estimate can be further simplified to use the estimated coefficients, <span class="math display">\[\begin{eqnarray*}
\tilde{y}_n^{(m)}
&amp; \approx &amp;
\frac{1}{M} \sum_{m = 1}^M
  \left( \alpha^{(m)} + \beta^{(m)} \cdot \tilde{x}_n \right)
\\[4pt]
&amp; = &amp; \frac{1}{M} \sum_{m = 1}^M \alpha^{(m)}
      + \frac{1}{M} \sum_{m = 1}^M (\beta^{(m)} \cdot \tilde{x}_n)
\\[4pt]
&amp; = &amp; \frac{1}{M} \sum_{m = 1}^M \alpha^{(m)}
      + \left( \frac{1}{M} \sum_{m = 1}^M \beta^{(m)}\right) \cdot \tilde{x}_