Reparametrization Invariant Statistical Inference and Gravity

Bialek, Callan, and Strong have recently given a solution of the problem of determining a continuous probability distribution from a finite set of experimental measurements by formulating it as a one-dimensional quantum field theory. This Letter gives a reparametrization invariant solution of the problem, obtained by coupling to gravity. The case of a large number of dimensions is also formulated geometrically, and solved, in terms of metrics of vanishing Weyl curvature.