Type-S posterior contracts in quotient log-density coordinates without a tail restriction on a log-Lipschitz prior

Quotient out constants before using likelihood curvature. Constants belong to the fixed low scaling space. An orthogonal change within that finite space separates $T=\theta +c1$; integrate $c$ out of the prior and write $q_L$ for the resulting Lebesgue density on $E_L$. The likelihood depends only on $\theta$. In S1 its product density before integration obeys $p(\theta +u,c)\mathrm{/}p(\theta ,c)\in [e^{-B_L\mathrm{\parallel }u{\mathrm{\parallel }}_2},e^{B_L\mathrm{\parallel }u{\mathrm{\parallel }}_2}]$, where $B_L^2\le C{\sum }_{l\le L}{2}^l{\sigma }_l^{-2}\le C{d}^{2\alpha +2}$. Integrating preserves both inequalities for $q_L$; it is positive, finite almost everywhere, and the log-Lipschitz inequality supplies its positive continuous finite version everywhere. In S2 the marginal is a nondegenerate Gaussian on $E_L$, with precision $Q_L$ equal to ${\sigma }_l^{-2}$ on high wavelets and a fixed positive definite low block; hence $\mathrm{\parallel }{Q}_L\mathrm{\parallel }\le C{d}^{2r+1}$.

Let $f_{\ast }=e^{{\theta }_{\ast }}\mathrm{/}\int e^{{\theta }_{\ast }}$. [F1] gives $\mathrm{\parallel }{f}_{\ast }-f_0{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{d}^{-\alpha }$. For $\mathrm{\ell }(\theta )=n(P_n\theta -\log \int e^{\theta })$, the centered score at ${\theta }_{\ast }$ has expected squared Euclidean norm at most $Cnd$: sum the coordinate variances, each bounded by $D_0$ by orthonormality. The score's deterministic mean is the coefficient vector of $n(f_0-f_{\ast })$, with norm at most $Cn{d}^{-\alpha }$. Consequently [F3] gives $\mathrm{\parallel }\mathrm{\nabla }\mathrm{\ell }({\theta }_{\ast }){\mathrm{\parallel }}_2=O_P(n{e}_n)$. On every ball $\mathrm{\parallel }u{\mathrm{\parallel }}_2\le R{e}_n$ with fixed $R$, [F1] and $\sqrt{d}{e}_n\to 0$ make $f_{{\theta }_{\ast }+u}$ bounded uniformly above and away from zero. For $v\in E_L$, ${\mathrm{Var}}_f(v)={\inf }_a\int (v-a{)}^{2}f$ lies between $c\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$ and $C\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$, using $\int v=0$. Thus $-{\mathrm{\nabla }}^2\mathrm{\ell }$ is between $cnI$ and $CnI$ on these balls. Convexity of $-\mathrm{\ell }$ holds globally.

In S1, $B_L\le Cn{e}_n$ by either cutoff. In S2 the Hölder coefficient bound gives $G_L^2:=\mathrm{\parallel }{Q}_L{\theta }_{\ast }{\mathrm{\parallel }}_2^2\le C[1+{\sum }_{l\le L}{2}^{l(4r+2-2\alpha )}]$, so $G_L\le Cn{e}_n$, and $\mathrm{\parallel }{Q}_L\mathrm{\parallel }\mathrm{/}n\to 0$ because $r<\alpha$. The Gaussian log-density increment is exactly $-⟨Q_L{\theta }_{\ast },u⟩-⟨Q_{L}u,u⟩\mathrm{/}2$. In particular its radial upper slope is at most $G_L$, and on the $e_n$ ball it is bounded below by $-G_L{e}_n-\mathrm{\parallel }{Q}_L\mathrm{\parallel }{e}_n^2\mathrm{/}2$.

Fix a score event $\mathrm{\parallel }\mathrm{\nabla }\mathrm{\ell }({\theta }_{\ast }){\mathrm{\parallel }}_2\le An{e}_n$. Put $r_0=R{e}_n$, with $R$ a large fixed constant. Taylor's formula on the $r_0$ ball bounds $\mathrm{\ell }({\theta }_{\ast }+r_0\omega )-\mathrm{\ell }({\theta }_{\ast })$ above by $An{e}_n{r}_0-cn{r}_0^2\mathrm{/}2$ and its radial derivative above by $An{e}_n-cn{r}_0$. Global convexity of $-\mathrm{\ell }$ extends this derivative bound to every $r\ge r_0$. Add the prior bound $B_{L}r$ in S1, or $G_{L}r$ with its nonpositive quadratic term in S2. For $R$ large enough the log unnormalized posterior ratio $H(r\omega )$ relative to ${\theta }_{\ast }$ is at most $-c^{\mathrm{\prime }}n{r}_{0}r$ for all $r\ge r_0$. On $r\le e_n$, the lower Taylor bound and the prior lower bounds instead give $H\ge -C_{A}n{e}_n^2$.

Polar integration in dimension $m=d-1$ bounds the tail/normalizer ratio by $e^{C_{A}n{e}_n^2}m{\int }_R^{\mathrm{\infty }}{t}^{m-1}{e}^{-c^{\mathrm{\prime }}nR{e}_n^{2}t}dt$. Here $n{e}_n^2\ge d>m$. Choose $R$ so large that $\log t\le c^{\mathrm{\prime }}Rt\mathrm{/}4$ for $t\ge R$ and $C_A\le c^{\mathrm{\prime }}{R}^2\mathrm{/}4$. The integral is then at most $[m\mathrm{/}(c^{\mathrm{\prime }\mathrm{\prime }}nR{e}_n^2)]e^{-c^{\mathrm{\prime }\mathrm{\prime }}n{R}^2{e}_n^2}$ after decreasing $c^{\mathrm{\prime }\mathrm{\prime }}>0$, and tends to zero. The denominator used here is the integral on the $e_n$ ball, at least its volume times $e^{-C_{A}n{e}_n^2}$ times the positive unnormalized value at ${\theta }_{\ast }$. First increase $A$ to make its score event arbitrarily probable, then increase $R$. This proves tightness. For any $b_n\to \mathrm{\infty }$, eventual containment of every fixed $R{e}_n$ ball in the $b_n{e}_n$ ball yields the stated equivalent formulation.

Every modeled density is everywhere positive. Its prior predictive product density is positive and integrates to one by [F2], so it is finite Lebesgue-almost everywhere on the sample cube. The law $P_0^n$ has a bounded Lebesgue density, hence the same exceptional set is $P_0^n$-null. This proves finite positive evidence on the asserted fibers independently of the score event.