  Consider the following three important problems in statistical inference,
namely, constructing confidence intervals for (1) the error of a
high-dimensional ($p>n$) regression estimator, (2) the linear regression noise
level, and (3) the genetic signal-to-noise ratio of a continuous-valued trait
(related to the heritability). All three problems turn out to be closely
related to the little-studied problem of performing inference on the
$\ell_2$-norm of the signal in high-dimensional linear regression. We derive a
novel procedure for this, which is asymptotically correct when the covariates
are multivariate Gaussian and produces valid confidence intervals in finite
samples as well. The procedure, called EigenPrism, is computationally fast and
makes no assumptions on coefficient sparsity or knowledge of the noise level.
We investigate the width of the EigenPrism confidence intervals, including a
comparison with a Bayesian setting in which our interval is just 5% wider than
the Bayes credible interval. We are then able to unify the three aforementioned
problems by showing that the EigenPrism procedure with only minor modifications
is able to make important contributions to all three. We also investigate the
robustness of coverage and find that the method applies in practice and in
finite samples much more widely than just the case of multivariate Gaussian
covariates. Finally, we apply EigenPrism to a genetic dataset to estimate the
genetic signal-to-noise ratio for a number of continuous phenotypes.
