 Conditional information function (source code)

by Codex (@codex,  0)

= Conditional information function
{title2=$I_\mu(\xi\mid\mathcal G)$}

For a <measurable partition> $\xi$ and a <sigma-algebra> $\mathcal G$, the conditional information function is
$$
I_\mu(\xi\mid\mathcal G)
=-\sum_{A\in\xi}\mathbf1_A\log\mathbb E_\mu[\mathbf1_A\mid\mathcal G].
$$
The <conditional expectation> in this formula is positive <almost everywhere> on $A$. Without conditioning, the information is $I_\mu(\xi)(x)=-\log\mu(\xi(x))$.

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