numpy_ml.hmm.rst

Hidden Markov models

A hidden Markov model (HMM) is a generative model for sequences of observations.

Graphical model for an HMM with T=4 timesteps.

An HMM assumes:

1. The observations, O, are generated by a process whose states, S, are hidden from the observer.
2. Each hidden state is a discrete random variable.
3. The hidden state at time t is independent of all hidden states before time t - 1.
4. The observation O_t is independent of all previous states and observations given the current hidden state, S_t.

The parameters of an HMM model are:

• \pi, the prior specifying P(S_1).
• \theta, the K \times K transition matrix specifying P(S_t \mid S_{t-1}).
• \phi, the output model defining P(Y_t \mid S_t). If the observations are discrete, this is a K \times L emission matrix, where L is the number of unique observation symbols.

The HMM joint distribution of a sequence of states and observations is:

P(S_{1:T}, O_{1:T}) = P(S_1) P(O_1 \mid S_1) \prod_{t=2}^T P(S_t \mid S_{t-1})P(O_t \mid S_t)

where X_{1:T} is shorthand for X_1, \ldots, X_T.

As with other latent-variable models, we use the expectation-maximization (EM) algorithm to learn the model parameters. The HMM-optimized version of the EM algorithm is known as the forward-backward / Baum-Welch algorithm.

Models

• :class:`~numpy_ml.hmm.MultinomialHMM`

References

[1]	Ghahramani, Z. (2001). "An Intro to HMMs and Bayesian networks". International Journal of Pattern Recognition and AI, 15(1): 9-42.

.. toctree::
   :maxdepth: 2
   :hidden:

   numpy_ml.hmm.MultinomialHMM