For concreteness, we also work out a specific case. The reference potential U q (x) deviates from the assumed true potential by the addition of a linear term, U p (x) = U q (x) -b x, with U q (x) = 3(x 2 -1) 2 as above. A slope of b = 1.217165 was chosen so that ⟨x⟩ p = 0.8 = Y exactly. The corresponding force-field error is var p (∆u) ≈ 0.51. For this error, we expect a value of θ = 2/var p (∆u) ≈ 4 to give a balanced fit, as will be tested. As a more realistic example, we refine a twodimensional polymer model 33 using synthetic onedimensional data (Figure ). The Boltzmann distribution is given by a product of von Mises distributions acting on the angle differences between neighboring bonds. In this phantom-chain model, the beads do not interact. We set the mean values of the angle differences to zero. All bonds have length one and we apply the same stiffness parameter κ = 10. We randomly generate conformations of polymers with 100 beads. We sample N = 10000 independent conformations. As observable, we use the end-to-end distance. We set Y = 75 for the experimental value and an error of σ = 5.