  Hypothesis testing and other statistical inference procedures are most
efficient when a reliable low-dimensional parametric family can be specified.
We propose a method that learns such a family when one exists but its form is
not known a priori, by examining samples from related populations and fitting a
low-dimensional exponential family that approximates all the samples as well as
possible. We propose a computationally efficient spectral method that allows us
to carry out hypothesis tests that are valid whether or not the fit is good,
and recover asymptotically optimal power if it is. Our method is
computationally efficient and can produce substantial power gains in simulation
and real-world A/B testing data.
