已知：$$p(x)=N\left(x | \mu, \Lambda^{-1}\right)$$ $$p(y | x)=N\left(y | A x+b, L^{-1}\right)$$
求：$p(y)$，$p(x|y)$
$$
\begin{aligned}
&y=A x+b+\varepsilon\\
&\varepsilon \sim N\left(0, L^{-1}\right)
\end{aligned}
$$
$$
\begin{aligned}
E[y] &=E[A x+b+\varepsilon]=E[A x+b]+E[\varepsilon] \\
&=A \mu+b
\end{aligned}
$$
$$\operatorname{Var}[y]=\operatorname{Var}[A x+b+\varepsilon]=
\operatorname{Var}[A x+b]+\operatorname{Var} [\varepsilon]
=A \Lambda^{-1} A^{\top}+L^{-1} $$
$$y \sim N\left(A \mu+b, L^{-1}+A \Lambda^{-1} A^{\top}\right)$$
$$
z=\left(\begin{array}{c}
{x} \\
{y}
\end{array}\right) \sim N\left(\left[\begin{array}{c}
{\mu} \\
{A \mu+b}
\end{array}\right],\left[\begin{array}{cc}
{\lambda^{-1}} & {\Delta} \\
{\Delta^T} & {L^{-1}+A \Lambda^{-1} A^{T}}
\end{array}\right]\right)
$$
直接套用上一篇博客条件概率公式，变可以得出$p(x|y)$。
下面求$\Delta$的值：
$$
\begin{aligned}
\Delta &=\operatorname{Cov}(x, y) \\
&=E[(x-E[x]) \cdot(y-E[y])^{T}] \\
&=E[(x-\mu)(y-A \mu-b)^{T}] \\
&=E\left[(x-\mu)({A x-A \mu}+\varepsilon)^{T}\right] \\
&=E\left[(x-\mu)({A x-A \mu}+\varepsilon)^{T}\right] \\
&=E\left[(x-\mu)(A x-A\mu)^{T}\right]+E\left[(x-\mu) \varepsilon^{T}\right] \\
&=E\left[(x-\mu)(A x-A\mu)^{T}\right]+E\left[(x-\mu) \varepsilon^{T}\right] \\
&=E\left[(x-\mu)(A x-A \mu)^{T}\right]\\
&=E\left[(x-\mu)(x-\mu)^{T} \cdot A^{T}\right]\\
&\begin{array}{l}
{=E\left[(x-\mu)(x-\mu)^{T}\right] \cdot A^{T}} \\
{=\operatorname{Var}[x] \cdot A^{T}}
\end{array}\\
&=\Lambda^{-1} A^{T}
\end{aligned}
$$
