</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">\(\hat{\theta}_n\)</span> is given by the posterior standard deviation divided by the square root of the effective sample size. This standard 