I'm trying to understand the paper "Direct Uncertainty Prediction for Medical Second Opinions" [Raghu et. al., 2019] but I am missing some small intermediate steps to fully understand their equation. On page 3 - left column, third paragraph-, they present the marginal $f_O$, with $f_O = \int_\textbf{y} f(O, \textbf{y})$. A few lines below they present the uncertainty formula
$$U\big(\int_\textbf{y} \textbf{y} \cdot f(\textbf{Y = y} | O) \big)$$
But where does exactly this come from? I thought it might be the product rule from the joint probability, but then it would be something like $U\big(\int_\textbf{y} \cdot f(\textbf{Y = y} | O) \cdot f_O \big)$, where for me $f_O \neq \textbf{y}$.
I think there is only one step I am somehow missing to get the relationships of these equations.
Thx in advance.
The uncertainly formula is using $E(Y|O)=\int_y y\cdot f(Y|O)$ by definition.
Separately, $f(Y|O)=\frac{f(O,y)}{f(O)}$ and $f(O)= f_O=\int_y f(O,y)$.