msmtools.estimation.tmatrix_cov

msmtools.estimation.tmatrix_cov(C, k=None)

Covariance tensor for non-reversible transition matrix posterior.

Parameters:

• C ((M, M) ndarray or scipy.sparse matrix) – Count matrix
• k (int (optional)) – Return only covariance matrix for entires in the k-th row of the transition matrix

Returns:

cov – Covariance tensor for transition matrix posterior

Return type:

(M, M, M) ndarray

Notes

The posterior of non-reversible transition matrices is

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

Each row in the transition matrix is distributed according to a Dirichlet distribution with parameters given by the observed transition counts \(c_{ij}\).

The covariance tensor \(\text{cov}[p_{ij},p_{kl}]=\Sigma_{i,j,k,l}\) is zero whenever \(i \neq k\) so that only \(\Sigma_{i,j,i,l}\) is returned.