Tuesday, February 18, 2014

Bayes' theorem

I wasn't there. Here's a short summary of what we learned.

1Conditional probability¶

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

1.1Bayes' theorem¶

$$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$$

1.2Independence¶

$A$ and $B$ are independent if $P(A|B) = P(A)$ and $P(B|A) = P(A)$.

Equivalently: when $P(A\cap B) = P(A) \cdot P(B)$.

Note that when considering a set of $\geq 2$ events, events within the set can be pairwise independent but not independent overall.

1.3Simpson's paradox¶

A pattern/correlation present when looking at subgroups can disappear when looking at the big picture. Due to the sizes of the subgroups.

Example: Democrats/Republicans + civil rights votes

1.4Berkson's paradox¶

Independent events can appear dependent when groups are combined.