Function-space pCN and invariant posterior kernels — Examples and Counterexamples

By [A1] and [F1], take a countable independent sequence $(Z_n{)}_{n\ge 1}$ with law $N(0,1)$ and put $X_j=Z_{2j-1}$, $Y_j=Z_{2j}$. Each subsequence is again IID standard normal. The odd and even coordinate sigma-algebras are independent by [F4]. Apply [F3] with $H_0=H_1=H$, $w_j=1$, and ${\lambda }_j=2^{-j}$ to each subsequence. Since ${\sum }_j{2}^{-j}=1$, both series converge almost surely in $H$ and define centered Borel Gaussian random elements. Their finite coordinate laws agree: by [F3], each finite coordinate vector is obtained by multiplying independent standard normals by the same coefficients $2^{-j\mathrm{/}2}$, so each vector has the same product normal law by [F5] and [F9]. AC implies Countable Choice by [A1], so [F13] identifies their Borel laws. Each limit is measurable with respect to its own coordinate sigma-algebra: its partial sums are block-measurable, the event that they are Cauchy is a countable combination of measurable norm inequalities, and the limit, set to zero on the null complement, is therefore block-measurable. In detail, for a proper open $O\subset H$ with closed complement $C$, on the convergence event the preimage of $O$ is the condition ${\mathrm{lim\hspace{0.17em}inf}}_N\mathrm{dist}(S_N,C)>0$; the distance function is 1-Lipschitz, and the zero value assigned off that event is included exactly when $0\in O$. This is a measurable condition on the block. The odd and even coordinate sigma-algebras are independent by [F4], hence $U$ and $\xi$ are independent. Denote their common law by ${\mu }_0$.

For $(u,v)\in H\times H$, set $f_n(u,v):=\frac{1}{n}{\sum }_{j=1}^n{2}^j(v_j^2-u_j^2),A_{\beta }:=\{(u,v):f_n(u,v)\to {\beta }^2\}\mathrm{.}$ Each coordinate map is continuous since $\mathrm{\mid }{u}_j-v_j\mathrm{\mid }\le \mathrm{\parallel }u-v{\mathrm{\parallel }}_2$ by [F2]; finite arithmetic operations make $f_n$ Borel by [F10]. Explicitly, $A_{\beta }={\bigcap }_{k=1}^{\mathrm{\infty }}{\bigcup }_{N=1}^{\mathrm{\infty }}{\bigcap }_{n=N}^{\mathrm{\infty }}\{\mathrm{\mid }{f}_n-{\beta }^2\mathrm{\mid }<1\mathrm{/}k\},$ so $A_{\beta }$ is Borel by [F10]. The limit criterion follows because for every $\epsilon >0$ one can choose $k$ with $1\mathrm{/}k<\epsilon$.

For each $u\in H$, $Q_{\beta }(u,\cdot )$ is the pushforward of ${\mu }_0$ under the continuous map $z\mapsto u+\beta z$, hence is a probability measure. By [F11], $H$ is second-countable and the product Borel sigma-algebra is $\mathcal{B}(H)\otimes \mathcal{B}(H)$. For Borel $B\subseteq H$, addition and scalar multiplication make $(u,z)\mapsto u+\beta z$ continuous, so $(u,z)\mapsto {\mathbf{1}}_B(u+\beta z)$ is product-measurable. Integrating it against the constant kernel $u\mapsto {\mu }_0$ is measurable by [F11], so $Q_{\beta }$ is a probability kernel. Its joint flow $\eta$ exists by [F8]. By the product joint law [F9] and the proposal pushforward, $\eta$ is the law of $(U,U+\beta \xi )$.

For each $j$, define $D_j:=2^j((U+\beta \xi {)}_j^2-U_j^2)=(X_j+\beta Y_j{)}^2-X_j^2=2\beta X_j{Y}_j+{\beta }^2{Y}_j^2\mathrm{.}$ The pairs $(X_j,Y_j)$ are independent by grouping the independent coordinates, and have the same product standard-normal law by [F9]. Applying the same polynomial map to each pair therefore makes $(D_j)$ IID by [F4] and [F9]. By [F5]–[F6], $\mathbb{E}[X_j{Y}_j]=\mathbb{E}[X_j]\mathbb{E}[Y_j]=0$, $\mathbb{E}[Y_j^2]=1$, and $\mathbb{E}[D_j]={\beta }^2\mathrm{.}$ They are square-integrable: using $(a+b{)}^2\le 2a^2+2b^2$, $\mathbb{E}[D_j^2]\le 8{\beta }^2\mathbb{E}[X_j^2{Y}_j^2]+2{\beta }^4\mathbb{E}[Y_j^4]=8{\beta }^2+6{\beta }^4<\mathrm{\infty },$ where independence gives $\mathbb{E}[X_j^2{Y}_j^2]=1$ and [F5] gives $\mathbb{E}[Y_j^4]=3$. By [F7], $\frac{1}{n}{\sum }_{j=1}^n{D}_j⟶{\beta }^2\text{almost surely}\mathrm{.}$

Under the forward flow $\eta$, the pair has the law of $(U,V)$ from step 2.1, and $f_n(U,V)=n^{-1}{\sum }_{j=1}^n{D}_j$. Step 2.2 therefore gives $\eta (A_{\beta })=1$. Under the transpose ${\eta }^{\mathrm{\top }}$, the pair has the law of $(V,U)$, and $f_n(V,U)=-f_n(U,V)\to -{\beta }^2$. Since $\beta >0$, $-{\beta }^2\mathrm{\ne }{\beta }^2$, hence ${\eta }^{\mathrm{\top }}(A_{\beta })=0$.

Thus $A_{\beta }$ and its complement are disjoint Borel sets with $\eta (A_{\beta })=1$ and ${\eta }^{\mathrm{\top }}(A_{\beta }^c)=1$. This is mutual singularity of the two probability measures, and in particular neither is absolutely continuous with respect to the other. By [F12], the pCN reference proposal flow at this same scale equals its transpose.

Apply [F1] with ${\mu }_0=0$, ${\tau }_0^2=1$, ${\sigma }^2=1$, $n=1$, and $y=0$. The posterior variance is $(1+1{)}^{-1}=1\mathrm{/}2$, and its mean is $(1+1{)}^{-1}(0+0)=0$. The evidence is finite and positive, so the posterior is the proper law $N(0,1\mathrm{/}2)$.

By [F2], the likelihood density at $y=0$ is $(2\pi {)}^{-1\mathrm{/}2}{e}^{-u^2\mathrm{/}2}$. Its constant factor cancels when the likelihood is normalized against the prior, so the potential $\mathrm{\Phi }(u)=u^2\mathrm{/}2$ gives exactly the posterior from step 1.1. It is finite on $D=H=\mathbb{R}$, and the pCN target assumptions are satisfied.

For the proposal in [F3], $\mathrm{\Phi }(u)-\mathrm{\Phi }(v)=\frac{u^2-v^2}{2},$ so the stated acceptance probability is precisely the algorithm's pCN acceptance rule for this target.

Step 1.1 identifies the normalized target, and step 3.1 identifies the kernel as the pCN accept-reject kernel for that target. Applying [F4] proves that $N(0,1\mathrm{/}2)$ is invariant for every $\beta \in [0,1]$.