I’m implementing a backwards HMM algorithm. I used this link as reference. This link contains the results of the numerical example used (I am attempting to implement that and compare my generated results to it). * Page 3, section 2. Backward probability* , there is a table containing the calculated results.

Here is my code:

```
# Initial Transition matrix as shown in page 2 of above link
A = np.array([[0.6, 0.4], [0.3, 0.7]])
A = torch.from_numpy(A)
# Initial State Probability (page 2)
pi = np.array([0.8, 0.2])
pi = torch.from_numpy(pi)
# Output probabilities (page 2)
emission_matrix = np.array([[0.3, 0.4, 0.3, 0.3], [0.4, 0.3, 0.3, 0.3]])
emission_matrix = torch.from_numpy(emission_matrix)
# Initialize empty 2x4 matrix (dimensions of emission matrix)
backward = torch.zeros(emission_matrix.shape, dtype=torch.float64)
# Backward algorithm
def _backward(emission_matrix):
# Initialization: A(i, j) * B(T, i) * B(Ot+1, j) , where B(Ot+1, j) = 1
backward[:, -1] = torch.matmul(A, emission_matrix[:, -1])
# I reversed the emission matrix so as to start from the last column
rev_emission_mat = torch.flip(emission_matrix[:, :-1], [1])
# I transposed the reversed emission matrix such that each iterable in the for
# loop is the observation sequence probability
T_rev_emission_mat = torch.transpose(rev_emission_mat, 1, 0)
# This step is so that I assign a reverse index enumeration to each iterable in the
# emission matrix starts from time T to 0, rather than the opposite
zipped_cols = list(zip(range(len(T_rev_emission_mat)-1, -1, -1), T_rev_emission_mat))
for i, obs_prob in zipped_cols:
# Induction: Σ A(i, j) * B(j)(Ot+1) * β(t+1, j)
if i != 0:
backward[:, i] = torch.matmul(A * obs_prob, backward[:, i+1])
# Termination: Σ π(i) * bi * β(1, i)
backward[:, 0] = torch.matmul(pi * obs_prob, backward[:, 1])
# run backward algorithm
_backward(emission_matrix)
# check results, backward is an all zero matrix that was initialized above
print(backward)
>>> tensor([[0.0102, 0.0324, 0.0900, 0.3000],
[0.0102, 0.0297, 0.0900, 0.3000]], dtype=torch.float64)
```

As you can see, the 0-th index does not match the result in page 3 of the previous link. What did I do wrong? If there is anything I can clarify, please let me know. Thanks in advance!