# Predictive distribution of SPGP

Eq(8) in Sparse Gaussian Processes using Pseudo-inputs states that

\begin{align*} "p(y^*|x^*,D,\bar{X})=\int{p(y^*|x^*,\bar{X},\bar{f})p(\bar{f}|D,\bar{X})d\bar{f}}" \end{align*}

which can be derived from

\begin{align*} p(y^*|x^*,D,\bar{X})&=\int{p(y^*,\bar{f}|x^*,D, \bar{X})d\bar{f}} \\&=\int{p(y^*|x^*,D, \bar{X},\bar{f})p(\bar{f}|x^*,D,\bar{X})d\bar{f}} \\&= \int{p(y^*|x^*,D, \bar{X},\bar{f})p(\bar{f}|D,\bar{X})d\bar{f}} \\&\stackrel{?}{=} \int{p(y^*|x^*, \bar{X},\bar{f})p(\bar{f}|D,\bar{X})d\bar{f}} \end{align*}

What I can't figure out is the last equation: why $$D$$ can be removed from conditions given pseudo data $$\bar{X}$$ and $$\bar{f}$$?

We know the predictive distribution $$p(y^*|x^*,D)$$ explicitly demonstrates the dependence of $$y^*$$ on $$D$$. Then why the given $$\bar{X}$$ and $$\bar{f}$$ can eliminates the effects of $$D$$?

Currently the only reason I can think of is the pseudo data $$\bar{X}$$ and $$\bar{f}$$ is "good" enough to represent $$D$$ "approximately", so the last equation should be "$$\approx$$" instead of "$$=$$".

"Suppose now that $$f_m$$ is a sufficient statistic for the parameter $$f$$ in the sense that $$z$$ and $$f$$ are independent given $$f_m$$, i.e. it holds $$p(z|f_m,f) = p(z|f_m)$$".
"$$\bar{f}$$ is a sufficient statistic for $$D$$, so the equation holds $$p(y^*|x^*,D,\bar{X},\bar{f})=p(y^*|x^*,\bar{X},\bar{f})$$"