Posterior contraction rate in a Gaussian sequence inverse experiment

Let ${\mathbb{N}}_{>0}=\{1,2,3,\dots \}$. For each positive noise level $ϵ$, let ${\mathrm{\Pi }}_{0,ϵ}$ be a proper Borel prior on ${\mathrm{\ell }}^2({\mathbb{N}}_{>0},\mathbb{R})$ in the Gaussian sequence inverse experiment, and let ${\mathbb{P}}_{{\mathrm{\Pi }}_{0,ϵ}}$ be the resulting prior-predictive joint law of parameter and data. A posterior kernel is a data-measurable assignment $y\mapsto {\mathrm{\Pi }}_{ϵ}(\cdot \mid y)$ of Borel probability measures on ${\mathrm{\ell }}^2$ satisfying ${\mathbb{P}}_{{\mathrm{\Pi }}_{0,ϵ}}(\theta \in A,Y\in B)={\int }_B{\mathrm{\Pi }}_{ϵ}(A\mid y){\mathbb{P}}_{{\mathrm{\Pi }}_{0,ϵ}}^Y(dy)$ for every Borel $A\subseteq {\mathrm{\ell }}^2$ and every product-Borel $B\subseteq {\mathbb{R}}^{{\mathbb{N}}_{>0}}$, where ${\mathbb{P}}_{{\mathrm{\Pi }}_{0,ϵ}}^Y$ is the prior-predictive data marginal. Fix a truth ${\theta }^{†}\in {\mathrm{\ell }}^2$ and write ${\mathbb{P}}_{{\theta }^{†}}^{ϵ}$ for its data law. Since a conditional-law version is specified only up to a prior-predictive null set, any contraction claim under ${\mathbb{P}}_{{\theta }^{†}}^{ϵ}$ refers to the stated version. A positive function $r_{ϵ}\to 0$ is a posterior contraction rate in the ${\mathrm{\ell }}^2$ norm at ${\theta }^{†}$ if, for every positive function $M_{ϵ}\to \mathrm{\infty }$, ${\mathbb{E}}_{{\theta }^{†}}^{ϵ}\text{\hspace{0.17em}⁣}\left[{\mathrm{\Pi }}_{ϵ}\text{\hspace{0.17em}⁣}(\{\theta \in {\mathrm{\ell }}^2:\mathrm{\parallel }\theta -{\theta }^{†}{\mathrm{\parallel }}_2>M_{ϵ}{r}_{ϵ}\}\mid Y)\right]⟶0\text{as }ϵ\downarrow 0.$

The parameter is square-summable, the data space has its product Borel sigma-algebra, and the signal coordinates use positive singular values.

A posterior version is a data-indexed conditional law under the prior-predictive joint distribution.

Knapik, van der Vaart, and van Zanten, Bayesian inverse problems with Gaussian priors, §4, Theorem 4.1

Johnstone, Gaussian estimation: Sequence and wavelet models, §2.3