Figure 8.

MaxEnt approaches to ensemble refinement as special cases of BioEn. For BioEn optimal ensembles, we plot the reduced χ² and the relative entropy S_KL parametrized by the confidence parameter θ (blue). The solution of Gull–Daniell-type⁶⁰ methods is given by the intersection of this curve with χ² = 1 (orange), of traditional MaxEnt methods^20,22,61,62 by the intersection with χ² = 0 (green), and of the method of Cesari et al.^25,30 by the BioEn solution for θ = 1 (red). For a simple model (M × N = 1 × 2, y = (0,1), σ = 0.14), we vary the experimental value Y, top to bottom. (A) All methods provide a solution for an experimental value Y = 0.09 within the range of calculated observables. (B) Traditional MaxEnt methods fail to give a solution for Y values outside this range (Y = 1.08). (C) Both traditional MaxEnt and Gull–Daniell-type methods fail to give a solution where a reduced χ² ≤ 1 cannot be realized by reweighting (Y = 1.16). The vertical gray lines indicate the maximum value S_KL = ln(2) for a two-state system.