Epistemic status: just a review of a well known math theorem and a brief rant about terminology.
Yesterday I saw another example of this: is just a normalizing constant for the posterior probability, and it's really hard (impossible?) to calculate, so let's switch to log odd probabilities, which are easier and the pesky term is canceled.
Except it's not: , the second term is exactly what you need to get the probability in odd form, and if you have it you can very well calculate the prior for the data.
So please, whatever you write, stop saying that odds are easier. They are possibly more intuitive to manipulate, but they need exactly the same amount of information.