Gaussian white noise

Gaussian white noise is a generalized Gaussian process with zero mean and covariance proportional to a Dirac delta function in time, and possibly in space. It is the formal time derivative of Brownian motion and has independent fluctuations on disjoint time intervals.

Observe $Y(t)={\int }_0^{t}f(s)ds+\sigma W(t)$ on $[0,1]$, where $f\in L^2[0,1]$ is an unknown deterministic drift and $W$ is standard Brownian motion. The observation against a deterministic test function is $Y(h)=⟨f,h⟩+\sigma W(h)$, with covariance ${\sigma }^2⟨h,g⟩$. Gaussian white noise is represented by an isonormal Gaussian process on test functions rather than a pointwise noise function. Taking $\sigma =n^{-1/2}$ is the usual statistical scaling.

Generalized Gaussian white noise over a real Hilbert space $H=L^2$ is an isonormal Gaussian process $W(h)$ with covariance $⟨h,g{⟩}_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued Gaussian measure because it is not a trace-class operator. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with Radon-Nikodym derivativeIn an orthonormal basis, $W_m(h)={\sum }_j{m}_j{h}_j$ is a stochastic series with total variance $\Vert h{\Vert }_H^2$, not an inner product of two $H$-valued observations. The formula follows from the Cameron-Martin theorem for a Gaussian measure in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On ${\mathbb{R}}^2$, generalized noise can be realized on tempered distributions; an unweighted global negative Sobolev space is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.

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 identifies a centered, scaled posterior distribution with the Gaussian law suggested by an efficient likelihood approximation. In an infinite-dimensional model, this conclusion requires control of prior-induced bias; a positive marginal prior density alone does not state enough assumptions.

A concrete calculation shows the issue. In the Gaussian sequence model, let $Y_k={\theta }_{0k}+n^{-1/2}{g}_k$, with ${\theta }_0\in {\ell }^2$, and assign independent priors ${\theta }_k\sim N(0,{\tau }_k^2)$ with every ${\tau }_k^2>0$ and ${\sum }_k{\tau }_k^2<\infty$. Put $a_{nk}=n{\tau }_k^2/(1+n{\tau }_k^2)$. For a nonzero coefficient vector $l\in {\ell }^2$, the linear functional $L(\theta )={\sum }_k{l}_k{\theta }_k$ has the exact posterior lawThe efficient observation statistic $L(Y)$ exists as a Gaussian noise pairing even though $Y$ itself is not in ${\ell }^2$. dominated convergence gives ${\sum }_k{l}_k^2{a}_{nk}\to \Vert l{\Vert }_2^2$. To center at $L(Y)$, however, one also needsThe remaining noise part of the scaled centering error has variance ${\sum }_k{l}_k^2(1-a_{nk}{)}^2\to 0$. Under this displayed bias condition, the conditional law of $\sqrt{n}[L(\theta )-L(Y)]$ converges in total variation distance, in sampling probability, to $N(0,\Vert l{\Vert }_2^2)$. Without bias control, convergence of posterior variances is insufficient. The calculation is specific to conjugate product Gaussian priors; general nonparametric results impose their own prior, truth and functional regularity hypotheses.^[ref]

A Gaussian net test turns a finite covering of an alternative into a test with an explicit covering-number cost. In the Gaussian sequence model $Y=\theta +n^{-1/2}g$, compare a null ${\theta }_0$ with centers ${\theta }_1,\dots ,{\theta }_N$ in L2 space satisfying $\Vert {\theta }_j-{\theta }_0{\Vert }_2\ge r>0$. Put ${\delta }_j={\theta }_j-{\theta }_0$ and reject if anyEach pairing is well-defined as an isonormal Gaussian process pairing even though the noise sequence itself is not in L2 space. Under the null, it is centered normal with variance $\Vert {\delta }_j{\Vert }_2^2/n$ before subtracting the threshold. The Gaussian tail bound and union bound giveIf $\Vert \theta -{\theta }_j{\Vert }_2\le r/4$ for some center, the mean of that center's displayed statistic is at least $\Vert {\delta }_j{\Vert }_2^2/4$, by the Cauchy-Schwarz inequality. ThusA radius-$r/4$ net with centers in the separated alternative supplies such a test for its entire covered set. To make the first error small, the squared separation $n{r}^2$ must dominate $\log N$. This is why metric complexity, rather than just individual Gaussian tails, enters composite testing.

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.