Functional BvM limits and credible-set calibration — Examples

For data with $k$ successes, the likelihood is ${\theta }^k(1-\theta {)}^{n-k}$. It is bounded by $1$ and positive for $0<\theta <1$, so its integral against the uniform prior is finite and strictly positive. [F2] therefore gives the posterior density proportional to that likelihood. The sample space is finite for fixed $n$, so the interval rule's membership graph is a finite union of measurable slices. The posterior density is positive on $(0,1)$, making its CDF continuous and strictly increasing there; the posterior probability between its two quantiles is $(1-\alpha \mathrm{/}2)-\alpha \mathrm{/}2=1-\alpha$. Thus the rule is credible by [F1].

At ${\theta }_0=0$, the data are all zero with probability one, so the likelihood is $(1-\theta {)}^n$. Its normalizing integral is ${\int }_0^1(1-\theta {)}^{n}d\theta =1\mathrm{/}(n+1)$, giving posterior density $(n+1)(1-\theta {)}^n$ and CDF $F_n(t)=1-(1-t{)}^{n+1}$. Solving $F_n(q_u)=u$ yields $q_u=1-(1-u{)}^{1\mathrm{/}(n+1)}$. In particular $q_{\alpha \mathrm{/}2}>0$, so $0\mathrm{\notin }{C}_n$ for every $n$.

Because the truth is zero and all data are zero almost surely, [L1] gives frequentist coverage $P_0\{0\in C_n(X_1,\dots ,X_n)\}=0$. This contradicts the asserted lower bound $1-\alpha >0$ and exhibits failure at the boundary identified in [A1].

Since $\log f_0=0$ and $f_0=1$, its wavelet coefficient smoothness norm is finite at every order and it is bounded above and away from zero. For type S1 choose $\alpha =2$ and the Laplace density ${\phi }_H(x)=\frac{1}{2}{e}^{-\mathrm{\mid }x\mathrm{\mid }}$, whose logarithm is Lipschitz; its scales are $2^{-l(2+1\mathrm{/}2)}$. For type S2 choose $\alpha =2$ and $r=1$, which satisfies $0<1<2-1\mathrm{/}4$.

For type H choose truth smoothness $\alpha =1$, hyperparameters $a=c_1=c_2=1$, and $a_{k,L}=1$ for every cell. Then $2^{-L}\le a_{k,L}\le 1$, and the uniform truth belongs to $C_{\psi }^1$.

The sequence $w_l=\max (1,\sqrt{l}\log (l+1))$ is nondecreasing, since both entries in the maximum are nondecreasing for $l\ge 1$. It satisfies $w_l\mathrm{/}\sqrt{l}\to \mathrm{\infty }$. Also ${\sum }_l{w}_l{2}^{-l\mathrm{/}2}<\mathrm{\infty }$ because the ratio of successive terms $\sqrt{l}\log (l+1)2^{-l\mathrm{/}2}$ tends to $2^{-1\mathrm{/}2}<1$. Thus it is admissible and meets the CDF primitive summability condition. Steps 1.1–1.2 verify every regime and truth premise of [F3], so its posterior BvM and credible-band conclusions apply to these examples.

Expanding the inner product gives $\mathrm{\Psi }(f_0+h)=\mathrm{\parallel }{f}_0{\mathrm{\parallel }}_2^2+2⟨f_0,h{⟩}_{L^2}+\mathrm{\parallel }h{\mathrm{\parallel }}_2^2$. Thus the derivative is $D{\mathrm{\Psi }}_{f_0}[h]=2⟨1,h{⟩}_{L^2}$, which is nonzero, and the remainder equals $\mathrm{\parallel }h{\mathrm{\parallel }}_2^2$.

For integers $n\ge 1$ and $h_n=n^{-1\mathrm{/}4}v$, $\mathrm{\parallel }{h}_n{\mathrm{\parallel }}_2=n^{-1\mathrm{/}4}$ and $\sqrt{n}\mathrm{\parallel }{h}_n{\mathrm{\parallel }}_2^2=1$. Hence the scaled quadratic remainder does not tend to zero along this sequence, so the linear approximation needed in [F1] is not justified at this radius.

The standard-normal coefficient density is bounded, continuous, strictly positive at every real value, and has finite second moment. Since $f_0=0$, every normalized true coefficient is zero; choose any $0<M<\tau$ with $\tau >0$. The coefficient bound and continuity at each normalized true coefficient required by [F3] hold, and the scale sequence has the theorem's required form with $\alpha =\eta >0$.

By [A1], all wavelet coefficients of $g=1$ above a fixed finite level vanish; the remaining levels have finitely many coefficients. Therefore ${\sum }_{l,k}{a}_l^{2s}\mathrm{\mid }{g}_{lk}{\mathrm{\mid }}^2<\mathrm{\infty }$ for every finite $s$, in particular for some $s>1\mathrm{/}2$. Also $\mathrm{\parallel }g{\mathrm{\parallel }}_2=1$ and its pairing is $L_1(f)={\int }_0^{1}f$.

Under $f_0=0$ and countable choice [F4], $\sqrt{n}{L}_1(X^{(n)})=L_1(W)$ is exactly $N(0,1)$. Applying [F2] with $g=1$ gives the scalar posterior BvM limit $N(0,1)$ and limiting coverage $1-\alpha$.