Gaussian sequence model

The Gaussian sequence model observeswhere the $g_k$ are independent standard normal variables and the unknown signal commonly lies in ${\ell }^2$. It is the coordinate representation of a Gaussian white-noise experiment.

A Bernstein-von Mises theorem says that a suitably centered and scaled posterior distribution converges to the normal distribution prescribed by the local likelihood. For a fixed linear functional in a Gaussian sequence model and a prior with locally flat positive density, the limiting posterior variance is the squared ${\ell }^2$ norm of the functional's coefficient vector.

A Gaussian net test covers a composite alternative by finitely many metric balls and takes the maximum of the likelihood-ratio tests against their centers. Gaussian tail bounds control each test, while a union bound costs the logarithm of the covering number.

Because Gaussian white noise is not an ${\ell }^2$ vector, least squares over $\Theta \subset {\ell }^2$ is defined by maximizingover $\theta \in \Theta$, with the noise pairing interpreted through an isonormal Gaussian process. Differences of this contrast equal the formal differences of squared residual norms.