# Implementation of the Viterbi Algorithm in Python

Viterbi Algorithm is used for finding the most likely state sequence with the maximum a posteriori probability. It is a dynamic programming-based algorithm. This article will talk about how we can implement the Viterbi Algorithm using Python. We will use `numpy`

for the implementation.

## Python implementation of the Viterbi Algorithm

The following code implements the Viterbi Algorithm in Python. It is a function that accepts 4 parameters which are as follows -

`y`

: This is the observation state sequence.`A`

: This is the state transition matrix.`B`

: This is the emission matrix.`initial_probs`

: These are the initial state probabilities.

And the function returns 3 values as follows -

`x`

: Maximum a posteriori probability estimate of hidden state trajectory, conditioned on observation sequence y under the model parameters`A`

,`B`

,`initial_probs`

.`T1`

: The probability of the most likely path.`T2`

: The probability of the most likely path.

```
import numpy as np
def viterbi(y, A, B, initial_probs = None):
K = A.shape[0]
initial_probs = initial_probs if initial_probs is not None else np.full(K, 1 / K)
T = len(y)
T1 = np.empty((K, T), 'd')
T2 = np.empty((K, T), 'B')
T1[:, 0] = initial_probs * B[:, y[0]]
T2[:, 0] = 0
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)
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
```

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