 \alpha^{(m)}, \beta^{(m)}),
\]</span> for parameters <span class="math inline">\(\alpha^{(m)}\)</span> and <span class="math inline">\(\beta^{(m)}\)</span> drawn from the posterior, <span class="math display">\[
(\alpha^{(m)}, \beta^{(m)}) \sim p(\alpha, \beta \mid x, y).
\]</span></p>
<p>In the linear regression case, two stages of simplification can be carried out, the first of which helpfully reduces the variance of the estimator. First, rather than averaging draws <span class="math inline">\(\tilde{y}_