I am trying to understand the maths behind scikit learn's Gaussian process classifier. There is a link to the book from which the algorithm was taken. It is a bit involed and there is a particular formula here that I don't understand. Namely : $$\mathbb{V}_q[f_*| X,y,x_*] = \mathbb{E}_{p(f_*|X,x_*,f)}[(f_*-\mathbb{E}[f_*|X,x_*,f])^2]+\mathbb{E}_{q(f|X,y)}[(\mathbb{E}[f_*|X,x_*,f]-\mathbb{E}[f_*|X,y,x_*])^2]$$
If I write the definition of the conditional variance I get : $$\mathbb{V}_q[f_*| X,y,x_*] = \mathbb{E}_q\left((f_*-\mathbb{E}_q(f_*|X,y,x_*))^2|X,y,x_*\right)$$ but then I don't know how to obtain the formula. Also what does the notation $\mathbb{E}_{p(f_*|X,x_*,f)}(\cdot)$ exactly mean?
