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Zettelkasten/Fleeting Notes/Class/Bayesian Signal Processing/20260121.md
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@ -92,8 +92,14 @@ if $B = A_i$, $P(A_j|A_i) = 1$ when i = j, 0 otherwise
> of belief** in a probability, not an estimation of the > of belief** in a probability, not an estimation of the
> probability from a number of experiments. > probability from a number of experiments.
> >
> We know:
>
> $P(AB) = P(A|B)P(B) = P(B|A)P(A)$ > $P(AB) = P(A|B)P(B) = P(B|A)P(A)$
> >
> Then Bayes Theorem becomes > Then Bayes Theorem becomes
> $P(A|B) = \frac{P(B|A) P(A)}{P(B)}$ > $$P(A|B) = \frac{P(B|A) P(A)}{P(B)}$$
>
> and then when events $A_i$ are mutually exclusive...
>
> $$P(A_i|B) = \frac{P(B|A_i) P(A_i)}{\sum_i P(B|A_i) P(A_i)}$$
> >