Another Way to Write Conditional Probability

The \(p(\cdot | B)\) notation is convenient sometimes, but other times it obscures what’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 \(\sum\limits_A p'(A)\).

Related Posts

NTK reparametrization

Kate from Vancouver, please email me

ChatGPT Session: Emotions, Etymology, Hyperfiniteness

Some ChatGPT Sessions

2016 ML thoughts

My biggest takeaway from Redwood Research REMIX

finite, actual infinity, potential infinity

Actions and Flows

PSA: reward is part of the habit loop too

a kernel of lie theory