  Incredible progress on the theoretical uncertainty of the spatial
correlations of the stochastic gravitational wave (GW) background were recently
made. However, it remains to realize the impact of this theoretical uncertainty
on PTA cross correlations analysis. This paper pushes forward in this
direction, as a proof--of--principle: showing the potential role that
theoretical uncertainty has on unburying the stochastic GW background signal in
noisy PTA cross correlation measurements. We consider both a mock data set and
the noise--marginalized 12.5 years NANOGrav spatial correlation measurements,
and find optimistic conclusions regardless of the physical content of the GW
background and the nature of the noise in the data. Very briefly, we show
through various cases a modest result that looking out for a stochastic signal
is better when two of its moments are utilized. Or, in terms of GWs, we show
that the theoretical uncertainty can play a substantial role in the hunt for
the stochastic GW background.
