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 . (c) Sampled inverse temperatures . 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 .

Application of the Monte Carlo algorithm to a maximum entropy fitting problem. The one-dimensional toy system [] is analyzed using our sequential Monte Carlo sampler. (a) Evolution of the mean position (black line); the dashed red (gray) line indicates the observed average value . (b) Effective potential energy functions during Monte Carlo sampling. The light gray potential is gradually deformed by the addition of . With increasing number of iterations the effective potential energy curves are shown by increasingly darker lines. Furthermore, the reference potential and the target potential 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.

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 ( black, 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).