I'm reading through a lesson on Bayes theorem and I saw this slide: enter image description here

How can model & data be == (data | model) x P(model)?

Does the & not imply multiplication? Otherwise it implies that data == data | model.

  • $\begingroup$ Here $(A|B)$ means event A happens conditional on the event B happens. $(A \& B)$ means both event $A$ and $B$ happen. $\endgroup$ – user158565 May 25 '17 at 5:20

Both equations are correct. First one is the definition of conditional probability in terms of joint probability

$ P(Y = y \mid X = x) = \frac{P(X=x\ \,\cap\, Y=y)}{P(X=x)} $

What is incorrect, is calling it Bayes theorem, since it is just the definition of conditional probability.

The & sign is not multiplication, since in mathematics we do not use "&" to denote multiplication. It is what it is, it means probability of observing $X=x$ and $Y=y$ together. It leads to multiplication only when $X$ and $Y$ are independent.

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  • $\begingroup$ Minor quibble. It still doesn't mean multiplication, even when X and Y are independent (it still means and). It just results in multiplication algorithmically when the probability of the conjunction event is calculated in case the events are independent. $\endgroup$ – Matthew Drury May 26 '17 at 4:59
  • $\begingroup$ @MatthewDurdy good point, I changed the wording. $\endgroup$ – Tim May 26 '17 at 5:23

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