Information law: for a row drawn from a prior and any proposal, E[log(1+χ²)] ≥ I(J; row) and E[1+χ²] ≥ exp I(J; row), where J ~ π is a key drawn from the row

For all $\htmlData{mx=1}{I, J \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=3}{\mathit{lam} : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=5}{\pi : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=3}{r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, if $\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{i})}}$, $\htmlData{mx=3}{\sum_{i} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{i})}} \mathrel{\htmlData{mx=4}{=}} 1$, $\htmlData{mx=5}{\forall i, j,\ 0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{i}, \htmlData{mx=2}{j})}}$, $\htmlData{mx=6}{\forall i,\ \htmlData{mx=4}{\sum_{j} \htmlData{mx=3}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{i}, \htmlData{mx=2}{j})}} \mathrel{\htmlData{mx=5}{=}} 1}$, $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ and $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{I}}$, $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}$), then