  Bayesian statistical inference is a powerful tool for model-data comparisons
and extractions of physical parameters that are often unknown functions of
system variables. Existing Bayesian analyses often rely on explicit
parametrizations of the unknown function. It can introduce long-range
correlations that impose fictitious constraints on physical parameters in
regions of the variable space that are not probed by the experimental data. We
develop an information field (IF) approach to modeling the prior distribution
of the unknown function that is free of long-range correlations. We apply the
IF approach to the first global Bayesian inference of the jet transport
coefficient $\hat q$ as a function of temperature ($T$) from all existing
experimental data on single-inclusive hadron, di-hadron and $\gamma$-hadron
spectra in heavy-ion collisions at RHIC and LHC energies. The extracted $\hat
q/T^3$ exhibits a strong $T$-dependence as a result of the progressive
constraining power when data from more central collisions and at higher
colliding energies are incrementally included. The IF method guarantees that
the extracted $T$-dependence is not biased by a specific functional form.
