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#!/usr/bin/env python2.7
# ----------------------------------
#
# Module pylib_probabilities.py
#
# Functions and classes for computing probability estimates and
# entropy for various datatypes.
#
# Copyright 2003 Stephan Schulz, schulz@informatik.tu-muenchen.de
#
# This code is part of the support structure for the equational
# theorem prover E. Visit
#
# http://www4.informatik.tu-muenchen.de/~schulz/WORK/eprover.html
#
# for more information.
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program ; if not, write to the Free Software
# Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA 02111-1307 USA
#
# The original copyright holder can be contacted as
#
# Stephan Schulz (I4)
# Technische Universitaet Muenchen
# Institut fuer Informatik
# Boltzmannstrasse 3
# Garching bei Muenchen
# Germany
#
# or via email (address above).
from types import *
import pylib_basics
def rel_frequency(subset_size, total_size, omega_size=None):
"""
Estimate probability of a subset, given the the total
number of draws and the subset size, and the size of omega (the
probability space).
"""
return subset_size/float(total_size)
def rule_of_succession(subset_size, total_size, omega_size):
"""
Same thing, but use rule of succession instead of relative
frequency.
"""
return (subset_size+1)/float((total_size+omega_size))
def prob_vector(abs_freq_vector, sample_size, omega_size):
"""
Compute vector of probability estimates for a vector of absolute
frequencies.
"""
assert type(abs_freq_vector) == ListType
pv = map(lambda x:rule_of_succession(x,sample_size,omega_size), abs_freq_vector)
#print "prob_vector:", abs_freq_vector
#print "prob_vector:", pv
#print "prob_vector:", sum(pv)
return pv
def info_content(p):
"""
Return the information content of an event with probability p.
"""
if p==0:
return 0 # Hack for bad estimators
return -pylib_basics.log2(p)
def compute_entropy(prob_distrib):
"""
Compute the entropy for a list of relative frequencies.
"""
assert type(prob_distrib) == ListType
info = map(info_content, prob_distrib)
weighted_info = map(lambda x,y:x*y,prob_distrib,info)
return reduce(lambda x,y:x+y, weighted_info, 0)
def compute_entropy_absdistrib(distrib):
"""
As above, but use a list of absolute frequencies, one entry per
elementary event!
"""
omega_size = len(distrib)
sample_size = sum(distrib)
prob_distrib = prob_vector(distrib,sample_size, omega_size)
return compute_entropy(prob_distrib)
def remainder_entropy(sample, estimator=rel_frequency):
"""
Return the remainder entropy H(A|B) of a sample, where a sample
is a list of lists, outer lists represent the result of B, inner
list the results of A|B
"""
assert type(sample) == ListType
sample_size = reduce(lambda x,y:x+y, map(sum, sample),0)
outer_omega_size = len(sample)
#print "remainder_entropy:", sample
outer_abs_freq = map(sum, sample)
outer_probs = prob_vector(outer_abs_freq,estimator)
#print "remainder_entropy:", outer_abs_freq
#print "remainder_entropy:", outer_probs, sum(outer_probs)
weighted_entropies= map(lambda x, y: (y,
compute_entropy(x, estimator)),
sample,
outer_probs)
#print weighted_entropies
c_ent = reduce(lambda x,y:x+y, map(lambda x:x[0]*x[1], weighted_entropies),0)
return c_ent
def compute_entropies(distrib):
"""
Distrib is a list of lists.
Assume two experiments, A with outcomes A1...An and B with
outcomes B1...Bn. We give the distribution of results in a sample
as follows: The elements with Ai are in the ith list of
distrib. The elements with Bj are represented by the jth entry in
all the Ai. Return the entropy of A, the entropy of B, and the
remainder entropy of B|A.
"""
omega_size = len(distrib*len(distrib[0]))
sample_size = sum(map(sum, distrib))
# print sample_size, omega_size
prob_distrib = map(lambda x:prob_vector(x,sample_size, omega_size),distrib)
part_distrib = map(sum, prob_distrib)
part_entropy = compute_entropy(part_distrib)
# print "part_distrib:", part_distrib, sum(part_distrib)
class_distrib = reduce(lambda l1, l2:map(lambda x,y:x+y, l1, l2),
prob_distrib)
class_entropy = compute_entropy(class_distrib)
# print "Class_distrib:", class_distrib, sum(class_distrib)
remainder_entropy = 0;
for i in range(0, len(distrib)):
remainder_entropy += (part_distrib[i]*
compute_entropy_absdistrib(distrib[i]))
return part_entropy,class_entropy,remainder_entropy
def info_gain(a_priori_entropy, remainder_entropy):
"""
Compute the expected information gain given a-priory entropy of a
distribution and remainder entropy.
"""
#assert a_priori_entropy>=remainder_entropy
if a_priori_entropy < remainder_entropy:
# Allow for rounding errors
if pylib_basics.verbose():
print "Warning: Remainder enropy > a-priori-entropy",
print remainder_entropy, ">", a_priori_entropy
remainder_entropy = a_priori_entropy
return a_priori_entropy - remainder_entropy
def rel_info_gain(a_priori_entropy, remainder_entropy, test_cost_entropy):
"""
Compute the relative information gain given a-priory entropy,
remainder entropy, and the entropy of the test used to parition
the sample.
"""
g = info_gain(a_priori_entropy, remainder_entropy)
if g == test_cost_entropy:
return pylib_basics.Infinity
else:
return g/(test_cost_entropy - g)
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