Schemes illustrating the volume element changes upon reversible element shift (left) and row shift (right).
Illustration of sampling of transition probability matrices for the observation Panels (a), (b), and (c) show the probability distribution on the off-diagonal matrix elements. The color encodes the probability density, with and . Each density was scaled such that its maximum is equal to 1. (a) Analytic density of stochastic matrices. (b) Sampled density of stochastic matrices (these matrices automatically fulfill detailed balance). (c) Stationary probability of the first state . When sampling with respect to a fixed stationary probability distribution , the ensemble is fixed to the line . (d) Sampled and exact density of of reversible matrices with fixed stationary distribution .
Visualization of the probability density of transition matrices to the observation . Different two-dimensional marginal distributions are shown in the rows. The analytic and sampled distributions for stochastic matrices are shown in columns 1 and 2, respectively. Column 3 shows the sampled distribution for stochastic matrices fulfilling detailed balance.
Convergence of the transition matrix sampling Algorithm 1. One sample every moves was recorded and the convergence is shown for 5000 samples in total. The left panels show the values, for each sample, of (a) the self-transition probabilities of states 1 and , (b) eigenvalues 2, 3, and 4, and (c) the free energy differences of states 1 and with respect to state . The right panels show the current estimate for the standard deviation of each of these observables. The standard deviation can be considered converged after about 1000 samples ( moves).
Mean uncertainties of (a) the diagonal and (b) the off-diagonal elements of the transition matrix for different simulation lengths. The uncertainties are shown for the ensembles of transition matrices, transition matrices with detailed balance, and for a fixed stationary distribution.
Mean uncertainties of the free energy differences with respect to the starting state of the simulation for different simulation lengths. The uncertainties are shown for the ensembles of transition matrices and transition matrices with detailed balance.
Distributions of the eigenvalue spectrum of for different simulation lengths. The distributions are shown for the ensembles of transition matrices [no detailed balance (DB)], transition matrices with DB, and fixed stationary distribution .
Algorithm 1. Metropolis Monte Carlo sampling of stochastic matrices.
Algorithm 2. Metropolis Monte Carlo sampling of reversible stochastic matrices.
Algorithm 3. Initial reversible transition matrix with given stationary distribution .
Algorithm 4. Metropolis Monte Carlo sampling of reversible stochastic matrices.
Constraints for transition matrices.
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