Logistic Regression Probabilities

Question

I have performed a logistic regression on some data, and the function is:

$p(x,y) = \frac{\exp(4.5 + 3.5x - 0.1y)}{1+\exp(4.5 + 3.5x - 0.1y)}$

This also gives you the probability of observing some event $(x,y)$ as I understand it. My question is, what does it mean in this case that probabilities should sum to 1? I.e., if I take various inputs $(x,y)$, and get lets' say 0.4916,0.3379,0.3381,0.289, etc... these don't add to 1, but they are probabilities, so how does one interpret the whole "summing to 1" in this case?

Thanks!

Answer 1

You are misunderstanding the outcome of logistic regression. A logistic regression will model $P(Y=1 | X)$. Thus since there are 2 classes, 0 and 1 for instance, this is what should sum to 1: $$ P(Y=1 | X) + P(Y=0|X) = 1 $$ And you can see that this is always true.

Note : the function cannot contains y's. So I am not sure of what is the formula that you wrote