Type-H histogram posterior weak Bernstein–von Mises theorem

Abbreviate $m=m_n$, $a_k=a_{k,L_n}$, $A={\sum }_k{a}_k$, ${\beta }_k=N_{nk}+a_k$, and $B=n+A$. The original cutoff conditions and $\alpha >1\mathrm{/}2$ give $m\mathrm{/}\sqrt{n}\to 0$ and $m\log n\mathrm{/}n\to 0$, and $A\le c_{2}m$. On the positive simplex, the full-data histogram likelihood is $m^n{\prod }_k{\omega }_k^{N_{nk}}$. Multiplication by the prior density yields a constant times ${\prod }_k{\omega }_k^{{\beta }_k-1}$, with every shape positive; its normalization is finite positive by [F2]. Therefore the full-data posterior is Dirichlet$({\beta }_k{)}_k$, regardless of whether the actual sampling truth is itself a histogram. By [F1] realize it by ${\omega }_k=Z_k\mathrm{/}S$. Write $R_n$ for the posterior mean histogram measure, identified with its density in coefficient differences, with cell masses ${\beta }_k\mathrm{/}B$. For a cell-constant function $q$, set $T(q)={\sum }_k{q}_k(Z_k-{\beta }_k)$. Direct subtraction gives $\sqrt{n}(\omega q-R_{n}q)=(\sqrt{n}\mathrm{/}S)T(q-R_{n}q)$.

Fix constants $t>0$ and $C$ large enough that $c:=tC-D_0(e^t-1)>0$. Since $N_{nk}$ is Binomial$(n,p_k)$ with $p_k=P_0{I}_{L_n,k}\le D_0\mathrm{/}m$, independence of the observations gives $\mathbb{E}{e}^{t{N}_{nk}}=(1+p_k(e^t-1){)}^n\le e^{n{D}_0(e^t-1)\mathrm{/}m}$. Hence $E_n=\{{\max }_k{N}_{nk}\le Cn\mathrm{/}m\}$ satisfies $P_0^n(E_n^c)\le m{e}^{-cn\mathrm{/}m}\to 0$, by exponential Markov and the union bound. On $E_n$, for all sufficiently large $n$, ${\beta }_k\le C^{\mathrm{\prime }}n\mathrm{/}m$, $B\le 2n$, and both the empirical histogram density and the mean histogram density are bounded by a fixed constant $C^{\mathrm{\prime }}$ independent of $n$.

Fix finitely many Haar coordinates. Choose a fixed coarse dyadic partition of $r$ cells on which all their functions are constant. For large $n$ its cells are unions of level-$L_n$ cells. By [F1], the corresponding grouped Gamma variables $Z_{nj}$ are independent Gamma$(b_{nj},1)$ with $b_{nj}=N_n(J_j)+{\sum }_{I_{L_n,k}\subset J_j}{a}_k$. Their summed shape is still $B$. Write $p_j=P_0(J_j)>0$. Each coarse count satisfies $\mathbb{E}[N_n(J_j)\mathrm{/}n]=p_j$ and variance at most $1\mathrm{/}(4n)$, by observation independence. Markov applied to squared deviations, together with $A\mathrm{/}n\to 0$, proves $b_{nj}\mathrm{/}n\to p_j$ in $P_0^n$-probability simultaneously for the fixed finite partition.

On $E_n$, take any Haar detail $\psi ={\psi }_{lk}$ with $l<L_n$ and put $q=\psi -R_n\psi$. Then $\mathrm{\mid }q\mathrm{\mid }\le 22^{l\mathrm{/}2}$ and $V={\sum }_k{\beta }_k{q}_k^2=B{\mathrm{Var}}_{R_n}(\psi )\le B{R}_n{\psi }^2\le C^{\mathrm{\prime }}n$, using $\int {\psi }^2=1$ and the density bound. By [F1] and [F7], $\mathrm{Pr}(S<B\mathrm{/}2\mid X^n)\le 4\mathrm{/}B$, because $\mathrm{Var}(S)=B$. On the complement, step 1.1 yields $\mathrm{\mid }\sqrt{n}(\omega \psi -R_n\psi )\mathrm{\mid }\le 2\mathrm{\mid }T(q)\mathrm{\mid }\mathrm{/}\sqrt{n}$. Thus, apart from the single event $\{S<B\mathrm{/}2\}$, the conditional probability that this absolute coefficient exceeds $x$ is at most $2\exp \{-c_0{x}^2\mathrm{/}(1+2^{l\mathrm{/}2}x\mathrm{/}\sqrt{n})\}$ for a fixed $c_0>0$. The zero-variance case has no such failure event, by [F1].

Consider any deterministic data sequence along which $b_{nj}\mathrm{/}n\to p_j$ for every $j$. Under its conditional Gamma product laws put $Y_{nj}=(Z_{nj}-b_{nj})\mathrm{/}\sqrt{n}$. For every fixed $t\in {\mathbb{R}}^r$, [F1] and the logarithm series give $\log \mathbb{E}{e}^{t\cdot Y_n}={\sum }_j{b}_{nj}[-\log (1-t_j\mathrm{/}\sqrt{n})-t_j\mathrm{/}\sqrt{n}]\to \frac{1}{2}{\sum }_j{p}_j{t}_j^2$. The remainder is $O(n^{-3\mathrm{/}2}{\sum }_j{b}_{nj}\mathrm{\mid }{t}_j{\mathrm{\mid }}^3)=O(n^{-1\mathrm{/}2})$. The means of $Y_n$ are zero and their total second moments are $B\mathrm{/}n=O(1)$, so their laws are tight by the squared-norm Markov bound and compactness of finite-dimensional closed balls. By [F5], any subsequence has a further weak limit. For any fixed $t$, the same expansion at $2t$ uniformly bounds $\mathbb{E}{e}^{2t\cdot Y_n}$; hence $e^{t\cdot Y_n}$ is uniformly integrable. Truncation of this continuous exponential therefore passes its expectation through each weakly convergent subsequence. The limit law has transform $\exp (\frac{1}{2}{\sum }_j{p}_j{t}_j^2)$ and an exponential moment on a neighborhood of zero: $e^{\delta {\sum }_j\mathrm{\mid }{y}_j\mathrm{\mid }}\le {\sum }_{ϵ\in \{-1,1{\}}^r}{e}^{\delta ϵ\cdot y}$ and the corresponding transforms are finite. The independent centered normal vector with variances $p_j$ has this same transform, by completing the square in each normal density. Transform uniqueness [F5] identifies every subsequential limit, proving $Y_n\Rightarrow N_r(0,\mathrm{diag}(p))$.

Choose a fixed $K>0$ sufficiently large. Uniformly over $l<L_n$, $2^{l\mathrm{/}2}K\sqrt{l+1}\mathrm{/}\sqrt{n}\le K\sqrt{m(L_n+1)\mathrm{/}n}\to 0$. Summing the exponential bounds of step 3.1 over levels $J<l<L_n$ and their $2^l$ coefficients therefore gives, for all large $n$, ${\mathrm{\Pi }}_n\{{\max }_{J<l<L_n,k}\mathrm{\mid }\sqrt{n}(f-R_n{)}_{lk}\mathrm{\mid }\mathrm{/}\sqrt{l+1}>K\mid X^n\}\le 4\mathrm{/}B+{\sum }_{l>J}{2}^{l+1}{e}^{-c_1{K}^2(l+1)}$ on $E_n$, with fixed $c_1>0$. The probability $4\mathrm{/}B$ is counted only once, since the same $S$ normalizes every coordinate. Choose $K$ with $c_1{K}^2>2\log 2$; the displayed series tail then tends to zero with $J$. For any $\epsilon >0$, admissibility gives $K\sqrt{l+1}\le \epsilon w_l$ for all sufficiently large $l$. Both $f$ and $R_n$ are histograms, so all their detail coordinates at $l\ge L_n$ vanish. Consequently their conditional $M(w)$ tails vanish in the iterated limit of [F4], first $n\to \mathrm{\infty }$, then $J\to \mathrm{\infty }$, since $P_0^n(E_n^c)\to 0$.

For a Haar detail $\psi ={\psi }_{lk}$, exact subtraction gives $\sqrt{n}(R_n\psi -P_n\psi )=\sqrt{n}[{\sum }_k{a}_k{\psi }_k-A{P}_n\psi ]\mathrm{/}B$. Haar cell values satisfy $m^{-1}{\sum }_k\mathrm{\mid }{\psi }_k\mathrm{\mid }=\int \mathrm{\mid }\psi \mathrm{\mid }=2^{-l\mathrm{/}2}\le 1$. Hence $\mathrm{\mid }{\sum }_k{a}_k{\psi }_k\mathrm{\mid }\le c_{2}m$, while on $E_n$, $\mathrm{\mid }{P}_n\psi \mathrm{\mid }\le C\int \mathrm{\mid }\psi \mathrm{\mid }\le C$. This bounds every such mean shift by $C^{\mathrm{\prime }\mathrm{\prime }}m\mathrm{/}\sqrt{n}\to 0$. The scaling-coordinate shift is exactly zero, and all coordinates at $l\ge L_n$ vanish for both centers. Since $w_l\ge 1$, $\mathrm{\parallel }\sqrt{n}(R_n-h_n){\mathrm{\parallel }}_{M(w)}\to 0$ uniformly on $E_n$. In particular step 4.1 also supplies the required posterior tails about $h_n$.

Here $(S-B)\mathrm{/}\sqrt{n}={\sum }_j{Y}_{nj}$, $S\mathrm{/}n\to 1$ in conditional probability by its variance $B\mathrm{/}{n}^2\to 0$ and mean $B\mathrm{/}n\to 1$. Exact algebra gives $\sqrt{n}(Z_{nj}\mathrm{/}S-b_{nj}\mathrm{/}B)=(n\mathrm{/}S)[Y_{nj}-(b_{nj}\mathrm{/}B){\sum }_h{Y}_{nh}]$. By [F6] and step 3.2, this vector converges to $(U_j-p_j{\sum }_h{U}_h{)}_j$, with $U_j$ independent $N(0,p_j)$. Its covariance is $\mathrm{diag}(p)-p{p}^T$. Applying the fixed linear map from coarse-cell masses to the chosen Haar coordinates yields covariance exactly $P_0[(\psi -P_0\psi )({\psi }^{\mathrm{\prime }}-P_0{\psi }^{\mathrm{\prime }})]$. Step 5.1 removes the mean shift from $R_n$ to $h_n$ on $E_n$. Thus along every deterministic data sequence with the ratios in step 2.2 converging and $E_n$ eventually occurring, the finite-coordinate posterior converges weakly, hence in BL by [F6], to the corresponding projection of [F3].

The preceding deterministic-sequence conclusion implies the finite-coordinate BL convergence in $P_0^n$-probability. To see this without a simultaneous conditional null-set assertion, if that convergence failed, there would be $\epsilon ,\eta >0$ and infinitely many $n$ with probability at least $\eta$ that the distance exceeds $\epsilon$. Along a subsequence, step 2.2 and $P_0^n(E_n^c)\to 0$ make the probability of a ratio error above $1\mathrm{/}q$ or $E_n^c$ smaller than $\eta \mathrm{/}2$ at its $q$th chosen index. Thus at each such index there exists data with distance above $\epsilon$, ratio error below $1\mathrm{/}q$, and $E_n$ true. AC selects a deterministic sequence of those data, contradicting step 6.1. This proves the asserted convergence in probability for each fixed finite projection.

The posterior histogram construction [F8] and its finite-dimensional simplex kernel make $Q_n$ a data-indexed Borel probability on $M_0(w)$: each draw and $h_n$ has only finitely many Haar coefficients, and its map from simplex coordinates is continuous. The target is the Borel bridge law [F3]. Step 7.1 supplies every fixed finite-projection BL limit, and steps 4.1, 5.1 supply its posterior-tail condition. Apply [F4] to obtain the stated $M_0(w)$ BL convergence. Every original type-H assumption has been retained; in particular the lower parameter bound is compatible with, but is not needed beyond, positivity in this conjugate proof.