Hidden Markov models

#################### Hidden Markov models ####################

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

.. _hidden Markov model : https://en.wikipedia.org/wiki/Hidden_Markov_model

.. figure:: img/hmm_model.png :scale: 25 % :align: center

Graphical model for an HMM with :math:`T=4` timesteps.

An HMM assumes:

1. The observations, `O`, are generated by a process whose states,
   :math:`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 :math:`t - 1`.

4. The observation :math:`O_t` is independent of all previous states and
   observations given the current hidden state, :math:`S_t`.

The parameters of an HMM model are:

- :math:`\pi`, the prior specifying :math:`P(S_1)`.

- :math:`\theta`, the :math:`K \times K` transition matrix specifying
  :math:`P(S_t \mid S_{t-1})`.

- :math:`\phi`, the output model defining :math:`P(Y_t \mid S_t)`. If the
  observations are discrete, this is a :math:`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:

.. math::

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 :math:X_{1:T} is shorthand for :math: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_.

.. _expectation-maximization (EM) algorithm : https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm .. _forward-backward: https://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithm .. _Baum-Welch algorithm: https://en.wikipedia.org/wiki/Baum%E2%80%93Welch_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