  We study the problem of detecting the presence of a single unknown spike in a
rectangular data matrix, in a high-dimensional regime where the spike has fixed
strength and the aspect ratio of the matrix converges to a finite limit. This
setup includes Johnstone's spiked covariance model. We analyze the likelihood
ratio of the spiked model against an "all noise" null model of reference, and
show it has asymptotically Gaussian fluctuations in a region below---but in
general not up to---the so-called BBP threshold from random matrix theory. Our
result parallels earlier findings of Onatski et al.\ (2013) and
Johnstone-Onatski (2015) for spherical spikes. We present a probabilistic
approach capable of treating generic product priors. In particular, sparsity in
the spike is allowed. Our approach is based on Talagrand's interpretation of
the cavity method from spin-glass theory. The question of the maximal parameter
region where asymptotic normality is expected to hold is left open. This region
is shaped by the prior in a non-trivial way. We conjecture that this is the
entire paramagnetic phase of an associated spin-glass model, and is defined by
the vanishing of the replica-symmetric solution of Lesieur et al.\ (2015).
