hmmlearn: how to get the prediction for the hidden state probability at time T+1, given a full observation sequence 1:T

Question

I'm using hmmlearn's GaussianHMM to train a Hidden Markov Model with Gaussian observations. Each hidden state k has its corresponding Gaussian parameters: mu_k, Sigma_k.

After training the model, I would like to calculate the following quantity:

P(z_{T+1} = j | x_{1:T}),

where j = 1, 2, ... K, K is the number of hidden states.

The above probability is basically the one-step-ahead hidden state probabilities, given a full sequence of observations: x_1, x_2, ..., x_T, where x_i, i=1,...,T are used to train the HMM model.

I read the documentation , but couldn't find a function to calculate this probability. Is there any workaround?

Answer 1

The probability you are looking for is simply one row of the transition matrix. The n-th row of the transition matrix gives the probability of transitioning to each state at time t+1 knowing the state the system is at time t .

In order to know in which state the system is at time t given a sequence of observations x_1,...,x_t one can use the Viterbi algorithm which is the default setting of the method predict in hmmlearn .

model = hmm.GaussianHMM(n_components=3, covariance_type="full", n_iter=100)  # Viterbi is set by default as the 'algorithm' optional parameter.
model.fit(data)
state_sequence = model.predict(data)
prob_next_step = model.transmat_[state_sequence[-1], :]

I suggest you give a closer look at this documentation that shows concrete use examples.

Answer 2

Once the an HMM model is trained, you can get the t+1 state given 1:t observations X as following:

import numpy as np
from sklearn.utils import check_random_state
sates = model.predict(X)
transmat_cdf = np.cumsum(model.transmat_, axis=1)
random_sate = check_random_state(model.random_state)
next_state = (transmat_cdf[states[-1]] > random_state.rand()).argmax()

the t+1 state is generated according to the t state and the transmat_