HMM state posteriors with forward-backward
Advanced
Probability of each hidden state at each time
Viterbi gives one path. Often you need the probability that the hidden state is i at time t, given the whole observed sequence. That is the smoothed posterior gamma_t(i) = p(s_t = i | obs), computed with the forward-backward algorithm.
- Forward:
alpha_0(i) = pi_i B[i, o_0],alpha_t(j) = (sum_i alpha_(t-1)(i) A[i, j]) B[j, o_t]. - Backward:
beta_(T-1)(i) = 1,beta_t(i) = sum_j A[i, j] B[j, o_(t+1)] beta_(t+1)(j). - Posterior:
gamma_t(i)is proportional toalpha_t(i) beta_t(i), normalized overi.
Implement state_posteriors(pi, A, B, obs). It returns a (T, K) array whose rows sum to 1.
Constraints
- Inputs are validated as for a forward-algorithm HMM: stochastic rows, consistent shapes, observations in
0..M-1. - The observation sequence must not be empty. Otherwise raise
ValueError. - Do not modify the inputs.
Hints
Hint 1
Store every forward vector, run the backward pass from the end, and combine them row by row. Normalize each row so rounding does not accumulate.
Keywords
classical-mlhmmforward-backwardsmoothingposteriors
Python
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