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 $\mu_D$ measuring the likelihood of observing a material state in phase space; and another $\mu_E$ 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 $\mu_\infty$ of entropic regularizations $\mu_\beta$, or thermalizations, of the product measure $\mu = \mu_D\times \mu_E$ as $\beta \to +\infty$. We also supply conditions under which $\mu_\infty$ can be obtained as the athermal limit of carefully thermalized $(\mu_{h,\beta_h})$ sequences of empirical data sets $(\mu_h)$ approximating weakly an unknown likelihood function $\mu$. In particular, we find that the cooling sequence $\beta_h \to +\infty$ must be slow enough, corresponding to quenching, in order for the proper limit $\mu_\infty$ to be delivered. Finally, we derive explicit analytic expressions for expectations $\mathbb{E}[f]$ 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.