Probability notation in logistic regression

Question

I found these expressions for the probability of an outcome $y$ given variables $x$ and parameter $W$. $\theta$ is the logistic function.

$p(y \mid x,W) = Bernoulli(y \mid \theta(W^\intercal X) ) )$

adapted from [1]

$p(y \mid x,W) = \theta(y W^\intercal X) $ [1]

adapted from [2]

I presume both are correct. How can interpret the first one where the argument of the Bernoulli distribution has a conditional.

[2] youtube/qSTHZvN8hzs?t=44m1s

Answer 1

The first equation cannot be correct. The left hand size is a number, and the right hand side is a distribution (so it does not type check). The correct way to write what the first equation is getting at is

$$ y \mid X, W \sim Bernoulli(y, p = \theta(W^\intercal X) ) ) $$

where $\sim$ is pronounced "is distributed as".

The second equation is correct as written*.

* Assuming that $\theta$ is the logistic function.