Estimation theory and gravity

It is shown that if the Euclidean path integral measure of a minimally coupled free quantum scalar field on a classical metric background is interpreted as probability of observing the field configuration given the background metric then the maximum likelihood estimate of the metric satisfies Euclidean Einstein field equations with the stress-energy tensor of the observed field as the source. In the case of a slowly varying metric the maximum likelihood estimate is very close to its actual value. Then by virtue of the asymptotic normality of the maximum likelihood estimate the fluctuations of the metric are Gaussian and governed by the Fisher information bi-tensor. Cramer-Rao bound can be interpreted as uncertainty relations between metric and stress-energy tensor. A plausible prior distribution for the metric fluctuations in a Bayesian framework is introduced. Our approach can be interpreted as a formulation of Euclidean version of stochastic gravity in the language of estimation theory.