Hi Sam!



 shartzog:

How, then, should I calculate a “HomeRecovery” likelihood after the fact?


I’m not really sure what you are asking, but from the perspective of
your dataset, you can simply count:
Empirically observed probability = number of samples having both
“Not Hospitalized” and “Not Deceased” divided by total number of
samples.


I can’t really assume independence between “Hospitalized” and “Deceased”, so simple methods e.g.
P(HomeRec) = 1 - P(Hosp U Deceased)

won’t work.


On the contrary.  If I interpret the U as meaning “union” or “or,” this
is correct.
Using the trivial identity:
1 = total_samples / total_samples

your expression becomes:
P(HomeRec) = (total_samples - samples_with_Hosp_or_Deceased) / total_samples
           = samples_with_Not_Hosp_and_Not_Deceased / total_samples

This doesn’t assume that “Hospitalized” and “Deceased” are
independent.  Indeed, the “union” (your U) is what accounts for any
interdependency.
Let’s rewrite your probability in terms of an “intersection” (“and”):
P(HomeRec) = P(NotHosp and NotDeceased)

What you can’t do (without"Hospitalized" and “Deceased” being
probabilistically independent) is the following:
P(NotHosp and NotDeceased) = P(NotHosp) * P(NotDeceased)

That is, you can’t factor the joint probability into a product of two
individual probabilities unless the two conditions are independent.
Best.
K. Frank