> f_11 <white> f12 ... f_1n-1 <white> f_1n <cr>
.                          .
.                          .
.                          .
freq_i <white> f_i1 <white> fi2 ... f_in-1 <white> f_in <cr>
.                          .
.                          .
.                          .
freq_m <white> f_m1 <white> fm2 ... f_mn-1 <white> f_mn

(m events, n features) The feature names are separated by tabs, not white space. The line containing the feature names will be split on tabs; this implies that (non-tab) white space may be part of the feature names. The distinction between binary and integer feature functions is a matter of interpretation. If vector type "binary" is used, nonzero values are interpreted as 1.

## PARAMETERS FILE (input/output)

Syntax of the initial parameters file; one parameter per line:

    par_1 <cr>
.
.
.
par_i <cr>
.
.
.
par_n

The syntax of the output distribution is the same. The alternative procedure for saving parameters to a file write_parameters_with_names writes files that have the following syntax

    n <cr>
name_1 <tab> par_1 <cr>
.
.
.
name_i <tab> par_i <cr>
.
.
.
name_n <tab> par_n <cr>
bitmask

where bitmask can be used to tell other programs what features to use in computing probabilities. Features that were ignored during scaling or because they are constant functions, receive a 0 bit.

## DUMP FILE (input/output)

A dump file contains the event space (which is a hash blessed into class Statistics::MaxEntropy) as a Perl expression that can be evaluated with eval.

It's slow.

# DIAGNOSTICS

The module dies with an appropriate message if

• it cannot open a specified events file;

• if you specified a constant feature function (in the events file or the candidates file);

• If the events file, candidates file, or the parameters file is not consistent. Possible causes are (a.o.):

• insufficient or too many features for some event;

• inconsistent candidate lines;

• insufficient, or to many event lines in the candidates file.

• it is tried to do feature induction with integer feature functions.

The module captures SIGQUIT and SIGINT. On a SIGINT (typically <CONTROL-C> it will dump the current event space(s) and die. If a SIGQUIT (<CONTROL-BACKSLASH>) occurs it dumps the current event space as soon as possible after the first iteration it finishes.

# REFERENCES

(Abney 1997)

Steven P. Abney, Stochastic Attribute Value Grammar, Computational Linguistics 23(4).

(Darroch and Ratcliff 1972)

J. Darroch and D. Ratcliff, Generalised Iterative Scaling for log-linear models, Ann. Math. Statist., 43, 1470-1480, 1972.

(Jaynes 1983)

E.T. Jaynes, Papers on probability, statistics, and statistical physics. Ed.: R.D. Rosenkrantz. Kluwer Academic Publishers, 1983.

(Jaynes 1997)

E.T. Jaynes, Probability theory: the logic of science, 1997, unpublished manuscript. URL:http://omega.math.albany.edu:8008/JaynesBook.html

(Della Pietra et al. 1997)

Stephen Della Pietra, Vincent Della Pietra, and John Lafferty, Inducing features of random fields, In: Transactions Pattern Analysis and Machine Intelligence, 19(4), April 1997.

Version 1.0.

# AUTHOR

Statistics::MaxEntropy` comes with ABSOLUTELY NO WARRANTY and may be copied only under the terms of the GNU Library General Public License (version 2, or later), which may be found in the distribution.