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.. currentmodule:: msmbuilder.hmm

Hidden Markov models (HMMs)

_static/kde-vs-histogram.png

KDE is to histogram as HMM is to MSM

One potential weakness of :ref:`MSMs <msm>` is that they use discrete states with "hard" cutoffs. In some sense, an MSM forms an approximation to the transfer operator in the same sense that a histogram is an approximation a probability density function. When a probability density function is smooth, better performance can often be achieved with kernel density estimators, or mixture models, which smooth the data in a potentially more natural way.

A Gaussian hidden Markov model (HMM) is one way of applying this same logic to probabilistic models of dynamics of molecular systems. Like :ref:`MSMs <msm>`, the HMM also models the dynamics of the system as a first-order Markov jump process between a discrete set of states. The difference is that the states in the HMM are not associated with discrete non-overlapping regions of phase space defined by clustering -- instead, the states are Gaussian distributions. Because the Gaussian distribution has infinite support, there is no unique and unambiguous mapping from a given conformation to a single state. Each state is a distribution over all conformations.

HMMs have been widely used in many fields, from speech processing to bioinformatics. Many good reviews have been written, such as [1].

HMMs for Biophysics

In [2], McGibbon et. al. introduced a reversible Gaussian HMM for studying protein dynamics. The class :class:`GaussianHMM` implements the algorithm described in that paper. Compared to a "vanilla" HMM:

  • The transition matrix is constrained to obey detailed balance. Detailed balance is a necessary condition for the the model to satisfy the 2nd law of thermodynamics. It also aids analysis, because models that don't satisfy detailed balance don't necessarily have a unique equilibrium distribution that they relax to in the limit of infinite time.
  • We added a penalty term (tunable via the fusion_prior hyperparameter) on the pairwise difference between the means of each of the states. This helps encourage a sense of sparisty where each state might be different from the other states along only a subset of the coordinates.

Algorithms

.. autosummary::
    :toctree: _hmm/
    :template: class.rst

    GaussianHMM
    VonMisesHMM

References

[1]Rabiner, Lawrence, and Biing-Hwang Juang. An introduction to hidden Markov models ASSP Magazine, IEEE 3.1 (1986): 4-16.
[2]McGibbon, Robert T. et al., Understanding Protein Dynamics with L1-Regularized Reversible Hidden Markov Models Proc. 31st Intl. Conf. on Machine Learning (ICML). 2014.