Python Implementation of Viterbi Algorithm

Here's mine. Its paraphrased directly from the psuedocode implemenation from wikipedia. It uses numpy for conveince of their ndarray but is otherwise a pure python3 implementation.

import numpy as np

def viterbi(y, A, B, Pi=None):
    """
    Return the MAP estimate of state trajectory of Hidden Markov Model.

    Parameters
    ----------
    y : array (T,)
        Observation state sequence. int dtype.
    A : array (K, K)
        State transition matrix. See HiddenMarkovModel.state_transition  for
        details.
    B : array (K, M)
        Emission matrix. See HiddenMarkovModel.emission for details.
    Pi: optional, (K,)
        Initial state probabilities: Pi[i] is the probability x[0] == i. If
        None, uniform initial distribution is assumed (Pi[:] == 1/K).

    Returns
    -------
    x : array (T,)
        Maximum a posteriori probability estimate of hidden state trajectory,
        conditioned on observation sequence y under the model parameters A, B,
        Pi.
    T1: array (K, T)
        the probability of the most likely path so far
    T2: array (K, T)
        the x_j-1 of the most likely path so far
    """
    # Cardinality of the state space
    K = A.shape[0]
    # Initialize the priors with default (uniform dist) if not given by caller
    Pi = Pi if Pi is not None else np.full(K, 1 / K)
    T = len(y)
    T1 = np.empty((K, T), 'd')
    T2 = np.empty((K, T), 'B')

    # Initilaize the tracking tables from first observation
    T1[:, 0] = Pi * B[:, y[0]]
    T2[:, 0] = 0

    # Iterate throught the observations updating the tracking tables
    for i in range(1, T):
        T1[:, i] = np.max(T1[:, i - 1] * A.T * B[np.newaxis, :, y[i]].T, 1)
        T2[:, i] = np.argmax(T1[:, i - 1] * A.T, 1)

    # Build the output, optimal model trajectory
    x = np.empty(T, 'B')
    x[-1] = np.argmax(T1[:, T - 1])
    for i in reversed(range(1, T)):
        x[i - 1] = T2[x[i], i]

    return x, T1, T2

I found the following code in the example repository of Artificial Intelligence: A Modern Approach. Is something like this what you're looking for?

def viterbi_segment(text, P):
    """Find the best segmentation of the string of characters, given the
    UnigramTextModel P."""
    # best[i] = best probability for text[0:i]
    # words[i] = best word ending at position i
    n = len(text)
    words = [''] + list(text)
    best = [1.0] + [0.0] * n
    ## Fill in the vectors best, words via dynamic programming
    for i in range(n+1):
        for j in range(0, i):
            w = text[j:i]
            if P[w] * best[i - len(w)] >= best[i]:
                best[i] = P[w] * best[i - len(w)]
                words[i] = w
    ## Now recover the sequence of best words
    sequence = []; i = len(words)-1
    while i > 0:
        sequence[0:0] = [words[i]]
        i = i - len(words[i])
    ## Return sequence of best words and overall probability
    return sequence, best[-1]

I have just corrected the pseudo implementation of Viterbi in Wikipedia. From the initial (incorrect) version, it took me a while to figure out where I was going wrong but I finally managed it, thanks partly to Kevin Murphy's implementation of the viterbi_path.m in the MatLab HMM toolbox.

In the context of an HMM object with variables as shown:

hmm = HMM()
hmm.priors = np.array([0.5, 0.5]) # pi = prior probs
hmm.transition = np.array([[0.75, 0.25], # A = transition probs. / 2 states
                           [0.32, 0.68]])
hmm.emission = np.array([[0.8, 0.1, 0.1], # B = emission (observation) probs. / 3 obs modes
                         [0.1, 0.2, 0.7]])

The Python function to run Viterbi (best-path) algorithm is below:

def viterbi (self,observations):
    """Return the best path, given an HMM model and a sequence of observations"""
    # A - initialise stuff
    nSamples = len(observations[0])
    nStates = self.transition.shape[0] # number of states
    c = np.zeros(nSamples) #scale factors (necessary to prevent underflow)
    viterbi = np.zeros((nStates,nSamples)) # initialise viterbi table
    psi = np.zeros((nStates,nSamples)) # initialise the best path table
    best_path = np.zeros(nSamples); # this will be your output

    # B- appoint initial values for viterbi and best path (bp) tables - Eq (32a-32b)
    viterbi[:,0] = self.priors.T * self.emission[:,observations(0)]
    c[0] = 1.0/np.sum(viterbi[:,0])
    viterbi[:,0] = c[0] * viterbi[:,0] # apply the scaling factor
    psi[0] = 0;

    # C- Do the iterations for viterbi and psi for time>0 until T
    for t in range(1,nSamples): # loop through time
        for s in range (0,nStates): # loop through the states @(t-1)
            trans_p = viterbi[:,t-1] * self.transition[:,s]
            psi[s,t], viterbi[s,t] = max(enumerate(trans_p), key=operator.itemgetter(1))
            viterbi[s,t] = viterbi[s,t]*self.emission[s,observations(t)]

        c[t] = 1.0/np.sum(viterbi[:,t]) # scaling factor
        viterbi[:,t] = c[t] * viterbi[:,t]

    # D - Back-tracking
    best_path[nSamples-1] =  viterbi[:,nSamples-1].argmax() # last state
    for t in range(nSamples-1,0,-1): # states of (last-1)th to 0th time step
        best_path[t-1] = psi[best_path[t],t]

    return best_path

This is an old question, but none of the other answers were quite what I needed because my application doesn't have specific observed states.

Taking after @Rhubarb, I've also re-implemented Kevin Murphey's Matlab implementation (see viterbi_path.m), but I've kept it closer to the original. I've included a simple test case as well.

import numpy as np


def viterbi_path(prior, transmat, obslik, scaled=True, ret_loglik=False):
    '''Finds the most-probable (Viterbi) path through the HMM state trellis
    Notation:
        Z[t] := Observation at time t
        Q[t] := Hidden state at time t
    Inputs:
        prior: np.array(num_hid)
            prior[i] := Pr(Q[0] == i)
        transmat: np.ndarray((num_hid,num_hid))
            transmat[i,j] := Pr(Q[t+1] == j | Q[t] == i)
        obslik: np.ndarray((num_hid,num_obs))
            obslik[i,t] := Pr(Z[t] | Q[t] == i)
        scaled: bool
            whether or not to normalize the probability trellis along the way
            doing so prevents underflow by repeated multiplications of probabilities
        ret_loglik: bool
            whether or not to return the log-likelihood of the best path
    Outputs:
        path: np.array(num_obs)
            path[t] := Q[t]
    '''
    num_hid = obslik.shape[0] # number of hidden states
    num_obs = obslik.shape[1] # number of observations (not observation *states*)

    # trellis_prob[i,t] := Pr((best sequence of length t-1 goes to state i), Z[1:(t+1)])
    trellis_prob = np.zeros((num_hid,num_obs))
    # trellis_state[i,t] := best predecessor state given that we ended up in state i at t
    trellis_state = np.zeros((num_hid,num_obs), dtype=int) # int because its elements will be used as indicies
    path = np.zeros(num_obs, dtype=int) # int because its elements will be used as indicies

    trellis_prob[:,0] = prior * obslik[:,0] # element-wise mult
    if scaled:
        scale = np.ones(num_obs) # only instantiated if necessary to save memory
        scale[0] = 1.0 / np.sum(trellis_prob[:,0])
        trellis_prob[:,0] *= scale[0]

    trellis_state[:,0] = 0 # arbitrary value since t == 0 has no predecessor
    for t in xrange(1, num_obs):
        for j in xrange(num_hid):
            trans_probs = trellis_prob[:,t-1] * transmat[:,j] # element-wise mult
            trellis_state[j,t] = trans_probs.argmax()
            trellis_prob[j,t] = trans_probs[trellis_state[j,t]] # max of trans_probs
            trellis_prob[j,t] *= obslik[j,t]
        if scaled:
            scale[t] = 1.0 / np.sum(trellis_prob[:,t])
            trellis_prob[:,t] *= scale[t]

    path[-1] = trellis_prob[:,-1].argmax()
    for t in range(num_obs-2, -1, -1):
        path[t] = trellis_state[(path[t+1]), t+1]

    if not ret_loglik:
        return path
    else:
        if scaled:
            loglik = -np.sum(np.log(scale))
        else:
            p = trellis_prob[path[-1],-1]
            loglik = np.log(p)
        return path, loglik


if __name__=='__main__':
    # Assume there are 3 observation states, 2 hidden states, and 5 observations
    priors = np.array([0.5, 0.5])
    transmat = np.array([
        [0.75, 0.25],
        [0.32, 0.68]])
    emmat = np.array([
        [0.8, 0.1, 0.1],
        [0.1, 0.2, 0.7]])
    observations = np.array([0, 1, 2, 1, 0], dtype=int)
    obslik = np.array([emmat[:,z] for z in observations]).T
    print viterbi_path(priors, transmat, obslik)                                #=> [0 1 1 1 0]
    print viterbi_path(priors, transmat, obslik, scaled=False)                  #=> [0 1 1 1 0]
    print viterbi_path(priors, transmat, obslik, ret_loglik=True)               #=> (array([0, 1, 1, 1, 0]), -7.776472586614755)
    print viterbi_path(priors, transmat, obslik, scaled=False, ret_loglik=True) #=> (array([0, 1, 1, 1, 0]), -8.0120386579275227)

Note that this implementation does not use emission probabilities directly but uses a variable obslik. Generally, emissions[i,j] := Pr(observed_state == j | hidden_state == i) for a particular observed state i, making emissions.shape == (num_hidden_states, num_obs_states).

However, given a sequence observations[t] := observation at time t, all the Viterbi Algorithm requires is the likelihood of that observation for each hidden state. Hence, obslik[i,t] := Pr(observations[t] | hidden_state == i). The actual value the of the observed state isn't necessary.

I have modified @Rhubarb's answer for the condition where the marginal probabilities are already known (e.g by computing the Forward Backward algorithm).

def viterbi (transition_probabilities, conditional_probabilities):
    # Initialise everything
    num_samples = conditional_probabilities.shape[1]
    num_states = transition_probabilities.shape[0] # number of states

    c = np.zeros(num_samples) #scale factors (necessary to prevent underflow)
    viterbi = np.zeros((num_states,num_samples)) # initialise viterbi table
    best_path_table = np.zeros((num_states,num_samples)) # initialise the best path table
    best_path = np.zeros(num_samples).astype(np.int32) # this will be your output

    # B- appoint initial values for viterbi and best path (bp) tables - Eq (32a-32b)
    viterbi[:,0] = conditional_probabilities[:,0]
    c[0] = 1.0/np.sum(viterbi[:,0])
    viterbi[:,0] = c[0] * viterbi[:,0] # apply the scaling factor

    # C- Do the iterations for viterbi and psi for time>0 until T
    for t in range(1, num_samples): # loop through time
        for s in range (0,num_states): # loop through the states @(t-1)
            trans_p = viterbi[:, t-1] * transition_probabilities[:,s] # transition probs of each state transitioning
            best_path_table[s,t], viterbi[s,t] = max(enumerate(trans_p), key=operator.itemgetter(1))
            viterbi[s,t] = viterbi[s,t] * conditional_probabilities[s][t]

        c[t] = 1.0/np.sum(viterbi[:,t]) # scaling factor
        viterbi[:,t] = c[t] * viterbi[:,t]

    ## D - Back-tracking
    best_path[num_samples-1] =  viterbi[:,num_samples-1].argmax() # last state
    for t in range(num_samples-1,0,-1): # states of (last-1)th to 0th time step
        best_path[t-1] = best_path_table[best_path[t],t]
    return best_path