\[ \newcommand\l{\left} \newcommand\r{\right} \newcommand\cmt[1]{\class{Cmt}{\mbox{#1}}} \newcommand\b[1]{\class{Bold}{\mathrm{#1}}} \newcommand\bx{\b{x}} \newcommand\bxh{\widehat \bx} \newcommand\bX{\b{X}} \newcommand\bmu{\b{\mu}} \newcommand\bLambda{\b{\Lambda}} \newcommand\bpi{\b{\pi}} \newcommand\balpha{\b{\alpha}} \newcommand\bm{\b{m}} \newcommand\bM{\b{M}} \newcommand\bW{\b{W}} \newcommand\bI{\b{I}} \newcommand\bL{\b{L}} \newcommand\N{{\cal N}} \newcommand\W{{\cal W}} \newcommand\Dir{\operatorname{Dir}} \newcommand\B{\operatorname{B}} \newcommand\Tr{\operatorname{Tr}} \newcommand\St{\operatorname{St}} \newcommand\T{\mathrm T} \]

(10.80)より \[ \begin{align} p(\bxh|\bX)&\simeq\sum_{k=1}^K\iiint\pi_k\N(\bxh|\bmu_k,\bLambda_k^{-1})q(\bpi)q(\bmu_k,\bLambda_k)\,d\bpi d\bmu_k d\bLambda \\ &=\sum_{k=1}^K\int\pi_k q(\bpi)\,d\bpi\iint\N(\bxh|\bmu_k,\bLambda_k^{-1})q(\bmu_k,\bLambda_k)\,d\bmu_k d\bLambda \end{align} \] ここで \[ \begin{align} \int\pi_k q(\bpi)\,d\bpi &= \int\pi_k\Dir(\bpi|\balpha)\,d\bpi~~~(\because\ (10.57)) \\ &= {\alpha_k \over \widehat \alpha}~~~(\because\ (B.17)) \end{align} \] また \[ \small \begin{align} &\iint\N(\bxh|\bmu_k,\bLambda_k^{-1})q(\bmu_k,\bLambda_k)\,d\bmu_k d\bLambda \\ &~~~= \iint\N(\bxh|\bmu_k,\bLambda_k^{-1})\N(\bmu_k|\bm_k,(\beta_k\bLambda_k)^{-1})\W(\bLambda_k|\bW_k,\nu_k)\,d\bmu_k d\bLambda_k ~~~(\because\ (10.59)) \\ &~~~= \int\W(\bLambda_k|\bW_k,\nu_k)\underset{\cmt{※1}}{\l[\int\N(\bxh|\bmu_k,\bLambda_k^{-1})\N(\bmu_k|\bm_k,(\beta_k\bLambda_k)^{-1})\,d\bmu_k\r]} d\bLambda_k \\ &~~~= \int\W(\bLambda_k|\bW_k,\nu_k)\N(\bxh|\bm_k,(1+\beta_k^{-1})\bLambda_k^{-1})\,d\bLambda_k \\ &~~~= \int\B(\bW_k,\nu_k)|\bLambda_k|^{\nu_k-D-1 \over 2}\exp\l\{-{1\over 2}\Tr(\bW_k^{-1}\bLambda_k)\r\} \\ &~~~~~~~~~~\cdot{1\over(2\pi)^{D/2}}{1\over\l|(1+\beta_k^{-1})\bLambda_k^{-1}\r|^{1/2}}\exp\l\{-{1\over 2}(\bxh-\bm_k)^\T(1+\beta_k^{-1})^{-1}\bLambda_k(\bxh-\bm_k)\r\}\,d\bLambda_k ~~~(\because\ (B.78),(B.37)) \\ &~~~={1\over(2\pi)^{D/2}}{1\over(1+\beta_k^{-1})^{D/2}}\B(\bW_k,\nu_k)\int|\bLambda_k|^{\nu_k-D\over 2}\exp\bigg[-{1\over 2}\bigg[\Tr(\bW_k^{-1}\bLambda_k)+\underset{(\because\ スカラーの\Trと(C.9))}{\Tr\Big\{(1+\beta_k^{-1})^{-1}(\bxh-\bm_k)(\bxh-\bm_k)^\T\bLambda_k\Big\}} \bigg]\bigg]\,d\bLambda_k \\ &~~~={1\over(2\pi)^{D/2}}{1\over(1+\beta_k^{-1})^{D/2}}\B(\bW_k,\nu_k)\int|\bLambda_k|^{\nu_k-D\over 2}\exp\l[-{1\over 2}\Tr\l[\l\{\bW_k^{-1}+(1+\beta_k^{-1})^{-1}(\bxh-\bm_k)(\bxh-\bm_k)^\T)\r\}\bLambda_k\r]\r]\,d\bLambda_k \\ &~~~={1\over(2\pi)^{D/2}}{1\over(1+\beta_k^{-1})^{D/2}}\B(\bW_k,\nu_k)\int{1\over\B(\bM_k,\nu_k+1)}\W(\bLambda_k|\bM_k,\nu_k+1)\,d\bLambda_k~~~(\because\ (B.78)) \\ &~~~={1\over(2\pi)^{D/2}}{1\over(1+\beta_k^{-1})^{D/2}}{\B(\bW_k,\nu_k)\over\B(\bM_k,\nu_k+1)} \tag{1} \\ \end{align} \] である。ただし \[ \bM_k^{-1}=\bW_k^{-1}+(1+\beta_k^{-1})^{-1}(\bxh-\bm_k)(\bxh-\bm_k)^\T \] とする。ここで \[ \begin{align} {\B(\bW_k,\nu_k)\over\B(\bM_k,\nu_k+1)} &={|\bW_k|^{-{\nu_k\over2}}\l(2^{\nu_kD\over2}\pi^{D(D-1)\over4}\prod_{i=1}^D\Gamma({\nu_k+1-i\over2}) \r)^{-1}\over|\bM_k|^{-{\nu_k+1\over2}}\l(2^{(\nu_k+1)D\over2}\pi^{D(D-1)\over4}\prod_{i=1}^D\Gamma({\nu_k+2-i\over2}) \r)^{-1} } ~~~(\because\ (B.79))\\ &= {|\bW_k|^{-{\nu_k\over2}}\over|\bM_k|^{-{\nu_k+1\over2}}}2^{D\over2}{\prod_{i=1}^D\Gamma\l({\nu_k+2-i\over2}\r)\over\prod_{i=1}^D\Gamma\l({\nu_k+1-i\over2}\r)} \\ &=\underset{\cmt{※2}}{|\bW_k|^{1\over2}\l\{1+(1+\beta^{-1})^{-1}(\bxh-\bm_k)^\T\bW_k(\bxh-\bm_k)\r\}^{-{\nu_k+1\over2}} }2^{D\over2} \underset{\cmt{※3}}{ {\Gamma\l({\nu_k+1\over2}\r)\over\Gamma\l({\nu_k+1-D\over2}\r)} } \end{align} \] これを\(\ (1)\ \)に入れて \[ \small \begin{align} &\iint\N(\bxh|\bmu_k,\bLambda_k^{-1})q(\bmu_k,\bLambda_k)\,d\bmu_k d\bLambda \\ &~~~={1\over(2\pi)^{D/2}}{1\over(1+\beta_k^{-1})^{D/2}} |\bW_k|^{1\over2}\l\{1+(1+\beta_k^{-1})^{-1}(\bxh-\bm_k)^\T\bW_k(\bxh-\bm_k)\r\}^{-{\nu_k+1\over2}} 2^{D\over2} {\Gamma\l({\nu_k+1\over2}\r)\over\Gamma\l({\nu_k+1-D\over2}\r)} \\ &~~~={\Gamma\l({\nu_k+1\over2}\r)\over\Gamma\l({\nu_k+1-D\over2}\r)} {|\bW_k|^{1/2}\over\pi^{D/2}(1+\beta_k^{-1})^{D/2}} \l\{1+(1+\beta_k^{-1})^{-1}(\bxh-\bm_k)^\T\bW_k(\bxh-\bm_k)\r\}^{-{\nu_k+1\over2}} \\ &~~~={\Gamma\l({\nu_k+1-D\over2}+{D\over2}\r)\over\Gamma\l({\nu_k+1-D\over2}\r)} {\l|{\nu_k+1-D\over1+\beta_k^{-1}}\bW_k\r|^{1/2}\over\l\{\pi(\nu_k+1-D)\r\}^{D/2}} \l\{1+(\bxh-\bm_k)^\T{\nu_k+1-D\over1+\beta_k^{-1}}(\bxh-\bm_k){1\over\nu_k+1-D}\r\}^{-{\nu_k+1-D\over2}-{D\over2}} \\ &~~~={\Gamma\l({\nu_k+1-D\over2}+{D\over2}\r)\over\Gamma\l({\nu_k+1-D\over2}\r)} {\l|\bL_k\r|^{1/2}\over\l\{\pi(\nu_k+1-D)\r\}^{D/2}} \l\{1+(\bxh-\bm_k)^\T\bL_k(\bxh-\bm_k){1\over\nu_k+1-D}\r\}^{-{\nu_k+1-D\over2}-{D\over2}} \\ &~~~={\Gamma\l({\nu_k+1-D\over2}+{D\over2}\r)\over\Gamma\l({\nu_k+1-D\over2}\r)} {\l|\bL_k\r|^{1/2}\over\l\{\pi(\nu_k+1-D)\r\}^{D/2}} \l(1+{\Delta^2\over\nu_k+1-D}\r)^{-{\nu_k+1-D\over2}-{D\over2}} \\ &~~~=\St(\bxh|\bm_k,\bL_k,\nu_k+1-D)~~~(\because\ (B.68)) \end{align} \] となる。ただし \[ \begin{align} \bL_k&={\nu_k+1-D\over1+\beta_k^{-1}}\bW_k \\ \Delta^2&=(\bxh-\bm_k)^\T\bL_k(\bxh-\bm_k) \end{align} \] とする。これらを最初の式に入れて \[ \begin{align} p(\bxh|\bX)\simeq\sum_k{\alpha_k\over\widehat \alpha}\St(\bxh|\bm_k,\bL_k,\nu_k+1-D) \tag{10.81} \end{align} \] を得る。