Alternative Equation Formats

$$\begin{array}{rcl}p({\mathbf{Y}}_{\text{pred}}|\mathbf{Y},\mathbf{X})& =& \int <!--\; \int \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>p({\mathbf{Y}}_{\text{pred}},\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}|\mathbf{Y},\mathbf{X})d\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}\\ & =& \int <!--\; \int \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>p({\mathbf{Y}}_{\text{pred}}|\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>},\mathbf{Y},\mathbf{X})p(\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}|\mathbf{Y},\mathbf{X})d\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}\end{array}$$

$$\begin{array}{rcl}p({\mathbf{Y}}_{\text{pred}}|\mathbf{Y},\mathbf{X})& =& \int <!--\; \int \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>p({\mathbf{Y}}_{\text{pred}},\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}|\mathbf{Y},\mathbf{X})d\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}\\ & =& \int <!--\; \int \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>p({\mathbf{Y}}_{\text{pred}}|\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>},\mathbf{Y},\mathbf{X})p(\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}|\mathbf{Y},\mathbf{X})d\mathit{\theta <!--\; \theta \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}\end{array}$$

\begin{eqnarray*} p(\mb{Y}_{\mbox{pred}} | \mb{Y}, \mb{X}) & =& \int p(\mb{Y}_{\mbox{pred}}, \btheta | \mb{Y}, \mb{X}) d\btheta \\ & =& \int p(\mb{Y}_{\mbox{pred}} | \btheta , \mb{Y}, \mb{X}) p(\btheta | \mb{Y}, \mb{X}) d\btheta \\ \end{eqnarray*}