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total = 0
argmax = None
valmax = 0
for state in X:
(prob, v_path, v_prob) = T[state]
p = ep[state][output] * tp[state][next_state]
prob *= p
v_prob *= p
total += prob
if v_prob > valmax:
argmax = v_path + [next_state]
valmax = v_prob
U[next_state] = (total, argmax, valmax)
T = U
## apply sum/max to the final states:
total = 0
argmax = None
valmax = 0
for state in X:
(prob, v_path, v_prob) = T[state]
total += prob
if v_prob > valmax:
argmax = v_path
valmax = v_prob
return (total, argmax, valmax)
The function <code>forward_viterbi</code> takes the following arguments: <code>y</code> is the sequence of observations, e.g. <code>['walk', 'shop', 'clean']</code>; <code>X</code> is the set of hidden states; <code>sp</code> is the start probability; <code>tp</code> are the transition probabilities; and <code>ep</code> are the emission probabilities.
The algorithm works on the mappings <code>T</code> and <code>U</code>. Each is a mapping from a state to a triple <code>(prob, v_path, v_prob)</code>, where <code>prob</code> is the total probability of all paths from the start to the current state, <code>v_path</code> is the Viterbi path up to the current state, and <code>v_prob</code> is the probability of the Viterbi path up to the current state. The mapping <code>T</code> holds this information for a given point ''t'' in time, and the main loop constructs <code>U</code>, which holds similar information for time ''t''+1. Because of the [[Markov property]], information about any point in time prior to ''t'' is not needed.
The algorithm begins by initializing ''T'' to the start probabilities: the total probability for a state is just the start probability of that state; and the Viterbi path to a start state is the singleton path consisting only of that state; the probability of the Viterbi path is the same as the start probability.
The main loop considers the observations from <code>y</code> in sequence. Its [[loop invariant]] is that <code>T</code> contains the correct information up to but excluding the point in time of the current observation. The algorithm then computes the triple <code>(prob, v_path, v_prob)</code> for each possible next state. The total probability of a given next state, <code>total</code> is obtained by adding up the probabilities of all paths reaching that state. More precisely, the algorithm iterates over all possible source states. For each source state, <code>T</code> holds the total probability of all paths to that state. This probability is then multiplied by the emission probability of the current observation and the transition probability from the source state to the next state. The resulting probability <code>prob</code> is then added to <code>total</code>. The probability of the Viterbi path is computed in a similar fashion, but instead of adding across all paths one performs a discrete maximization. Initially the maximum value <code>valmax</code> is zero. For each source state, the probability of the Viterbi path to that state is known. This too is multiplied with the emission and transition probabilities and replaces <code>valmax</code> if it is greater than its current value. The Viterbi path itself is computed as the corresponding [[argmax]] of that maximization, by extending the Viterbi path that leads to the current state with the next state. The triple <code>(prob, v_path, v_prob)</code> computed in this fashion is stored in <code>U</code> and once <code>U</code> has been computed for all possible next states, it replaces <code>T</code>, thus ensuring that the loop invariant holds at the end of the iteration.
In the end another summation/maximization is performed (this could also be done inside the main loop by adding a pseudo-observation after the last real observation).
In the running example, the forward/Viterbi algorithm is used as follows:
def example():
return forward_viterbi(['walk', 'shop', 'clean'],
states,
start_probability,
transition_probability,
emission_probability)
This reveals that the total probability of <code>['walk', 'shop', 'clean']</code> is 0.033612 and that the Viterbi path is <code>['Sunny', 'Rainy', 'Rainy', 'Rainy']</code>. The Viterbi path contains four states because the third observation was generated by the third state and a transition to the fourth state. In other words, given the observed activities, it was most likely sunny when your friend went for a walk and then it started to rain the next day and kept on raining.
==Extensions==
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