I am looking for a reference for something like : a probability density put on the ADM phase space, thereby making the metric probabilistic in the classical probability sense. But not exactly this. I need this for a general spacetime, i.e. without the assumption of ADM that you can foliate spacetime into space-like slices. The Hamiltonian formulation will go away if you throw away this assumption.

So I need probability densities implemented on GR in a covariant way, i.e. time should be treated equally as space. Things like time loops should be allowed in principle. The formulation should be agnostic to the nature of time.