Example of implementation of Baum-Welch
I\'m trying to learn about Baum-Welch algorithm(to be used with a hidden markov model). I understand the basic theory of forward-backward models, but it would be nice for
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Here's some code that I wrote several years ago for a class, based on the presentation in Jurafsky/Martin (2nd edition, chapter 6, if you have access to the book). It's really not very good code, doesn't use numpy which it absolutely should, and it does some crap to have the arrays be 1-indexed instead of just tweaking the formulae to be 0-indexed, but, well, maybe it'll help. Baum-Welch is referred to as "forward-backward" in the code.
The example/test data is based on Jason Eisner's spreadsheet that implements some HMM-related algorithms. Note that the implemented version of the model uses an absorbing END state which other states have transition probabilities to, rather than assuming a pre-existing fixed sequence length.
(Also available as a gist if you prefer.)
hmm.py, half of which is testing code based on the following files:#!/usr/bin/env python """ CS 65 Lab #3 -- 5 Oct 2008 Dougal Sutherland Implements a hidden Markov model, based on Jurafsky + Martin's presentation, which is in turn based off work by Jason Eisner. We test our program with data from Eisner's spreadsheets. """ identity = lambda x: x class HiddenMarkovModel(object): """A hidden Markov model.""" def __init__(self, states, transitions, emissions, vocab): """ states - a list/tuple of states, e.g. ('start', 'hot', 'cold', 'end') start state needs to be first, end state last states are numbered by their order here transitions - the probabilities to go from one state to another transitions[from_state][to_state] = prob emissions - the probabilities of an observation for a given state emissions[state][observation] = prob vocab: a list/tuple of the names of observable values, in order """ self.states = states self.real_states = states[1:-1] self.start_state = 0 self.end_state = len(states) - 1 self.transitions = transitions self.emissions = emissions self.vocab = vocab # functions to get stuff one-indexed state_num = lambda self, n: self.states[n] state_nums = lambda self: xrange(1, len(self.real_states) + 1) vocab_num = lambda self, n: self.vocab[n - 1] vocab_nums = lambda self: xrange(1, len(self.vocab) + 1) num_for_vocab = lambda self, s: self.vocab.index(s) + 1 def transition(self, from_state, to_state): return self.transitions[from_state][to_state] def emission(self, state, observed): return self.emissions[state][observed - 1] # helper stuff def _normalize_observations(self, observations): return [None] + [self.num_for_vocab(o) if o.__class__ == str else o for o in observations] def _init_trellis(self, observed, forward=True, init_func=identity): trellis = [ [None for j in range(len(observed))] for i in range(len(self.real_states) + 1) ] if forward: v = lambda s: self.transition(0, s) * self.emission(s, observed[1]) else: v = lambda s: self.transition(s, self.end_state) init_pos = 1 if forward else -1 for state in self.state_nums(): trellis[state][init_pos] = init_func( v(state) ) return trellis def _follow_backpointers(self, trellis, start): # don't bother branching pointer = start[0] seq = [pointer, self.end_state] for t in reversed(xrange(1, len(trellis[1]))): val, backs = trellis[pointer][t] pointer = backs[0] seq.insert(0, pointer) return seq # actual algorithms def forward_prob(self, observations, return_trellis=False): """ Returns the probability of seeing the given `observations` sequence, using the Forward algorithm. """ observed = self._normalize_observations(observations) trellis = self._init_trellis(observed) for t in range(2, len(observed)): for state in self.state_nums(): trellis[state][t] = sum( self.transition(old_state, state) * self.emission(state, observed[t]) * trellis[old_state][t-1] for old_state in self.state_nums() ) final = sum(trellis[state][-1] * self.transition(state, -1) for state in self.state_nums()) return (final, trellis) if return_trellis else final def backward_prob(self, observations, return_trellis=False): """ Returns the probability of seeing the given `observations` sequence, using the Backward algorithm. """ observed = self._normalize_observations(observations) trellis = self._init_trellis(observed, forward=False) for t in reversed(range(1, len(observed) - 1)): for state in self.state_nums(): trellis[state][t] = sum( self.transition(state, next_state) * self.emission(next_state, observed[t+1]) * trellis[next_state][t+1] for next_state in self.state_nums() ) final = sum(self.transition(0, state) * self.emission(state, observed[1]) * trellis[state][1] for state in self.state_nums()) return (final, trellis) if return_trellis else final def viterbi_sequence(self, observations, return_trellis=False): """ Returns the most likely sequence of hidden states, for a given sequence of observations. Uses the Viterbi algorithm. """ observed = self._normalize_observations(observations) trellis = self._init_trellis(observed, init_func=lambda val: (val, [0])) for t in range(2, len(observed)): for state in self.state_nums(): emission_prob = self.emission(state, observed[t]) last = [(old_state, trellis[old_state][t-1][0] * \ self.transition(old_state, state) * \ emission_prob) for old_state in self.state_nums()] highest = max(last, key=lambda p: p[1])[1] backs = [s for s, val in last if val == highest] trellis[state][t] = (highest, backs) last = [(old_state, trellis[old_state][-1][0] * \ self.transition(old_state, self.end_state)) for old_state in self.state_nums()] highest = max(last, key = lambda p: p[1])[1] backs = [s for s, val in last if val == highest] seq = self._follow_backpointers(trellis, backs) return (seq, trellis) if return_trellis else seq def train_on_obs(self, observations, return_probs=False): """ Trains the model once, using the forward-backward algorithm. This function returns a new HMM instance rather than modifying this one. """ observed = self._normalize_observations(observations) forward_prob, forwards = self.forward_prob( observations, True) backward_prob, backwards = self.backward_prob(observations, True) # gamma values prob_of_state_at_time = posat = [None] + [ [0] + [forwards[state][t] * backwards[state][t] / forward_prob for t in range(1, len(observations)+1)] for state in self.state_nums()] # xi values prob_of_transition = pot = [None] + [ [None] + [ [0] + [forwards[state1][t] * self.transition(state1, state2) * self.emission(state2, observed[t+1]) * backwards[state2][t+1] / forward_prob for t in range(1, len(observations))] for state2 in self.state_nums()] for state1 in self.state_nums()] # new transition probabilities trans = [[0 for j in range(len(self.states))] for i in range(len(self.states))] trans[self.end_state][self.end_state] = 1 for state in self.state_nums(): state_prob = sum(posat[state]) trans[0][state] = posat[state][1] trans[state][-1] = posat[state][-1] / state_prob for oth in self.state_nums(): trans[state][oth] = sum(pot[state][oth]) / state_prob # new emission probabilities emit = [[0 for j in range(len(self.vocab))] for i in range(len(self.states))] for state in self.state_nums(): for output in range(1, len(self.vocab) + 1): n = sum(posat[state][t] for t in range(1, len(observations)+1) if observed[t] == output) emit[state][output-1] = n / sum(posat[state]) trained = HiddenMarkovModel(self.states, trans, emit, self.vocab) return (trained, posat, pot) if return_probs else trained # ====================== # = reading from files = # ====================== def normalize(string): if '#' in string: string = string[:string.index('#')] return string.strip() def make_hmm_from_file(f): def nextline(): line = f.readline() if line == '': # EOF return None else: return normalize(line) or nextline() n = int(nextline()) states = [nextline() for i in range(n)] # <3 list comprehension abuse num_vocab = int(nextline()) vocab = [nextline() for i in range(num_vocab)] transitions = [[float(x) for x in nextline().split()] for i in range(n)] emissions = [[float(x) for x in nextline().split()] for i in range(n)] assert nextline() is None return HiddenMarkovModel(states, transitions, emissions, vocab) def read_observations_from_file(f): return filter(lambda x: x, [normalize(line) for line in f.readlines()]) # ========= # = tests = # ========= import unittest class TestHMM(unittest.TestCase): def setUp(self): # it's complicated to pass args to a testcase, so just use globals self.hmm = make_hmm_from_file(file(HMM_FILENAME)) self.obs = read_observations_from_file(file(OBS_FILENAME)) def test_forward(self): prob, trellis = self.hmm.forward_prob(self.obs, True) self.assertAlmostEqual(prob, 9.1276e-19, 21) self.assertAlmostEqual(trellis[1][1], 0.1, 4) self.assertAlmostEqual(trellis[1][3], 0.00135, 5) self.assertAlmostEqual(trellis[1][6], 8.71549e-5, 9) self.assertAlmostEqual(trellis[1][13], 5.70827e-9, 9) self.assertAlmostEqual(trellis[1][20], 1.3157e-10, 14) self.assertAlmostEqual(trellis[1][27], 3.1912e-14, 13) self.assertAlmostEqual(trellis[1][33], 2.0498e-18, 22) self.assertAlmostEqual(trellis[2][1], 0.1, 4) self.assertAlmostEqual(trellis[2][3], 0.03591, 5) self.assertAlmostEqual(trellis[2][6], 5.30337e-4, 8) self.assertAlmostEqual(trellis[2][13], 1.37864e-7, 11) self.assertAlmostEqual(trellis[2][20], 2.7819e-12, 15) self.assertAlmostEqual(trellis[2][27], 4.6599e-15, 18) self.assertAlmostEqual(trellis[2][33], 7.0777e-18, 22) def test_backward(self): prob, trellis = self.hmm.backward_prob(self.obs, True) self.assertAlmostEqual(prob, 9.1276e-19, 21) self.assertAlmostEqual(trellis[1][1], 1.1780e-18, 22) self.assertAlmostEqual(trellis[1][3], 7.2496e-18, 22) self.assertAlmostEqual(trellis[1][6], 3.3422e-16, 20) self.assertAlmostEqual(trellis[1][13], 3.5380e-11, 15) self.assertAlmostEqual(trellis[1][20], 6.77837e-9, 14) self.assertAlmostEqual(trellis[1][27], 1.44877e-5, 10) self.assertAlmostEqual(trellis[1][33], 0.1, 4) self.assertAlmostEqual(trellis[2][1], 7.9496e-18, 22) self.assertAlmostEqual(trellis[2][3], 2.5145e-17, 21) self.assertAlmostEqual(trellis[2][6], 1.6662e-15, 19) self.assertAlmostEqual(trellis[2][13], 5.1558e-12, 16) self.assertAlmostEqual(trellis[2][20], 7.52345e-9, 14) self.assertAlmostEqual(trellis[2][27], 9.66609e-5, 9) self.assertAlmostEqual(trellis[2][33], 0.1, 4) def test_viterbi(self): path, trellis = self.hmm.viterbi_sequence(self.obs, True) self.assertEqual(path, [0] + [2]*13 + [1]*14 + [2]*6 + [3]) self.assertAlmostEqual(trellis[1][1] [0], 0.1, 4) self.assertAlmostEqual(trellis[1][6] [0], 5.62e-05, 7) self.assertAlmostEqual(trellis[1][7] [0], 4.50e-06, 8) self.assertAlmostEqual(trellis[1][16][0], 1.99e-09, 11) self.assertAlmostEqual(trellis[1][17][0], 3.18e-10, 12) self.assertAlmostEqual(trellis[1][23][0], 4.00e-13, 15) self.assertAlmostEqual(trellis[1][25][0], 1.26e-13, 15) self.assertAlmostEqual(trellis[1][29][0], 7.20e-17, 19) self.assertAlmostEqual(trellis[1][30][0], 1.15e-17, 19) self.assertAlmostEqual(trellis[1][32][0], 7.90e-19, 21) self.assertAlmostEqual(trellis[1][33][0], 1.26e-19, 21) self.assertAlmostEqual(trellis[2][ 1][0], 0.1, 4) self.assertAlmostEqual(trellis[2][ 4][0], 0.00502, 5) self.assertAlmostEqual(trellis[2][ 6][0], 0.00045, 5) self.assertAlmostEqual(trellis[2][12][0], 1.62e-07, 9) self.assertAlmostEqual(trellis[2][18][0], 3.18e-12, 14) self.assertAlmostEqual(trellis[2][19][0], 1.78e-12, 14) self.assertAlmostEqual(trellis[2][23][0], 5.00e-14, 16) self.assertAlmostEqual(trellis[2][28][0], 7.87e-16, 18) self.assertAlmostEqual(trellis[2][29][0], 4.41e-16, 18) self.assertAlmostEqual(trellis[2][30][0], 7.06e-17, 19) self.assertAlmostEqual(trellis[2][33][0], 1.01e-18, 20) def test_learning_probs(self): trained, gamma, xi = self.hmm.train_on_obs(self.obs, True) self.assertAlmostEqual(gamma[1][1], 0.129, 3) self.assertAlmostEqual(gamma[1][3], 0.011, 3) self.assertAlmostEqual(gamma[1][7], 0.022, 3) self.assertAlmostEqual(gamma[1][14], 0.887, 3) self.assertAlmostEqual(gamma[1][18], 0.994, 3) self.assertAlmostEqual(gamma[1][23], 0.961, 3) self.assertAlmostEqual(gamma[1][27], 0.507, 3) self.assertAlmostEqual(gamma[1][33], 0.225, 3) self.assertAlmostEqual(gamma[2][1], 0.871, 3) self.assertAlmostEqual(gamma[2][3], 0.989, 3) self.assertAlmostEqual(gamma[2][7], 0.978, 3) self.assertAlmostEqual(gamma[2][14], 0.113, 3) self.assertAlmostEqual(gamma[2][18], 0.006, 3) self.assertAlmostEqual(gamma[2][23], 0.039, 3) self.assertAlmostEqual(gamma[2][27], 0.493, 3) self.assertAlmostEqual(gamma[2][33], 0.775, 3) self.assertAlmostEqual(xi[1][1][1], 0.021, 3) self.assertAlmostEqual(xi[1][1][12], 0.128, 3) self.assertAlmostEqual(xi[1][1][32], 0.13, 3) self.assertAlmostEqual(xi[2][1][1], 0.003, 3) self.assertAlmostEqual(xi[2][1][22], 0.017, 3) self.assertAlmostEqual(xi[2][1][32], 0.095, 3) self.assertAlmostEqual(xi[1][2][4], 0.02, 3) self.assertAlmostEqual(xi[1][2][16], 0.018, 3) self.assertAlmostEqual(xi[1][2][29], 0.010, 3) self.assertAlmostEqual(xi[2][2][2], 0.972, 3) self.assertAlmostEqual(xi[2][2][12], 0.762, 3) self.assertAlmostEqual(xi[2][2][28], 0.907, 3) def test_learning_results(self): trained = self.hmm.train_on_obs(self.obs) tr = trained.transition self.assertAlmostEqual(tr(0, 0), 0, 5) self.assertAlmostEqual(tr(0, 1), 0.1291, 4) self.assertAlmostEqual(tr(0, 2), 0.8709, 4) self.assertAlmostEqual(tr(0, 3), 0, 4) self.assertAlmostEqual(tr(1, 0), 0, 5) self.assertAlmostEqual(tr(1, 1), 0.8757, 4) self.assertAlmostEqual(tr(1, 2), 0.1090, 4) self.assertAlmostEqual(tr(1, 3), 0.0153, 4) self.assertAlmostEqual(tr(2, 0), 0, 5) self.assertAlmostEqual(tr(2, 1), 0.0925, 4) self.assertAlmostEqual(tr(2, 2), 0.8652, 4) self.assertAlmostEqual(tr(2, 3), 0.0423, 4) self.assertAlmostEqual(tr(3, 0), 0, 5) self.assertAlmostEqual(tr(3, 1), 0, 4) self.assertAlmostEqual(tr(3, 2), 0, 4) self.assertAlmostEqual(tr(3, 3), 1, 4) em = trained.emission self.assertAlmostEqual(em(0, 1), 0, 4) self.assertAlmostEqual(em(0, 2), 0, 4) self.assertAlmostEqual(em(0, 3), 0, 4) self.assertAlmostEqual(em(1, 1), 0.6765, 4) self.assertAlmostEqual(em(1, 2), 0.2188, 4) self.assertAlmostEqual(em(1, 3), 0.1047, 4) self.assertAlmostEqual(em(2, 1), 0.0584, 4) self.assertAlmostEqual(em(2, 2), 0.4251, 4) self.assertAlmostEqual(em(2, 3), 0.5165, 4) self.assertAlmostEqual(em(3, 1), 0, 4) self.assertAlmostEqual(em(3, 2), 0, 4) self.assertAlmostEqual(em(3, 3), 0, 4) # train 9 more times for i in range(9): trained = trained.train_on_obs(self.obs) tr = trained.transition self.assertAlmostEqual(tr(0, 0), 0, 4) self.assertAlmostEqual(tr(0, 1), 0, 4) self.assertAlmostEqual(tr(0, 2), 1, 4) self.assertAlmostEqual(tr(0, 3), 0, 4) self.assertAlmostEqual(tr(1, 0), 0, 4) self.assertAlmostEqual(tr(1, 1), 0.9337, 4) self.assertAlmostEqual(tr(1, 2), 0.0663, 4) self.assertAlmostEqual(tr(1, 3), 0, 4) self.assertAlmostEqual(tr(2, 0), 0, 4) self.assertAlmostEqual(tr(2, 1), 0.0718, 4) self.assertAlmostEqual(tr(2, 2), 0.8650, 4) self.assertAlmostEqual(tr(2, 3), 0.0632, 4) self.assertAlmostEqual(tr(3, 0), 0, 4) self.assertAlmostEqual(tr(3, 1), 0, 4) self.assertAlmostEqual(tr(3, 2), 0, 4) self.assertAlmostEqual(tr(3, 3), 1, 4) em = trained.emission self.assertAlmostEqual(em(0, 1), 0, 4) self.assertAlmostEqual(em(0, 2), 0, 4) self.assertAlmostEqual(em(0, 3), 0, 4) self.assertAlmostEqual(em(1, 1), 0.6407, 4) self.assertAlmostEqual(em(1, 2), 0.1481, 4) self.assertAlmostEqual(em(1, 3), 0.2112, 4) self.assertAlmostEqual(em(2, 1), 0.00016,5) self.assertAlmostEqual(em(2, 2), 0.5341, 4) self.assertAlmostEqual(em(2, 3), 0.4657, 4) self.assertAlmostEqual(em(3, 1), 0, 4) self.assertAlmostEqual(em(3, 2), 0, 4) self.assertAlmostEqual(em(3, 3), 0, 4) if __name__ == '__main__': import sys HMM_FILENAME = sys.argv[1] if len(sys.argv) >= 2 else 'example.hmm' OBS_FILENAME = sys.argv[2] if len(sys.argv) >= 3 else 'observations.txt' unittest.main()observations.txt, a sequence of observations for testing:2 3 3 2 3 2 3 2 2 3 1 3 3 1 1 1 2 1 1 1 3 1 2 1 1 1 2 3 3 2 3 2 2example.hmm, the model used to generate the data4 # number of states START COLD HOT END 3 # size of vocab 1 2 3 # transition matrix 0.0 0.5 0.5 0.0 # from start 0.0 0.8 0.1 0.1 # from cold 0.0 0.1 0.8 0.1 # from hot 0.0 0.0 0.0 1.0 # from end # emission matrix 0.0 0.0 0.0 # from start 0.7 0.2 0.1 # from cold 0.1 0.2 0.7 # from hot 0.0 0.0 0.0 # from end讨论(0)
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