cubefox comments on Logits, log-odds, and loss for parallel circuits

cubefox 27 Jan 2025 15:38 UTC
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As you say, the addition of logits is equivalent to the multiplication of odds. And odds are related to probability with $o = \frac{p}{1 - p}$ . (One can view them as different scalings of the same quantity. Probabilities have the range $[0, 1]$ , odds have the range $[0, + \infty]$ , and logits have the range $[- \infty, + \infty]$ .)

Now a well-known fact about multiplication of probabilities $P$ is this:
- $P (A) \times P (B) = P (A \land B)$ , when $A$ and $B$ are independent.
But there is a similar fact about the multiplication of odds $O$ , though not at all well-known:
- $O (A) \times O (B) = O (A \land B ∣ A \leftrightarrow B)$ , when $A$ and $B$ are independent.
That is, multiplying the odds of two independent events/propositions gives you the probability of their conjunction, given that their biconditional is true, i.e. given that they have the same truth values / that they either both happen or both don’t happen.

Perhaps this yields some more insight in how to interpret practical logit addition.