I was reading an article on probabilistic global optimization and came across following equation which defines probabilistic measure over function space.
$$p_{\min }(x) \equiv p[x=\arg \min f(x)]=\int_{f : I \rightarrow \mathbb{R}} p(f) \prod_{\tilde{x} \in I \atop \tilde{x} \neq x} \theta[f(\tilde{x})-f(x)] \mathrm{d} f$$
Now the more confusion part is that author mentions that the exact meaning of infinite product over the entire domain I in this equation should be intuitively clear
Can someone please explain why are we taking infinite product over the functions will determine probability that the value of f at x is lower than at any other ˜x ∈ I ?
This is the link to the equation 1 in journal.