A Hölder-density entropy bound gives the source-covered rate

Let $0<\alpha \le 1$ and let ${\mathcal{H}}^{\alpha }(L)$ be the class of measurable probability densities on $[0,1]$ satisfying $0\le p\le L$ and $\mathrm{\mid }p(x)-p(y)\mathrm{\mid }\le L\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$. Suppose a dominated IID model has truth $p_0\in {\mathcal{H}}^{\alpha }(L)$, density image ${\mathcal{F}}_n={\mathcal{H}}^{\alpha }(L)$ at each $n$, and a proper prior $\mathrm{\Pi }$ on a measurable parameterization of this class with a jointly measurable density family. Assume its Hellinger packing numbers obey $\log D(u,{\mathcal{F}}_n,h)\le C_H{u}^{-1\mathrm{/}\alpha }$ for all sufficiently small $u>0$. Fix $C>0$, choose $K\ge \max \{1,C_H^{\alpha \mathrm{/}(2\alpha +1)}\}$, and set ${ϵ}_0:=K$ and ${ϵ}_n:=K{n}^{-\alpha \mathrm{/}(2\alpha +1)}$ for $n\ge 1$. Assume that, for all sufficiently large $n$, $\mathrm{\Pi }\text{\hspace{0.17em}⁣}\{\theta :\mathrm{K}\mathrm{L}(P_0\mathrm{\parallel }{P}_{\theta })\le {ϵ}_n^2,\int {[\log \frac{p_{\theta }}{p_0}]}^2{p}_{0}d\mu \le {ϵ}_n^2\}\ge e^{-Cn{ϵ}_n^2}\mathrm{.}$ Here $\mu$ is Lebesgue measure on $[0,1]$, and the log-square expression is the second moment, with the weighted integrand zero where $p_0=0$ and $+\mathrm{\infty }$ where $p_0>0,p_{\theta }=0$, as in IID posterior contraction by entropy and prior mass. Take the full parameter space as the sieve, so its prior tail is zero. Then the entropy balance gives the source-covered rate ${ϵ}_n\asymp n^{-\alpha \mathrm{/}(2\alpha +1)}$ under the theorem's sufficient conditions.