Model-free Data-Driven Inference

We present a model-free data-driven inference method that enables inferences on system outcomes to be derived directly from empirical data without the need for intervening modeling of any type, be it modeling of a material law or modeling of a prior distribution of material states. We specifically consider physical systems with states characterized by points in a phase space determined by the governing field equations. We assume that the system is characterized by two likelihood measures: one µ_D measuring the likelihood of observing a material state in phase space; and another µ_E measuring the likelihood of states satisfying the field equations, possibly under random actuation. We introduce a notion of intersection between measures which can be interpreted to quantify the likelihood of system outcomes. We provide conditions under which the intersection can be characterized as the athermal limit µ∞ of entropic regularizations µ_β, or thermalizations, of the product measure µ = µ_D x µ_E as β → +∞. We also supply conditions under which µ∞ can be obtained as the athermal limit of carefully thermalized (µ_h,β_h) sequences of empirical data sets (µ_h) approximating weakly an unknown likelihood function µ. In particular, we find that the cooling sequence β_h → +∞ must be slow enough, corresponding to quenching, in order for the proper limit µ∞ to be delivered. Finally, we derive explicit analytic expressions for expectations E[f] of outcomes f that are explicit in the data, thus demonstrating the feasibility of the model-free data-driven paradigm as regards making convergent inferences directly from the data without recourse to intermediate modeling steps.