# msmtools.estimation.log_likelihood¶

msmtools.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 logL – Log-likelihood of the count matrix 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 msmtools.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

 [1] Prinz, J H, H Wu, M Sarich, B Keller, M Senne, M Held, J D Chodera, C Schuette and F Noe. 2011. Markov models of molecular kinetics: Generation and validation. J Chem Phys 134: 174105