line">\(\theta_n\)</span>
<span class="math display">\[
\hat{\theta}_n
= \mathrm{mean}(\theta^{(1)}_n, \ldots, \theta^{(m)}_n)
\]</span></p>
<p>is treated as an estimator of the true posterior mean,</p>
<p><span class="math display">\[
\mathbb{E}[\theta_n \mid y]
\ = \
\int_{-\infty}^{\infty}
    \, \theta \, p(\theta | y)
\, \mathrm{d}\theta_n,
\]</span></p>
<p>based the observed data <span class="math inline">\(y\)</span>.</p>
<p>The standard error for the estimator <span class="math inline">\(\hat{\theta