Baum-Welch实现示例

Question

我试图学习Baum-Welch算法（用于隐马尔可夫模型）。我了解前向-后向模型的基本理论，但是有人能用一些代码来帮助解释它是很好的（我发现读代码更容易，因为我可以到处玩来理解它）。我检查了github和bitbucket，没有发现任何容易理解的东西。

网络上有很多HMM教程，但是概率已经提供，或者在拼写检查的情况下，添加单词的出现来建立模型。如果有人能用仅有的观测数据来创建Baum-Welch模型，那就太酷了。例如，在http://en.wikipedia.org/wiki/Hidden_Markov_model#A_concrete_example中，如果您只有：

states = ('Rainy', 'Sunny')

observations = ('walk', 'shop', 'clean')

这只是一个例子，我认为任何能解释它的例子，我们都可以用好的来更好地理解它。我有一个我正在试图解决的特定问题，但我认为展示代码，让人们可以从中学习并应用到他们自己的问题上（如果这是不可接受的，我可以发布我自己的问题）可能更有价值。不过，如果可能的话，用python（或java）编写它会很好。

提前谢谢！

Answer 1

以下是几年前我为一门课写的一些代码，基于Jurafsky/Martin中的演示（第二版，第6章，如果你可以阅读这本书的话）。这真的不是很好的代码，没有使用它绝对应该使用的numpy，让数组被1索引而不是仅仅将公式调整为0索引是很糟糕的，但是，好吧，也许它会有帮助。Baum Welch在代码中被称为“向前向后”。

示例/测试数据基于实现一些HMM相关算法的Jason Eisner's spreadsheet。注意，模型的实现版本使用了吸收结束状态，而其他状态具有转换概率，而不是假设预先存在的固定序列长度。

（如果您愿意，也可以使用as a gist。）

hmm.py，其中一半是基于以下文件测试代码：

#!/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，用于测试的观察序列：

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
2

example.hmm，用于生成数据的模型

4 # 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