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As far as I understand, logistic regression takes the linear model from linear regression and uses it as input to the logistic formula (pardon my complete lack of technical accuracy):

$\frac{1}{1+exp(-(\beta_{0}+\beta_{1}x^{(i)}_{1}+\ldots+\beta_{p}x^{(i)}_{p}))}$

My question is: Does this formula calculate $P(y)$ or $P(y|x;\beta)$?

I have seen both.

user3629892
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  • of course this is a conditional probability, cause you could get similar probability with the different regressors. That s the reason why we look at condition. – Gregory Stelmashenko Oct 30 '21 at 19:58
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    Because values of the $x_i$ appear in the formula, it *must* be a conditional probability. – whuber Oct 30 '21 at 20:12
  • @whuber In the accepted answer in the link, they are using P(Y=1). – user3629892 Oct 30 '21 at 21:09
  • Yes: that's evidently an abbreviated notation, because on the right hand side the $x_i$ explicitly appear. – whuber Oct 30 '21 at 21:09
  • why the hell would one abbreviate that? isnt there a clear distinction between P(Y=1) and P(Y=1|X=x)? – user3629892 Oct 30 '21 at 21:12
  • There is indeed a distinction: but the context makes the meaning unambiguous, so let's turn the question around: why would one use a longer, baroque notation to convey a clear message? There are good answers to that (and, frankly, I would prefer seeing the conditional notation made explicit in that answer), but the point is that one mustn't insist on thoroughgoing pedantry in all answers. – whuber Oct 31 '21 at 15:33

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