Replica-Exchange Monte Carlo Scheme for Bayesian Data Analysis

We develop a sampling algorithm to explore the probability densities arising in Bayesian data analysis problems. Our algorithm is a multiparameter generalization of a replica-exchange Monte Carlo scheme. The strategy relies on gradual weighing of experimental data and on Tsallis generalized statistics. We demonstrate the effectiveness of the method on nuclear magnetic resonance data for a folded protein.

Figure 1

The effect of the replica parameters $q$ and $\lambda$ on the prior density and the likelihood function, respectively. In the upper panel, the effective potential energy $E(\theta ;q)$ is shown for $q=1+{10}^{-\alpha }$ varying from 1.0 ($\alpha =\infty$) to 1.001 ($\alpha =3$). The lower panel shows the effect of weighing the likelihood function by letting $\lambda$ range from 1.0 to 0.1. The curves are obtained by varying the torsion angle $\phi$ in the amino acid Alanine-122 of a Fyn SH3 domain in a folded configuration.

Figure 2

Samples of the backbone torsion angles $\phi$ and $\psi$ of three different replicas: high-temperature heat bath with $\lambda =0.1$, $q=1.1$ (left), Boltzmann ensemble with $\lambda =0.1$, $q=1.0$ (middle), and posterior distribution with $\lambda =1.0$, $q=1.0$ (right).

Figure 3

Upper panel: Dependency of the average radius of gyration $⟨R⟩$ on the position in the replica chain. The radius of gyration of the x-ray structure is 10.5 Å. Lower panel: Distributions of backbone rmsd values between conformations in the replica ensembles and the x-ray structure. Only replicas with fixed $q$ have been considered. With increasing $\lambda$ the rmsd distributions are shifted to smaller values. The rmsd histogram for the target distribution is shown as a shaded region.

Figure 4

${\mathrm{C}}^{\alpha }$ traces of the 20 most probable conformations, superimposed [24] onto the x-ray structure. Conformations are drawn in black, the backbone trace of the x-ray structure in gray. On the left-hand side, conformational samples of the posterior distribution ($\lambda =1.0$, $q=1.0$) are shown. The right-hand side shows structures that are distributed according to the Boltzmann ensemble ($\lambda =0.1$, $q=1.0$). Structures on the right-hand side are scaled in size compared to the ensemble on the left.