del, parameters for the intercept <span class="math inline">\(\alpha\)</span> and slope <span class="math inline">\(\beta\)</span>, along with predictors <span class="math inline">\(\tilde{x}_n\)</span>. The standard Bayesian estimate is the expected value of <span class="math inline">\(\tilde{y}\)</span> given the predictors and training data, <span class="math display">\[\begin{eqnarray*}
\hat{\tilde{y}}_n
&amp; = &amp; \mathbb{E}[\tilde{y}_n \mid \tilde{x}_n, x, y]
\\[4pt]
&amp; \approx &amp; \frac{1}{M}