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Published July 19, 2021 | Submitted
Report Open

Model-free Data-Driven Inference

Abstract

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.

Additional Information

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project 211504053 - SFB 1060; project 441211072 - SPP 2256; and project 390685813 - GZ 2047/1 - HCM.

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Created:
August 20, 2023
Modified:
October 23, 2023