  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. Using this distribution, we
calculate the decoherence functional acting on the field by integrating out the
metric fluctuations around flat space. Our approach can be interpreted as a
formulation of Euclidean version of stochastic gravity in the language of
estimation theory.
