Type-S log-density and type-H histogram prior families

For type S use a fixed sufficiently regular compactly supported orthonormal multiresolution wavelet basis $({\psi }_{lk})$ on $[0,1]$ as in Logarithmic negative Sobolev spaces for white noise, with boundary correction, regularity greater than the truth smoothness $\alpha$, and $\mathrm{\mid }{K}_l\mathrm{\mid }=2^l$. The spaces $V_L$ spanned by the retained levels $l\le L$ contain constants and have orthonormal scaling bases with uniformly bounded overlap, support diameters at most $C{2}^{-L}$, and suprema at most $C{2}^{L\mathrm{/}2}$; a fixed shift of the scaling-resolution index is absorbed in $C$. The wavelets likewise have uniformly bounded overlap, support diameters at most $C{2}^{-l}$, and suprema at most $C{2}^{l\mathrm{/}2}$. For an integer $m>\alpha$, the detail wavelets have vanishing moments through degree $m-1$, and derivatives through order $m$ satisfy $\mathrm{\parallel }{\psi }_{lk}^{(q)}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C_q{2}^{l(q+1\mathrm{/}2)}$, with uniformly bounded overlap of their supports; the fixed low scaling functions are $C^m$. These are the sufficiently regular compact multiresolution basis conditions in the source density theorem. For the density-model source results, every truth $f_0$ is a probability density satisfying $0<{\rho }_0\le f_0\le D_0<\mathrm{\infty }$ for constants ${\rho }_0,D_0$. For $n\ge 3$ and smoothness index $\alpha >0$, let $j_n$ and ${\mathrm{\ell }}_n$ be the largest integers satisfying $2^{j_n}\le n^{1\mathrm{/}(2\alpha +1)},2^{{\mathrm{\ell }}_n}\le (n\mathrm{/}\log n{)}^{1\mathrm{/}(2\alpha +1)}\mathrm{.}$ Choose one cutoff sequence $L_n\in \{j_n,{\mathrm{\ell }}_n\}$ for all $n\ge 3$; set any finite $L_n$ for $n=1,2$, which does not change asymptotic claims.

For a finite wavelet series $T_n={\sum }_{l\le L_n}{\sum }_{k\in K_l}{\sigma }_l{\xi }_{lk}{\psi }_{lk}$, define $c(T_n)=\log {\int }_0^1{e}^{T_n(x)}dx$ and the normalized log-density $f_n(x)=e^{T_n(x)-c(T_n)}$.

The type-S1 family takes IID coefficients ${\xi }_{lk}$ with a continuous density ${\phi }_H$ for which $\log {\phi }_H$ is Lipschitz, and uses ${\sigma }_l=2^{-l(\alpha +1\mathrm{/}2)}$. Its source theorem assumes $\log f_0$ has coefficient Hölder regularity $\alpha >1$ and $f_0$ is bounded above and bounded away from zero.

The type-S2 family takes IID standard-normal coefficients and uses ${\sigma }_l=2^{-l(r+1\mathrm{/}2)}$, where $\alpha >1\mathrm{/}2$ and $0<r<\alpha -1\mathrm{/}4$. Here the truth regime is $\log f_0\in C^{\alpha }$ with positive finite density bounds. Equation (24) in the source specifies this prior-parameter range, but its Theorem 3 assumes $\alpha >1$ for both type-S families. The broader Gaussian truth range $\alpha >1\mathrm{/}2$ is established separately by the local type-S posterior BvM theorem.

For the type-H family, use the Haar basis instead of the regular type-S wavelets: its scaling function is $1$ and ${\psi }_{lk}=2^{l\mathrm{/}2}(1_{I_{l+1,2k}}-1_{I_{l+1,2k+1}})$ for $l\ge 0$ and $0\le k<2^l$. The histogram space at resolution $L$ consists of the scaling coordinate and detail levels $l<L$; all detail coefficients at $l\ge L$ vanish. Partition $[0,1]$ into dyadic cells $I_{L_n,k}$ of length $2^{-L_n}$, draw $\omega =({\omega }_k{)}_{k<2^{L_n}}\sim \mathrm{Dirichlet}(a_{0,L_n},\dots ,a_{2^{L_n}-1,L_n})$, and set $f_n=2^{L_n}{\omega }_k$ on cell $I_{L_n,k}$. The source assumptions are $c_1{2}^{-L_{n}a}\le a_{k,L_n}\le c_2$ for fixed positive $a,c_1,c_2$, and $f_0\in C^{\alpha }$ for $1\mathrm{/}2<\alpha \le 1$, together with the common positive finite density bounds above. The posterior asymptotic conclusions for these families are separate results.

Here $C_{\psi }^{\alpha }$ denotes the coefficient class with norm $\mathrm{\parallel }g{\mathrm{\parallel }}_{C_{\psi }^{\alpha }}:={\sup }_{l,k}{2}^{l(\alpha +1\mathrm{/}2)}\mathrm{\mid }⟨g,{\psi }_{lk}⟩\mathrm{\mid }$. For sufficiently regular standard wavelets this is the usual Hölder scale (with the usual Zygmund interpretation at integer smoothness).