The conditional density p(x|a, b, c, d) for a = 8.0, b = 2.0, c = 4.0, and d = 30.0 (solid line) and the corresponding enveloping function g(x) (dashed line). The density was scaled to the mode point value p(x m |a, b, c, d) to fit it into the range [0, 1].
The picture shows the location of the optimal points and for p(x) (solid line) and the points x l and x u obtained from the Gaussian approximation (dashed line). The Gaussian approximation touches the density at the mode point.
Conditional density p(x) (solid line) for different parameters a, b, c, and d. The histograms show a sample of N = 106 random variates generated using the method outlined above. First row c = 4, d = 10: (a) a = 5, b = 0, (b) a = 5, b = 2, and (c) a = 0, b = 2. Second row c = 40, d = 100: (d) a = 100, b = 5, (e) a = 100, b = 100, and (f) a = 5, b = 100. Third row: (g) a = 0, b = 0, c = 4, d = 10, (h) a = 0.5, b = 0.2, c = 40, d = 100, and (i) a = 0, b = 0, c = 4 × 103, d = 104.
Results obtained for the model system with count matrix (23) and stationary distribution (24) . Standard deviation for estimated mean and variance of observables is plotted against the number of elementary sampling steps N. (a) Mean transition matrix element , (b) mean of the second largest implied time scale , (c) transition matrix element variance , and (d) variance of the second largest implied time scale . The Gibbs sampler introduced here (solid line) convergences faster than the Metropolis chain from Ref. 54 (dashed line) by almost two orders of magnitude. For the mean second largest time scale , (b), the achieved speedup is more than one order of magnitude.
Autocorrelation functions for the model system with count matrix (23) and stationary distribution (24) . (a) Autocorrelation function for the transition matrix element p 13 and (b) autocorrelation function for the second largest implied time scale t 2. The number of steps to take until samples are decorrelated n decorr is two orders of magnitude smaller for the Gibbs sampling method (solid line), n decorr = 3 for p 13 as well as for t 2, compared to the Metropolis sampling method (dashed line), n decorr = 123 for p 13 and n decorr = 135 for t 2.
Results obtained for the synthetic peptide MR121-GSGS-W. Standard deviation of Figure 6(a) the mean implied time scale and Figure 6(b) the implied time scale variance . The Gibbs sampler (solid line) shows a faster convergence than the Metropolis sampler (dashed line) for mean and variance of the second largest implied timescale t 2.
Autocorrelation function for the MR121-GSGS-W peptide count matrix. The number of steps to take until samples are decorrelated n decorr is an order of magnitude smaller for the Gibbs sampling method (solid line), n decorr = 4600, compared to the Metropolis sampling method (dashed line), n decorr = 33 000.
Histograms for implied time scale t 2 corresponding to second largest eigenvalue λ2. The histogram of timescales generated by incorporating knowledge about stationary probabilities (a) in comparison to the histogram of timescales generated by the non-reversible (b) and reversible method (c). It is clearly visible that the sample mean (dashed line) gives a more accurate prediction of the true value (solid line) due to the additional information about stationary probabilities.
Sample p(x|a, b, c, d).
Modified rejection algorithm.
Iterative maximum likelihood estimation with fixed stationary distribution.
Gibbs sampling of P with fixed stationary distribution π.
Gibbs sampling of P with uncertain stationary distribution.
Gibbs sampling of P with fixed stationary distribution π—Sparse version.
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