Countably partitioned sieves yield posterior contraction by weighted entropy

Put $x_n=n{ϵ}_n^2$, $y_n=n{\delta }_n^2$, and let $D_n$ be the likelihood denominator relative to $p_0^n$. By [F1], on an event ${\mathrm{\Omega }}_n$ with $P_0^n({\mathrm{\Omega }}_n^c)\le y_n^{-1}$, $D_n\ge e^{-(C_0+2)y_n}$.

Write $T_n={\sum }_j\sqrt{N_{n,j}{\pi }_{n,j}}$ and $s_n=e^{-x_n}{T}_n$, so $s_n\to 0$. Fix $K>2(C_0+3)$ and split indices into active $J_n^{+}=\{j:{\pi }_{n,j}\ge e^{-K{x}_n}\}$ and small-mass $J_n^{-}$. Since $N_{n,j}\ge 1$, ${\sum }_{J_n^{-}}{\pi }_{n,j}\le e^{-K{x}_n\mathrm{/}2}{T}_n=e^{(1-K\mathrm{/}2)x_n}{s}_n$. For active $j$, $\sqrt{N_{n,j}{\pi }_{n,j}}\le T_n=e^{x_n}{s}_n$, hence $\log (N_{n,j}\mathrm{/}{\pi }_{n,j})\le (2+2K)x_n$ for all sufficiently large $n$.

Fix a sufficiently large constant $M$. If $M{ϵ}_n\ge \sqrt{2}$, then the alternative is empty because Hellinger distance is at most $\sqrt{2}$, and the desired posterior mass is zero for this $n$. For the remaining $n$, choose, by [A1], a finite ${ϵ}_n$-cover with centers in each active stratum's density image, as allowed by [F9]. Discard centers whose closed ${ϵ}_n$-ball meets no density in the stratum with $h(p_{\theta },p_0)>M{ϵ}_n$; at most $N_{n,j}$ centers remain, and they cover that alternative. For every retained center $q$, the triangle inequality gives $h(q,p_0)>(M-1){ϵ}_n$. Set $m_n=\lceil \frac{2\log 8}{(M-1{)}^2{ϵ}_n^2}\rceil$. On a block of $m_n$ observations define $T_q=\mathbf{1}\{{\prod }_{i}q(X_i)>{\prod }_i{p}_0(X_i)\}$. On rejection, $p_0^{\otimes m_n}\le \sqrt{p_0^{\otimes m_n}{q}^{\otimes m_n}}$, so its type-I error is at most the product affinity; on acceptance, $q^{\otimes m_n}\le \sqrt{p_0^{\otimes m_n}{q}^{\otimes m_n}}$, so its type-II error under $q$ has the same bound. By [F2], this affinity is $(1-h^2(q,p_0)\mathrm{/}2{)}^{m_n}\le e^{-m_n{h}^2(q,p_0)\mathrm{/}2}\le 1\mathrm{/}8$.

If $h(p_{\theta },q)\le {ϵ}_n$, product affinity gives $h^2(P_{\theta }^{m_n},Q^{m_n})=2[1-{(1-h^2(p_{\theta },q)\mathrm{/}2)}^{m_n}]\le m_n{ϵ}_n^2$, using $1-(1-u{)}^m\le mu$ for $u\in [0,1]$; [F3] bounds their total variation by $\sqrt{m_n}{ϵ}_n$. Since $M{ϵ}_n<\sqrt{2}$, ${ϵ}_n^2<2\mathrm{/}{M}^2$ and $m_n{ϵ}_n^2\le \frac{2\log 8}{(M-1{)}^2}+\frac{2}{M^2}\le 1\mathrm{/}64$ for sufficiently large $M$. The total variation is then at most $1\mathrm{/}8$. Therefore the block test in step 1.3 rejects with probability at least $3\mathrm{/}4$ under every $p_{\theta }$ in the alternative and at most $1\mathrm{/}8$ under $p_0$.

Divide $L_n=\lfloor n\mathrm{/}{m}_n\rfloor$ disjoint blocks and let $Y_k$ be their block-test indicators. Under each IID law the indicators are independent by [F6]; their means are at most $a=1\mathrm{/}8$ under $P_0$ and at least $b=3\mathrm{/}4$ under every alternative by step 2.1. For independent Bernoulli variables with means at most $a$, exponential Markov [F7] gives $P_0(L_n^{-1}\sum Y_k\ge t)\le e^{-L_{n}D(t\mathrm{\parallel }a)}$; with means at least $b$, the negative-exponential bound gives $P_{\theta }(L_n^{-1}\sum Y_k<t)\le e^{-L_{n}D(t\mathrm{\parallel }b)}$, where $D(t\mathrm{\parallel }u)=t\log (t\mathrm{/}u)+(1-t)\log ((1-t)\mathrm{/}(1-u))$. These bounds follow by bounding each moment generating function by $1-u+u{e}^{\lambda }$ (using $u=a$ for $\lambda >0$, $u=b$ for $\lambda <0$) and optimizing over $\lambda$.

The function $D(t\mathrm{\parallel }a)-D(t\mathrm{\parallel }b)$ is affine and strictly increasing on $[a,b]$, is negative at $a$, and equals $D(b\mathrm{\parallel }a)>0$ at $b$. Let $t_{\ast }$ be its unique zero. By AC [A1], for each active $j$ choose $t_j\in [t_{\ast },b]$ such that $L_n\{D(t_j\mathrm{\parallel }a)-D(t_j\mathrm{\parallel }b)\}=\log (N_{n,j}\mathrm{/}{\pi }_{n,j})$. This is possible after increasing $M$: writing $A=2\log 8$, $M{ϵ}_n<\sqrt{2}$ implies $A\mathrm{/}((M-1{)}^2{ϵ}_n^2)>2$ for sufficiently large $M$, so the ceiling in $m_n$ costs at most a fixed factor; since $n{ϵ}_n^2\to \mathrm{\infty }$, the floor in $L_n$ also costs at most a fixed factor eventually. Hence $L_n\ge c{M}^2{x}_n$ for a universal $c>0$, while step 1.2 bounds the right side by $(2+2K)x_n$. Define ${\varphi }_{n,j}$ as the maximum of the block-majority tests over its retained centers (or zero if there are none), using threshold $t_j$. Continuity on $[t_{\ast },b]$ and positivity of both divergences at $t_{\ast }$ give a constant $c_{\ast }>0$ such that $(D(t\mathrm{\parallel }a)+D(t\mathrm{\parallel }b))\mathrm{/}2\ge c_{\ast }$ there. The null union bound and step 3.1 give $P_0^n({\varphi }_{n,j}=1)\le \sqrt{N_{n,j}{\pi }_{n,j}}{e}^{-c_{\ast }{L}_n}$; the same Chernoff bound for a center within ${ϵ}_n$ of each alternative gives ${\sup }_{\theta \in S_{n,j}:h(p_{\theta },p_0)>M{ϵ}_n}{P}_{\theta }^n({\varphi }_{n,j}=0)\le \sqrt{N_{n,j}\mathrm{/}{\pi }_{n,j}}{e}^{-c_{\ast }{L}_n}$ by step 2.1.

Let ${\varphi }_n$ be one on the countable union of the rejection events of ${\varphi }_{n,j}$ over $j\in J_n^{+}$ and zero otherwise; it is measurable as a countable union of measurable events. The union bound and step 4.1 give $P_0^n({\varphi }_n=1)\le T_n{e}^{-c_{\ast }{L}_n}$. For the tested likelihood numerator on an active stratum, Tonelli [F4], the jointly measurable likelihood interface [F5] and change of density give $E_0[(1-{\varphi }_n){\int }_{\{\theta \in S_{n,j}:h(p_{\theta },p_0)>M{ϵ}_n\}}{\prod }_i\frac{p_{\theta }(X_i)}{p_0(X_i)}{\mathrm{\Pi }}_n(d\theta )]\le {\pi }_{n,j}{\sup }_{\theta \text{ in the set}}{P}_{\theta }^n(1-{\varphi }_{n,j})\le \sqrt{N_{n,j}{\pi }_{n,j}}{e}^{-c_{\ast }{L}_n}\mathrm{.}$

Write $d_n=e^{-(C_0+2)y_n}$. For the active numerator $R_n^{+}$, step 5.1 gives $P_0^n({\varphi }_n=1)\le T_n{e}^{-c_{\ast }{L}_n}$ and $E_0[(1-{\varphi }_n)R_n^{+}]\le T_n{e}^{-c_{\ast }{L}_n}$. Thus, for every fixed $\zeta >0$, $P_0^n(R_n^{+}\mathrm{/}{D}_n>\zeta )\le P_0^n(D_n<d_n)+T_n{e}^{-c_{\ast }{L}_n}+{\zeta }^{-1}{d}_n^{-1}{T}_n{e}^{-c_{\ast }{L}_n}$. This tends to zero for sufficiently large $M$, since $T_n=e^{x_n}{s}_n$, $y_n\le x_n$, $L_n\ge c{M}^2{x}_n$, and $s_n\to 0$. For the small-mass numerator $R_n^{-}$, [F8] and Tonelli give $E_0[R_n^{-}]\le {\sum }_{J_n^{-}}{\pi }_{n,j}\le e^{-K{x}_n\mathrm{/}2}{T}_n$; hence its posterior mass tends to zero in probability by Markov and the denominator event, because $e^{(C_0+2)y_n}{e}^{-K{x}_n\mathrm{/}2}{T}_n\le s_n{e}^{(C_0+3-K\mathrm{/}2)x_n}\to 0$ when $K>2(C_0+3)$.

The posterior outside the $M{ϵ}_n$-ball is the union of its active-stratum part, its small-mass-stratum part, and the sieve complement. Step 6.1 sends the first two parts to zero in probability, and the sieve-complement term tends to zero by hypothesis.