\[ \newcommand\b[1]{\pmb{#1}} \newcommand\bx{\b{x}} \newcommand\bzero{\b{0}} \newcommand\bmu{\b{\mu}} \newcommand\bSigma{\b{\Sigma}} \newcommand\pdiff[2]{\frac{\partial #1}{\partial #2}} \newcommand\Tr[1]{\mathrm{Tr}\left[#1\right]} \newcommand\T{\mathrm{T}} \newcommand\Ex{\b{\mathrm{E}}} \newcommand\KL{\mathrm{KL}} \newcommand\comment[1]{\Tiny\mbox{※#1}} \] 上図のようなノードが３つのモデルについて \[ \leqalignno{ &p(x_1,x_2,x_3) = p(x_3 | x_1, x_2) p(x_1) p(x_2) } \] である。これより \[ \leqalignno{ \int p(x_1,x_2,x_3)\,dx_1 dx_2 dx_3 &= \int p(x_3 | x_1, x_2)p(x_1)p(x_2)\,dx_1 dx_2 dx_3 \\ &= \iint \underbrace{ \int p(x_3 | x_1, x_2)\,dx_3}_1 \:p(x_2)\,dx_2\: p(x_1)\,dx_1 \\ &= \underbrace{ \int \underbrace{\int p(x_2)\,dx_2}_1 \: p(x_1)\,dx_1 }_1 \\ &= 1 } \] 一般には \[ \leqalignno{ &p(\bx) = \prod_{k=1}^{K} p(x_k | pa_k) } \] より \[ \leqalignno{ \int p(\bx)\,d\bx &= \int \prod_{k=1}^{K}p(x_k | pa_k)\,dx_1 \dotsm dx_K \\ &= \int \dotsi \underbrace{\int p(x_K | pa_K)\,dx_K}_1 \prod_{k=1}^{K-1} p(x_k | pa_k)\,dx_1 \dotsm dx_{K-1} ~\comment{1} \\ &= \int \dotsi \int \prod_{k=1}^{K-1} p(x_k | pa_k)\,dx_1 \dotsm dx_{K-1} \\ &= \underbrace{~~~\dotsm~~~}_{\comment{2}} = 1 } \]