In logistic regression, we set the probability of predicting a target $y$ given a data $x$ as,

$Pr(Y = 1|X;w) = dfrac{exp(w^TX)}{(1+exp(w^TX))}$

What is exactly this probability distribution (or more accurately, conditional probability mass function) called?

I tried to look up logistic distribution but it doesn't look the same https://en.wikipedia.org/wiki/Logistic_distribution

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#### Best Answer

Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:

$$Y_i | mathbf{x}, mathbf{w} sim text{Bern} Bigg( text{Prob} = frac{exp(mathbf{w}^text{T} mathbf{x})}{1 + exp(mathbf{w}^text{T} mathbf{x})} Bigg).$$

One alternative way of looking at the logistic regression is to regard the observed response variable as a discretisation of an underlying "latent variable", where the latter has a logistic distribution. In this (equivalent) alternative formulation, we have an observed response variable $Y_i equiv mathbb{I}(tilde{Y}_i > 0)$, with the underlying latent response having the distribution:

$$tilde{Y}_i | mathbf{x}, mathbf{w} sim text{Logistic} Bigg( text{Location} = mathbf{w}^text{T} mathbf{x}, text{Scale} = 1 Bigg).$$

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