Hidden Markov Models for POS-tagging in Python

# Hidden Markov Models in Python

# Katrin Erk, March 2013 updated March 2016

# This HMM addresses the problem of part-of-speech tagging. It estimates

# the probability of a tag sequence for a given word sequence as follows:

# Say words = w1....wN

# and tags = t1..tN

# then

# P(tags | words) is_proportional_to product P(ti | t{i-1}) P(wi | ti)

# To find the best tag sequence for a given sequence of words,

# we want to find the tag sequence that has the maximum P(tags | words)

import nltk

import sys

from nltk.corpus import brown

# Estimating P(wi | ti) from corpus data using Maximum Likelihood Estimation (MLE):

# P(wi | ti) = count(wi, ti) / count(ti)

# We add an artificial "start" tag at the beginning of each sentence, and

# We add an artificial "end" tag at the end of each sentence.

# So we start out with the brown tagged sentences,

# add the two artificial tags,

# and then make one long list of all the tag/word pairs.

brown_tags_words = [ ]

for sent in brown.tagged_sents():

# sent is a list of word/tag pairs

# add START/START at the beginning

brown_tags_words.append( ("START", "START") )

# then all the tag/word pairs for the word/tag pairs in the sentence.

# shorten tags to 2 characters each

brown_tags_words.extend([ (tag[:2], word) for (word, tag) in sent ])

# then END/END

brown_tags_words.append( ("END", "END") )

# conditional frequency distribution

cfd_tagwords = nltk.ConditionalFreqDist(brown_tags_words)

# conditional probability distribution

cpd_tagwords = nltk.ConditionalProbDist(cfd_tagwords, nltk.MLEProbDist)

print("The probability of an adjective (JJ) being 'new' is", cpd_tagwords["JJ"].prob("new"))

print("The probability of a verb (VB) being 'duck' is", cpd_tagwords["VB"].prob("duck"))

# Estimating P(ti | t{i-1}) from corpus data using Maximum Likelihood Estimation (MLE):

# P(ti | t{i-1}) = count(t{i-1}, ti) / count(t{i-1})

brown_tags = [tag for (tag, word) in brown_tags_words ]

# make conditional frequency distribution:

# count(t{i-1} ti)

cfd_tags= nltk.ConditionalFreqDist(nltk.bigrams(brown_tags))

# make conditional probability distribution, using

# maximum likelihood estimate:

# P(ti | t{i-1})

cpd_tags = nltk.ConditionalProbDist(cfd_tags, nltk.MLEProbDist)

print("If we have just seen 'DT', the probability of 'NN' is", cpd_tags["DT"].prob("NN"))

print( "If we have just seen 'VB', the probability of 'JJ' is", cpd_tags["VB"].prob("DT"))

print( "If we have just seen 'VB', the probability of 'NN' is", cpd_tags["VB"].prob("NN"))

###

# putting things together:

# what is the probability of the tag sequence "PP VB TO VB" for the word sequence "I want to race"?

# It is

# P(START) * P(PP|START) * P(I | PP) *

# P(VB | PP) * P(want | VB) *

# P(TO | VB) * P(to | TO) *

# P(VB | TO) * P(race | VB) *

# P(END | VB)

# We leave aside P(START) for now.

prob_tagsequence = cpd_tags["START"].prob("PP") * cpd_tagwords["PP"].prob("I") * \

cpd_tags["PP"].prob("VB") * cpd_tagwords["VB"].prob("want") * \

cpd_tags["VB"].prob("TO") * cpd_tagwords["TO"].prob("to") * \

cpd_tags["TO"].prob("VB") * cpd_tagwords["VB"].prob("race") * \

cpd_tags["VB"].prob("END")

print( "The probability of the tag sequence 'START PP VB TO VB END' for 'I want to race' is:", prob_tagsequence)

prob_tagsequence = cpd_tags["START"].prob("PP") * cpd_tagwords["PP"].prob("I") * \

cpd_tags["PP"].prob("VB") * cpd_tagwords["VB"].prob("saw") * \

cpd_tags["VB"].prob("PP") * cpd_tagwords["PP"].prob("her") * \

cpd_tags["PP"].prob("NN") * cpd_tagwords["NN"].prob("duck") * \

cpd_tags["NN"].prob("END")

print( "The probability of the tag sequence 'START PP VB PP NN END' for 'I saw her duck' is:", prob_tagsequence)

prob_tagsequence = cpd_tags["START"].prob("PP") * cpd_tagwords["PP"].prob("I") * \

cpd_tags["PP"].prob("VB") * cpd_tagwords["VB"].prob("saw") * \

cpd_tags["VB"].prob("PP") * cpd_tagwords["PP"].prob("her") * \

cpd_tags["PP"].prob("VB") * cpd_tagwords["VB"].prob("duck") * \

cpd_tags["VB"].prob("END")

print( "The probability of the tag sequence 'START PP VB PP VB END' for 'I saw her duck' is:", prob_tagsequence)

#####

# Viterbi:

# If we have a word sequence, what is the best tag sequence?

# The method above lets us determine the probability for a single tag sequence.

# But in order to find the best tag sequence, we need the probability

# for _all_ tag sequence.

# What Viterbi gives us is just a good way of computing all those many probabilities

# as fast as possible.

# what is the list of all tags?

distinct_tags = set(brown_tags)

sentence = ["I", "want", "to", "race" ]

#sentence = ["I", "saw", "her", "duck" ]

sentlen = len(sentence)

# viterbi:

# for each step i in 1 .. sentlen,

# store a dictionary

# that maps each tag X

# to the probability of the best tag sequence of length i that ends in X

viterbi = [ ]

# backpointer:

# for each step i in 1..sentlen,

# store a dictionary

# that maps each tag X

# to the previous tag in the best tag sequence of length i that ends in X

backpointer = [ ]

first_viterbi = { }

first_backpointer = { }

for tag in distinct_tags:

# don't record anything for the START tag

if tag == "START": continue

first_viterbi[ tag ] = cpd_tags["START"].prob(tag) * cpd_tagwords[tag].prob( sentence[0] )

first_backpointer[ tag ] = "START"

print(first_viterbi)

print(first_backpointer)

viterbi.append(first_viterbi)

backpointer.append(first_backpointer)

currbest = max(first_viterbi.keys(), key = lambda tag: first_viterbi[ tag ])

print( "Word", "'" + sentence[0] + "'", "current best two-tag sequence:", first_backpointer[ currbest], currbest)

# print( "Word", "'" + sentence[0] + "'", "current best tag:", currbest)

for wordindex in range(1, len(sentence)):

this_viterbi = { }

this_backpointer = { }

prev_viterbi = viterbi[-1]

for tag in distinct_tags:

# don't record anything for the START tag

if tag == "START": continue

# if this tag is X and the current word is w, then

# find the previous tag Y such that

# the best tag sequence that ends in X

# actually ends in Y X

# that is, the Y that maximizes

# prev_viterbi[ Y ] * P(X | Y) * P( w | X)

# The following command has the same notation

# that you saw in the sorted() command.

best_previous = max(prev_viterbi.keys(),

key = lambda prevtag: \

prev_viterbi[ prevtag ] * cpd_tags[prevtag].prob(tag) * cpd_tagwords[tag].prob(sentence[wordindex]))

# Instead, we can also use the following longer code:

# best_previous = None

# best_prob = 0.0

# for prevtag in distinct_tags:

# prob = prev_viterbi[ prevtag ] * cpd_tags[prevtag].prob(tag) * cpd_tagwords[tag].prob(sentence[wordindex])

# if prob > best_prob:

# best_previous= prevtag

# best_prob = prob

this_viterbi[ tag ] = prev_viterbi[ best_previous] * \

cpd_tags[ best_previous ].prob(tag) * cpd_tagwords[ tag].prob(sentence[wordindex])

this_backpointer[ tag ] = best_previous

currbest = max(this_viterbi.keys(), key = lambda tag: this_viterbi[ tag ])

print( "Word", "'" + sentence[ wordindex] + "'", "current best two-tag sequence:", this_backpointer[ currbest], currbest)

# print( "Word", "'" + sentence[ wordindex] + "'", "current best tag:", currbest)

# done with all tags in this iteration

# so store the current viterbi step

viterbi.append(this_viterbi)

backpointer.append(this_backpointer)

# done with all words in the sentence.

# now find the probability of each tag

# to have "END" as the next tag,

# and use that to find the overall best sequence

prev_viterbi = viterbi[-1]

best_previous = max(prev_viterbi.keys(),

key = lambda prevtag: prev_viterbi[ prevtag ] * cpd_tags[prevtag].prob("END"))

prob_tagsequence = prev_viterbi[ best_previous ] * cpd_tags[ best_previous].prob("END")

# best tagsequence: we store this in reverse for now, will invert later

best_tagsequence = [ "END", best_previous ]

# invert the list of backpointers

backpointer.reverse()

# go backwards through the list of backpointers

# (or in this case forward, because we have inverter the backpointer list)

# in each case:

# the following best tag is the one listed under

# the backpointer for the current best tag

current_best_tag = best_previous

for bp in backpointer:

best_tagsequence.append(bp[current_best_tag])

current_best_tag = bp[current_best_tag]

best_tagsequence.reverse()

print( "The sentence was:", end = " ")

for w in sentence: print( w, end = " ")

print("\n")

print( "The best tag sequence is:", end = " ")

for t in best_tagsequence: print (t, end = " ")

print("\n")

print( "The probability of the best tag sequence is:", prob_tagsequence)

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