msmtools.estimation.tmatrix_sampler(C, reversible=False, mu=None, T0=None, nsteps=None, prior='sparse')

Generate transition matrix sampler object.

  • C ((M, M) ndarray or scipy.sparse matrix) – Count matrix
  • reversible (bool) – If true sample from the ensemble of transition matrices restricted to those obeying a detailed balance condition, else draw from the whole ensemble of stochastic matrices.
  • mu (array_like) – A fixed stationary distribution. Transition matrices with that stationary distribution will be sampled
  • T0 (ndarray, shape=(n, n) or scipy.sparse matrix) – Starting point of the MC chain of the sampling algorithm. Has to obey the required constraints.
  • nstep (int, default=None) – number of full Gibbs sampling sweeps per sample. This option is meant to ensure approximately uncorrelated samples for every call to sample(). If None, the number of steps will be automatically determined based on the other options and the matrix size. nstep>1 will only be used for reversible sampling, because nonreversible sampling generates statistically independent transition matrices every step.


Return type:

A :py:class:dense.tmatrix_sampler.TransitionMatrixSampler object that can be used to generate samples.


The transition matrix sampler generates transition matrices from the posterior distribution. The posterior distribution is given as a product of Dirichlet distributions

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

The method can generate samples from the posterior under the following constraints

Reversible sampling

Using a MCMC sampler outlined in .. [1] it is ensured that samples from the posterior are reversible, i.e. there is a probability vector \((\mu_i)\) such that \(\mu_i t_{ij} = \mu_j t_{ji}\) holds for all \(i,j\).

Reversible sampling with fixed stationary vector

Using a MCMC sampler outlined in .. [2] it is ensured that samples from the posterior fulfill detailed balance with respect to a given probability vector \((\mu_i)\).


[1]Noe, F. Probability distributions of molecular observables computed from Markov state models. J Chem Phys 128: 244103 (2008)
[2]Trendelkamp-Schroer, B., H. Wu, F. Paul and F. Noe: Estimation and uncertainty of reversible Markov models. J. Chem. Phys. (submitted)