A one-point logistic GP regression satisfies the L² link branch

Differentiation gives ${\mathrm{\Psi }}^{\mathrm{\prime }}(u)=\mathrm{\Psi }(u)(1-\mathrm{\Psi }(u))$, so $\mathrm{\mid }{\mathrm{\Psi }}^{\mathrm{\prime }}(u)\mathrm{\mid }\le 1\mathrm{/}4$ and $\mathrm{\mid }{\mathrm{\Psi }}^{\mathrm{\prime }}(u)\mathrm{/}[\mathrm{\Psi }(u)(1-\mathrm{\Psi }(u))]\mathrm{\mid }=1$. The true response probability is $\mathrm{\Psi }(0)=1\mathrm{/}2\in (0,1)$, and $w_0=0$ is in the support of the scalar standard-normal prior.

Here $L^2(G)$ is isometric to $\mathbb{R}$ and $\mathrm{\parallel }W{\mathrm{\parallel }}_{L^2(G)}=\mathrm{\mid }Z\mathrm{\mid }$. The RKHS approximation cost at zero is zero. By [F1], for $0<ϵ\le 1$, $\mathrm{Pr}(\mathrm{\mid }Z\mathrm{\mid }<ϵ)\ge 2\phi (1)ϵ$, so ${\varphi }_0({ϵ}_n)\le -\log (2\phi (1){ϵ}_n)$. With ${ϵ}_n=\sqrt{\log n\mathrm{/}n}$, this is at most $n{ϵ}_n^2=\log n$ for all sufficiently large n.

The joint Bernoulli density relative to $G\otimes \mathrm{\#}$ is bounded by one, and the constant-process evaluation is jointly measurable. Steps 1.1–1.2 verify the remaining link, truth-support and concentration conditions; applying the binary-regression branch of [F2] yields expected posterior contraction in $L^2(G)$.