  In a world devoid of axions, the smallness of the strong CP phase can have
implications for leptonic CP violation being probed by neutrino experiments.
For example, if nature adopted an axionless solution to the strong CP problem,
the same symmetries that set the strong CP phase to zero at the tree-level, may
also set the leptonic $CP$ phases to zero (mod \pi). This automatically happens
in the left-right symmetric model when it is extended minimally to solve the
strong CP problem by imposition of both P and CP. In the Nelson Barr solution
the needed symmetries can be assigned so that leptonic CP violation is not
generated. In the minimal left-right symmetric model with P (and not CP), where
the strong CP problem remains, leptonic CP phases radiatively generate the
strong CP phase in one loop, and therefore they may be absent (negligibly
small) in large regions of parameter space. All these results motivate us to
consider a Bayesian prior for leptonic Dirac CP phase \delta_{CP} of the PMNS
matrix that has delta function like peaks at CP conserving values of 0 and \pi
on top of a uniform distribution. We evaluate the posterior probability
distribution for \delta_{CP} using the current global fit to neutrino
experiments and find significant enhancements for the probability that
\delta_{CP} is at or negligibly close to \pi. We also provide useful tables for
the posterior probability considering present and future experimental
sensitivities.
