deeptime.markov.tools.estimation.log_likelihood

deeptime.markov.tools.estimation.log_likelihood(C, T)

Log-likelihood of the count matrix given a transition matrix.

Parameters:

  • C ((M, M) ndarray or scipy.sparse matrix) – Count matrix

  • T ((M, M) ndarray orscipy.sparse matrix) – Transition matrix

Returns:

logL – Log-likelihood of the count matrix

Return type:

float

Notes

The likelihood of a set of observed transition counts \(C=(c_{ij})\) for a given matrix of transition counts \(T=(t_{ij})\) is given by

\[L(C|P)=\prod_{i=1}^{M} \left( \prod_{j=1}^{M} p_{ij}^{c_{ij}} \right) \]

The log-likelihood is given by

\[l(C|P)=\sum_{i,j=1}^{M}c_{ij} \log p_{ij}. \]

The likelihood describes the probability of making an observation \(C\) for a given model \(P\).

Examples

>>> import numpy as np
>>> from deeptime.markov.tools.estimation import log_likelihood

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])

>>> C = np.array([[58, 7, 0], [6, 0, 4], [0, 3, 21]])
>>> logL = log_likelihood(C, T)
>>> logL 
-38.2808034725...

>>> C = np.array([[58, 20, 0], [6, 0, 4], [0, 3, 21]])
>>> logL = log_likelihood(C, T)
>>> logL 
-68.2144096814...

References