$$
\begin{array}{l} p (\bar {\mathbf {Y}} (\mathbf {x} _ {*}, 2) | \mathbf {Y} ^ {(2)}, \mathbf {Y} ^ {(1)}) \\ = \int \overbrace {p (\bar {\mathbf {Y}} (\mathbf {x} _ {*}, 2) | \mathbf {Y} ^ {(2)} , \vec {\mathbf {y}} ^ {(1)} , \hat {\mathbf {Y}} ^ {(1)})} ^ {\text {f i d e l i t y - 2 p o s t e r i o r}} \overbrace {p (\hat {\mathbf {Y}} ^ {(1)})} ^ {\text {f i d e l i t y - 1 p o s t e r i o r}} d \hat {\mathbf {Y}} ^ {(1)} \\ = \int \left\{2 \pi^ {- \frac {N _ {p} D}{2}} \times \left| \boldsymbol {\Sigma} _ {*} ^ {(2)} \right| ^ {- \frac {1}{2}} \times \exp - \frac {1}{2} \left[ \left(\mathbf {y} _ {*} ^ {(2)} - \bar {\mathbf {y}} ^ {(2)}\right) ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \left(\mathbf {y} _ {*} ^ {(2)} - \bar {\mathbf {y}} ^ {(2)}\right) \right] \right. \\ \times 2 \pi^ {- \frac {N _ {n} ^ {(2)} D}{2}} \times \left| \boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)} \right| ^ {- \frac {1}{2}} \times \exp \left[ - \frac {1}{2} \left(\hat {\mathbf {y}} ^ {(1)} - \bar {\mathbf {y}} ^ {(1)}\right) ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)}\right) ^ {- 1} \left(\hat {\mathbf {y}} ^ {(1)} - \bar {\mathbf {y}} ^ {(1)}\right) \right] \Bigg \} d \hat {\mathbf {Y}} ^ {(1)} \\ = 2 \pi^ {- \frac {\left(N _ {p} + N _ {n} ^ {(2)}\right) D}{2}} \times \left| \boldsymbol {\Sigma} _ {*} ^ {(2)} \right| ^ {- \frac {1}{2}} \times \left| \boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)} \right| ^ {- \frac {1}{2}} \times \exp \left[ - \frac {1}{2} \bar {\mathbf {y}} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \tilde {\mathbf {y}} - \frac {1}{2} \left(\bar {\mathbf {y}} ^ {(1)}\right) ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)}\right) ^ {- 1} \bar {\mathbf {y}} ^ {(1)} \right] \\ \times \int \exp \left[ \tilde {\mathbf {y}} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} \hat {\mathbf {y}} ^ {(1)} + \left(\bar {\mathbf {y}} ^ {(1)}\right) ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S}\right) ^ {- 1} \boldsymbol {\Gamma} \hat {\mathbf {y}} ^ {(1)} - \frac {1}{2} \left(\hat {\mathbf {y}} ^ {(1)}\right) ^ {\top} \boldsymbol {\Gamma} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \hat {\mathbf {y}} ^ {(1)} \right. \\ \left. - \frac {1}{2} \left(\hat {\mathbf {y}} ^ {(1)}\right) ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)}\right) ^ {- 1} \hat {\mathbf {y}} ^ {(1)} \right] d \hat {\mathbf {Y}} ^ {(1)} \\ = 2 \pi^ {- \frac {N _ {P} D}{2}} \times \left| \boldsymbol {\Sigma} _ {*} ^ {(2)} \right| ^ {- \frac {1}{2}} \times \left| \boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)} \right| ^ {- \frac {1}{2}} \times \left| \boldsymbol {\Gamma} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} + \boldsymbol {\Sigma} _ {*} ^ {(1)} \right| ^ {- \frac {1}{2}} \\ \times \exp \left\{- \frac {1}{2} \tilde {\mathbf {y}} ^ {\top} \left[ \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} - \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} \left(\boldsymbol {\Gamma} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} + \boldsymbol {\Sigma} _ {*} ^ {(1)}\right) ^ {- 1} \boldsymbol {\Gamma} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \right] \tilde {\mathbf {y}} \right\} \\ \times \exp \left\{- \frac {1}{2} (\bar {\mathbf {y}} ^ {(1)}) ^ {\top} \left[ \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)}\right) ^ {- 1} - \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \left(\boldsymbol {\Gamma} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} + \boldsymbol {\Sigma} _ {*} ^ {(1)}\right) ^ {- 1} \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)}\right) ^ {- 1} \right] (\bar {\mathbf {y}} ^ {(1)}) \right\} \\ \times \exp \left\{\tilde {\mathbf {y}} ^ {\top} \left[ \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} \left(\boldsymbol {\Gamma} ^ {\top} \left(\boldsymbol {\Sigma} _ {*} ^ {(2)}\right) ^ {- 1} \boldsymbol {\Gamma} + \boldsymbol {\Sigma} _ {*} ^ {(1)}\right) ^ {- 1} \left(\boldsymbol {\Sigma} _ {*} ^ {(1)} \otimes \mathbf {S} ^ {(1)}\right) ^ {- 1} \right] \left(\bar {\mathbf {y}} ^ {(1)}\right) \right\}. \tag {A.37} \\ \end{array}
$$