Bayesian approach to inverse statistical mechanics

Michael Habeck

Phys. Rev. E 89, 052113 – Published 9 May 2014

Abstract

Inverse statistical mechanics aims to determine particle interactions from ensemble properties. This article looks at this inverse problem from a Bayesian perspective and discusses several statistical estimators to solve it. In addition, a sequential Monte Carlo algorithm is proposed that draws the interaction parameters from their posterior probability distribution. The posterior probability involves an intractable partition function that is estimated along with the interactions. The method is illustrated for inverse problems of varying complexity, including the estimation of a temperature, the inverse Ising problem, maximum entropy fitting, and the reconstruction of molecular interaction potentials.

Received 27 December 2013

DOI:https://doi.org/10.1103/PhysRevE.89.052113

Authors & Affiliations

Michael Habeck^*

Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstrasse 7, 37077 Göttingen, Germany

^*mhabeck@gwdg.de

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 89, Iss. 5 — May 2014

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Flow chart of the sequential Monte Carlo algorithm. Superscripts and indices have been omitted for clarity. Details of the individual steps are given in Eqs. (15), (16), and (17).
Reuse & Permissions
Figure 2
Estimating the temperature of the Ising model: (a) The estimated feature distribution is shown as orange (thin gray) line. The black (thick) curve is the true density of states. Shown are the natural logarithms of the feature distributions. The red (gray) dashed line indicates the mean of the 10 energies from which the temperature was estimated. (b) Estimated posterior distributions $p^{(t)}$ . (c) Sampled inverse temperatures $β^{(t)}$ . The dashed red (gray) line is the true inverse temperature. (d) Final posterior distribution of the temperature after convergence. Black (thick) line: true posterior distribution; orange (thin gray) curve: estimated posterior distribution based on the final estimate of the DOS. The gray histogram shows the sampled $β^{(t)}$ .
Reuse & Permissions
Figure 3
Inverse Ising problem. (a) Evolution of the reconstruction error (root mean-squared error, MSE) (black line). The dashed line indicates the MSE of the mean interaction parameters. The inset shows the performance of the Bayesian estimators (posterior mean and maximum) in comparison to exact ML, PLM, and nMF. (b) Inverse Ising problem at different temperatures.
Reuse & Permissions
Figure 4
Application of the Monte Carlo algorithm to a maximum entropy fitting problem. The one-dimensional toy system [30] is analyzed using our sequential Monte Carlo sampler. (a) Evolution of the mean position ${〈 x 〉}_{λ^{(t)}}$ (black line); the dashed red (gray) line indicates the observed average value $- 0.127$ . (b) Effective potential energy functions during Monte Carlo sampling. The light gray potential $E_{0} (x) = - ln ρ (x)$ is gradually deformed by the addition of $λ x$ . With increasing number of iterations the effective potential energy curves are shown by increasingly darker lines. Furthermore, the reference potential $E_{0} (x)$ and the target potential $E (x) = E_{0} (x) + 10 x$ are highlighted by the dashed blue and red (gray) lines, respectively. (c) Histogram of the generated configurational samples; the inset shows the individual configurations generated after convergence. The correct maximum entropy distribution is shown as a red (gray) dashed line.
Reuse & Permissions
Figure 5
Estimation of a Lennard-Jones potential. (a) Sampled interaction parameters in the initial phase of the Monte Carlo algorithm. The inset shows the final estimated posterior distribution of the interaction parameters ( $λ_{1}$ black, $λ_{2}$ light red [gray]). (b) Reconstructed and correct energy function shown as orange (thin gray) and black (thick) curve, respectively. (c) Radial distribution function (RDF) obtained with the proposed Monte Carlo sampler (thin orange [gray] line) compared to the correct RDF obtained with a replica-exchange simulation of the fluid using the true LJ parameters (thick black line).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics