■ - White scenery @showyou, hatena

$lnq^*(\pi,\mu,\Lambda)=lnp(\pi)+\sum_{k=1}^K{lnp(\mu_k,\Lambda_k)}+E_z[lnp(Z|\pi)] + \sum_{k=1}^K \sum_{n=1}^NE[z_{nk}]ln\cal{N}(x_n|\mu_k,\Lambda_k^{-1})+const$ -(10.54)
μ,Λの項だけに注目すると
$lnq^*(\mu,\Lambda)=\sum_{k=1}^K{lnp(\mu_k,\Lambda_k)}+\sum_{k=1}^K \sum_{n=1}^NE[z_{nk}]ln\cal{N}(x_n|\mu_k,\Lambda_k^{-1})+const$
$lnq^*(\mu,\Lambda)=\sum_{k=1}^Kln[ \cal{N}(\mu_k|m_0,(\beta_0,\Lambda_k)^{-1})\cal{W}(\Lambda_k|W_0,\nu_0)] +\sum_{k=1}^K\sum_{n=1}^NE[z_{nk}]ln\cal{N}(x_n|\mu_k,\Lambda_k^{-1})$
$lnq^*(\mu_k,\Lambda_k)=ln[ \cal{N}(\mu_k|m_0,(\beta_0,\Lambda_k)^{-1})\cal{W}(\Lambda_k|W_0,\nu_0)] +\sum_{n=1}^NE[z_{nk}]ln\cal{N}(x_n|\mu_k,\Lambda_k^{-1})$ -(1)

$ln[ \cal{N}(\mu_k|m_0,(\beta_0,\Lambda_k)^{-1})\cal{W}(\Lambda_k|W_0,\nu_0)]$
$=ln[(2\pi)^{-D/2}(\beta_0\Lambda_k)^{1/2}exp(-1/2(\mu_k-m_0)^T\beta_0\Lambda_K(\mu_k-m_0))\cal{W}(\Lambda_k|W_0,\nu_0)]$
$=[-D/2ln(2\pi)+1/2ln(\beta_0\Lambda_k)-1/2(\mu_k-m_0)^T\beta_0\Lambda_K(\mu_k-m_0)+ln\cal{W}(\Lambda_k|W_0,\nu_0)]$
$=\frac{1}{2}[-Dln(2\pi)+ln(\beta_0\Lambda_k)-(\mu_k-m_0)^T\beta_0\Lambda_K(\mu_k-m_0)+2ln\cal{W}(\Lambda_k|W_0,\nu_0)]$

$\cal{W}(\Lambda_k|W_0,\nu_0)=\frac{1}{Z(W_0,\nu_0)}|\Lambda_k|^{\frac{\nu_0-1}{2}}exp[-\frac{1}{2}tr(W_0^{-1}\Lambda_k)]$
$ln\cal{W}(\Lambda_k|W_0,\nu_0)=-lnZ(W_0,\nu_0)+\frac{\nu_0-1}{2}|\Lambda_k|-\frac{1}{2}tr(W_0^{-1}\Lambda_k)$
$z=|W_0|^{\frac{\nu_0}{2}}2^{\frac{\nu_0D}{2}}\Gamma_D(\frac{\nu_0}{2})$
$lnz=\frac{\nu_0}{2}|W_0|+\frac{\nu_0D}{2}ln2+\frac{D(D-1)}{4}ln(\pi)+ln(\sum_{i=1}^D\Gamma_D(\frac{\nu_0+1-i}{2}))$
$ln\cal{W}(\Lambda_k|W_0,\nu_0)=-\frac{\nu_0}{2}|W_0|-\frac{\nu_0D}{2}ln2-\frac{D(D-1)}{4}ln(\pi)-ln(\sum_{i=1}^D\Gamma_D(\frac{\nu_0+1-i}{2}))+\frac{\nu_0-1}{2}|\Lambda_k|-\frac{1}{2}tr(W_0^{-1}\Lambda_k)$

$ln[ \cal{N}(\mu_k|m_0,(\beta_0,\Lambda_k)^{-1})\cal{W}(\Lambda_k|W_0,\nu_0)]$
$=\frac{1}{2}\sum_{k=1}^K[-Dln(2\pi)+ln(\beta_0\Lambda_k)-(\mu_k-m_0)^T\beta_0\Lambda_K(\mu_k-m_0)$
$-\nu_0|W_0|-\nu_0Dln2-\frac{D(D-1)}{2}ln(\pi)-2ln(\sum_{i=1}^D\Gamma_D(\frac{\nu_0+1-i}{2}))+(\nu_0-1)|\Lambda_k|-tr(W_0^{-1}\Lambda_k)]$

$\sum_{n=1}^NE[z_{nk}]ln\cal{N}(x_n|\mu_k,\Lambda_k^{-1})$
$=\frac{1}{2}\sum_{n=1}^N\gamma_{nk}[-Dln(2\pi)+ln(\Lambda_k)-(x_0-\mu_k)^T\Lambda_K(x_n-\mu_k)]$

(1)に代入
$lnq^*(\mu_k,\Lambda_k)=\frac{1}{2}[-Dln(2\pi)+ln(\beta_0\Lambda_k)-(\mu_k-m_0)^T\beta_0\Lambda_K(\mu_k-m_0)$
$-\nu_0|W_0|-\nu_0Dln2-\frac{D(D-1)}{2}ln(\pi)-2ln(\sum_{i=1}^D\Gamma_D(\frac{\nu_0+1-i}{2}))+(\nu_0-1)|\Lambda_k|-tr(W_0^{-1}\Lambda_k)]$
$+\frac{1}{2}\sum_{n=1}^N\gamma_{nk}[-Dln(2\pi)+ln(\Lambda_k)-(x_0-\mu_k)^T\Lambda_k(x_n-\mu_k)]+const$

μ_kに関する項を抜き出す
$lnq^*(\mu_k|\Lambda_k)=\frac{1}{2}[-(\mu_k-m_0)^T\beta_0\Lambda_k(\mu_k-m_0)]+\frac{1}{2}\sum_{n=1}^N\gamma_{nk}[-(x_0-\mu_k)^T\Lambda_k(x_n-\mu_k)]+const$
$=-\frac{1}{2}[\mu_k^T\beta_0\Lambda_k\mu_k-\mu_k^T\beta_0\Lambda_k\m_0-\m_0^T\beta_0\Lambda_k\mu_k+\sum_{n=1}^N\gamma_{nk}(\mu_k^T\Lambda_k\mu_k-\mu_k^T\Lambda_kx_n-x_n^T\Lambda_k\mu_k)]+const$
$=-\frac{1}{2}[\mu_k^T(\beta_0+\sum_{n=1}^N\gamma_{nk})\Lambda_k\mu_k-\mu_k^T\Lambda_k(\beta_0\m_0+\sum_{n=1}^N\gamma_{nk}x_n)-(\beta_0\m_0^T+\sum_{n=1}^N\gamma_{nk}x_n^T)\Lambda_k\mu_k]+const$

$\beta_0+\sum_{n=1}^N\gamma_{nk}=\beta_k, \frac{1}{\beta_k}(\beta_0\m_0+\sum_{n=1}^N\gamma_{nk}x_n)=m_k$ と置くと
$lnq^*(\mu_k|\Lambda_k)=-\frac{1}{2}[\mu_k^T\beta_k\Lambda_k\mu_k-\mu_k^T\Lambda_k(\beta_km_k)-(\beta_km_k)^T\Lambda_k\mu_k]+const$
両辺指数をとると、 $q^*(\mu_k|\Lambda_k)$ はeのなんとか乗になる。ガウス分布(N(μ_k|m_k,(β_kΛ_k)^-1))となる。

ここまでで力尽きた。mimeTex形式書きづらい・・