In the natural sciences, we frequently encounter the challenge of analyzing noisy and incomplete experimental data using theoretical models. The number of parameters in the models often exceeds the number of available data points. Image reconstruction is the textbook example of this class of ill-defined inverse problems. Another important class of problems is ensemble refinement, as performed in integrative structural biology and hybrid modelling. Experimentally averaged observables are used to refine an ensemble of biomolecular structural models. In such cases, we have to take into account statistical and systematic errors in the model and in the data, and balance the information provided by experiment and theory. While maximum entropy (MaxEnt) and Bayesian methods address these challenges, the balancing of information has proven to be a long-standing and difficult problem. The balancing of information from experiment and theory can be viewed as a Pareto optimization problem (Figure ). In a multi-objective optimization, we seek to minimize both the mean-squared deviations from the experimental data, χ 2 , with respect to the statistical weights in the refined model, and the deviation of these weights from a given reference model, as quantified by the Kullback-Leibler (KL) divergence S KL . In the plane spanned by S KL and χ 2 , the set of weights where one is optimal for a fixed value of the other defines a Pareto front. On this Pareto front, we then seek a particular solution for which we consider the trade-off between increases in S KL and χ 2 to be fair. To encode the relative value given to S KL and χ 2 , we introduce a parameter θ that sets the so-called a) Electronic mail: juergen.koefinger@biophys.mpg.de b) Electronic mail: gerhard.hummer@biophys.mpg.de FIG. . Bayesian/MaxEnt ensemble refinement as a Pareto optimization problem. The curve of optimal solutions in the plane spanned by the deviations from experiment, χ 2 , and the deviations from theory, SKL, determines the Pareto front or L-curve (red line). Sub-optimal solutions (orange shaded area) lie above this curve. We encode our prior knowledge by choosing a value of the so-called marginal rate of substitution, given by -2θ here. We use this rate to trade off the two deviations according to our prior experience, dχ 2 = -2θ dSKL, which gives a unique solution (blue disk). At this point, the slope of the Pareto front is exactly the marginal rate, as indicated by the blue line.