Dirichlet normal-mixture posterior contraction — Examples

For large $n$, $n^{-1\mathrm{/}4}\le {\sigma }_n\le n^{-1\mathrm{/}6}$ because $n^{1\mathrm{/}20}(\log n{)}^{2\mathrm{/}5}\to \mathrm{\infty }$ and $n^{-1\mathrm{/}30}(\log n{)}^{2\mathrm{/}5}\to 0$. Thus $0<a_2<a_1<1$ and $a_2=(4+\gamma {)}^{-1}$. The compactly supported $G$ satisfies the lower scale-tail bound since $G(s)=0$ for $s<1$, and the upper tail is zero for $s\ge 2$.

Direct calculation gives $(n{\sigma }_n{)}^{-1\mathrm{/}2}\log n=n^{-2\mathrm{/}5}(\log n{)}^{4\mathrm{/}5}$ and ${\sigma }_n^2\log n=n^{-2\mathrm{/}5}(\log n{)}^{4\mathrm{/}5}$. Hence [F1] yields this Hellinger contraction upper bound in $P_0^n$-probability. It is not a minimax claim.

The Pareto density integrates to one because ${\int }_2^{\mathrm{\infty }}2s^{-2}ds=1$, so $G$ is a probability measure. For $s<1$, $G(s)=0$, which satisfies every required lower-tail upper bound near zero.

For $s\ge 2$, the uniform component has no mass above $s$ and the Pareto survival function is $2\mathrm{/}s$; the mixture weight $1\mathrm{/}2$ therefore gives $1-G(s)=1\mathrm{/}s$.

For any $\beta >0$ and $\gamma >1$, $e^{\beta s^{\gamma }}\mathrm{/}s\to \mathrm{\infty }$, so $1\mathrm{/}s$ is not bounded above by a constant multiple of $e^{-\beta s^{\gamma }}$. Thus this prior is outside the theorem's upper-tail hypotheses. This example says nothing about whether its posterior contracts or is inconsistent.