In the reference Gaussian processes: iterative sparse approximations by Csató, Lehel (Csató, Lehel. Gaussian processes: iterative sparse approximations. Diss. Aston University, 2002), on page 20, chapter 2
Csató, Lehel, Sparse on-line Gaussian processes : iterative sparse approximations, Diss., Aston University (2002).
From Baysian law, the posterior distribution is given as, using $\boldsymbol{f}_D = {f(x_1),...,f(x_N)}$ for the random variables at the data positions, $$ p_{post}(\boldsymbol{f}) = \frac { P(D|\boldsymbol{f})p_0(\boldsymbol{f})} { \langle P(D|\boldsymbol{f}_D) \rangle_0}\\ =\frac { \int P(D|\boldsymbol{f})p_0(\boldsymbol{f}, \boldsymbol{f}_D)d\boldsymbol{f}_D} { \langle P(D|\boldsymbol{f}_D) \rangle_0}$$
where $p_0(\boldsymbol{f},\boldsymbol{f}_D)$ is the joint Gaussian distribution of the random variables at the training and sample locations, $<P(D|\boldsymbol{f}_D)_0>$ is the average of the likelihood with respect to the prior GP marginal, $P_0(\boldsymbol{f})$.
How did the author calculate the numerator of the integral form ($\int P(D|\boldsymbol{f})p_0(\boldsymbol{f}, \boldsymbol{f}_D)d\boldsymbol{f}_D$ ) from ordinary Bayesian law?
Did anyone see a similar form in the statistics society?
Thanks in advance.