The \(p(\cdot | B)\) notation is convenient sometimes, but other times it obscures whatâ€™s going on, like in the definition of conditional independence and the law of total probability (when you fix the conditioning event).

We can rewrite \(p(A|B)\) as \(p_B(A)\) to make clear that we are
deriving a *new* probability measure (\(p_B\)) *induced* from the old
one (\(p\)) using the knowledge that \(B\) occurred. Or we could shorten
it even further, from \(p_B\) to \(p'\).

## Conditional Independence

Conditional independence reduces to regular independence under the measure \(p'\). \(p(A, C | B) = p(A | B) p(C | B)\) becomes \(p'(A,C) = p'(A) p'(B)\).

## Total probability

\(\sum\limits_A p(A|B) = 1\), which is easier to see if we write it as \(\sum\limits_A p'(A)\).