What is the correct way of writing the log-odds in logistic regression?

Question

The most sources I know write the log-odds as $log[Pr(y=1|x)/Pr(y=0|x)]$, where

$Pr(y=1|x) = \frac{e^{\beta^T x}}{1+e^{\beta^T x}}$,

which eventually allows me to write the log-odds as

$\log\frac{Pr(y=1| x)}{Pr(y=0| x)} = \beta^T x$.

However, sometimes I found the reverse defintion. For example in the lecture notes of the Carnegie Mellon University, they define

$Pr(y=0|x) = \frac{e^{\beta^T x}}{1+e^{\beta^T x}}$

And hence get another odds-ratio. Bascially, they just swap the definitions. How is this to understand?

Answer 1

Answered in comments, copied below:

This is easy to resolve when you consider that odds is always the odds of some event. Conventionally we're interested in the odds of Y=1 but there's no reason why we can't look at Y=0 instead. – dsaxton

the result is the same, except that the signs of the coefficients switch (which is expected), see EDIT at the bottom of Logistic regression: what happens to the coefficients when we switch the labels (0/1) of the binary outcome