r of the posterior mean estimate, <span class="math display">\[\begin{eqnarray*}
  \hat{y}^{\textrm{test}} - y^{\textrm{test}}
&amp; = &amp;
\mathbb{E}\left[
  \hat{y}^{\textrm{test}}
  \mid x^{\textrm{test}}, x^{\textrm{train}}, y^{\textrm{train}}
\right]
  - y^{\textrm{test}}
\\[4pt]
&amp; = &amp;
\mathbb{E}\left[
  \hat{y}^{\textrm{test}} - y^{\textrm{test}}
  \mid x^{\textrm{test}}, x^{\textrm{train}}, y^{\textrm{train}}
\right]
\\[4pt]
&amp; \approx &amp;
\frac{1}{M} \sum_{m = 1}^M \hat{y}^{\textrm{tes