四元数において，可逆元とその逆元を左右から作用させる

$\mathbf{q}\mathbf{p}{\mathbf{q}}^{-1}$

$$\begin{array}{rcl}\mathbf{q}\mathbf{p}{\mathbf{q}}^{-1}& =& \mathbf{q}\mathbf{p}\frac{\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}=\frac{1}{|\mathbf{q}{|}^2}\mathbf{q}\mathbf{p}\bar{\mathbf{q}}\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w,\mathbf{V})(w_p,{\mathbf{V}}_p)(w,-\mathbf{V})\cdots \bar{\mathbf{q}}=w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}=(w,-\mathbf{V}),(\mathrm{実}\mathrm{部},\mathbf{i}\mathbf{j}\mathbf{k}\mathrm{部}),\mathbf{V}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w{w}_p-\mathbf{V}\cdot {\mathbf{V}}_p,w{\mathbf{V}}_p+w_p\mathbf{V}+\mathbf{V}\times {\mathbf{V}}_p)(w,-\mathbf{V})\cdots {\mathbf{q}}_1{\mathbf{q}}_2=(w_1,{\mathbf{V}}_1)(w_2,{\mathbf{V}}_2)=(w_1{w}_2-{\mathbf{V}}_1\cdot {\mathbf{V}}_2,w_1{\mathbf{V}}_2+w_2{\mathbf{V}}_1+{\mathbf{V}}_1\times {\mathbf{V}}_2)\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}((w{w}_p-\mathbf{V}\cdot {\mathbf{V}}_p)w-(w{\mathbf{V}}_p+w_p\mathbf{V}+\mathbf{V}\times {\mathbf{V}}_p)\cdot (-\mathbf{V}),(w{w}_p-\mathbf{V}\cdot {\mathbf{V}}_p)(-\mathbf{V})+w(w{\mathbf{V}}_p+w_p\mathbf{V}+\mathbf{V}\times {\mathbf{V}}_p)+(w{\mathbf{V}}_p+w_p\mathbf{V}+\mathbf{V}\times {\mathbf{V}}_p)\times (-\mathbf{V}))\cdots {\mathbf{q}}_1{\mathbf{q}}_2=(w_1,{\mathbf{V}}_1)(w_2,{\mathbf{V}}_2)=(w_1{w}_2-{\mathbf{V}}_1\cdot {\mathbf{V}}_2,w_1{\mathbf{V}}_2+w_2{\mathbf{V}}_1+{\mathbf{V}}_1\times {\mathbf{V}}_2)\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w^2{w}_p-w\mathbf{V}\cdot {\mathbf{V}}_p+w{\mathbf{V}}_p\cdot \mathbf{V}+w_p\mathbf{V}\cdot \mathbf{V}+(\mathbf{V}\times {\mathbf{V}}_p)\cdot \mathbf{V}),-w{w}_p\mathbf{V}+(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+w^2{\mathbf{V}}_p+w{w}_p\mathbf{V}+w\mathbf{V}\times {\mathbf{V}}_p+\mathbf{V}\times (w{\mathbf{V}}_p+w_p\mathbf{V}+\mathbf{V}\times {\mathbf{V}}_p))\\ & & \cdots \mathbf{A}\times \mathbf{B}=-\mathbf{B}\times \mathbf{A}\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w^2{w}_p+w_p|\mathbf{V}{|}^2+(\mathbf{V}\times \mathbf{V})\cdot {\mathbf{V}}_p,+(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+w^2{\mathbf{V}}_p+w\mathbf{V}\times {\mathbf{V}}_p+w\mathbf{V}\times {\mathbf{V}}_p+w_p\mathbf{V}\times \mathbf{V}+\mathbf{V}\times (\mathbf{V}\times {\mathbf{V}}_p))\\ & & \cdots (\mathbf{A}\times \mathbf{B})\cdot \mathbf{C}=(\mathbf{B}\times \mathbf{C})\cdot \mathbf{A}=(\mathbf{C}\times \mathbf{A})\cdot \mathbf{B},\mathbf{A}\times \mathbf{A}=0,\mathbf{A}\cdot \mathbf{A}=|\mathbf{A}{|}^2\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w_p(w^2+|\mathbf{V}{|}^2),(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+w^2{\mathbf{V}}_p+2w\mathbf{V}\times {\mathbf{V}}_p+(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}-(\mathbf{V}\cdot \mathbf{V}){\mathbf{V}}_p)\\ & & \cdots \mathbf{A}\times (\mathbf{B}\times \mathbf{C})=(\mathbf{A}\cdot \mathbf{C})\mathbf{B}-(\mathbf{A}\cdot \mathbf{B})\mathbf{C}\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w_p(w^2+|\mathbf{V}{|}^2),2(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+w^2{\mathbf{V}}_p+2w\mathbf{V}\times {\mathbf{V}}_p-|\mathbf{V}{|}^2{\mathbf{V}}_p)\\ & & \cdots \mathbf{A}\cdot \mathbf{A}=|\mathbf{A}{|}^2\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w_p(w^2+|\mathbf{V}{|}^2),2(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+(w^2-|\mathbf{V}{|}^2){\mathbf{V}}_p+2w\mathbf{V}\times {\mathbf{V}}_p)\end{array}$$

特殊なp, qを考える

ここで，$\mathbf{q}=(\cos \left(\frac{\theta }{2}\right),\sin \left(\frac{\theta }{2}\right)\mathbf{V})(\mathrm{た}\mathrm{だ}\mathrm{し}|\mathbf{V}|=1),\mathbf{p}=(0,{\mathbf{V}}_p)$とする． $$\begin{array}{rcl}\mathbf{q}\mathbf{p}{\mathbf{q}}^{-1}& =& \frac{1}{{\cos }^2\left(\frac{\theta }{2}\right)+{|\sin \left(\frac{\theta }{2}\right)\mathbf{V}|}^2}(0({\cos }^2\left(\frac{\theta }{2}\right)+{|\sin \left(\frac{\theta }{2}\right)\mathbf{V}|}^2),2(\sin \left(\frac{\theta }{2}\right)\mathbf{V}\cdot {\mathbf{V}}_p)\sin \left(\frac{\theta }{2}\right)\mathbf{V}+({\cos }^2\left(\frac{\theta }{2}\right)-{|\sin \left(\frac{\theta }{2}\right)\mathbf{V}|}^2){\mathbf{V}}_p+2\cos \left(\frac{\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)\mathbf{V}\times {\mathbf{V}}_p)\\ & =& \frac{1}{{\cos }^2\left(\frac{\theta }{2}\right)+\sin \left(\frac{\theta }{2}\right){\left|\mathbf{V}\right|}^2}(0,2{\sin }^2\left(\frac{\theta }{2}\right)(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+({\cos }^2\left(\frac{\theta }{2}\right)-{\sin }^2\left(\frac{\theta }{2}\right){\left|\mathbf{V}\right|}^2){\mathbf{V}}_p+2\cos \left(\frac{\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)\mathbf{V}\times {\mathbf{V}}_p)\\ & =& \frac{1}{{\cos }^2\left(\frac{\theta }{2}\right)+\sin \left(\frac{\theta }{2}\right)\cdot 1}(0,2{\sin }^2\left(\frac{\theta }{2}\right)(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+({\cos }^2\left(\frac{\theta }{2}\right)-{\sin }^2\left(\frac{\theta }{2}\right)\cdot 1){\mathbf{V}}_p+2\cos \left(\frac{\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)\mathbf{V}\times {\mathbf{V}}_p)\\ & =& \frac{1}{1}(0,2{\sin }^2\left(\frac{\theta }{2}\right)(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+({\cos }^2\left(\frac{\theta }{2}\right)-{\sin }^2\left(\frac{\theta }{2}\right)){\mathbf{V}}_p+2\cos \left(\frac{\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)\mathbf{V}\times {\mathbf{V}}_p)\\ & =& (0,(1-\cos \left(\theta \right))(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+\cos \left(\theta \right){\mathbf{V}}_p+\sin \left(\theta \right)\mathbf{V}\times {\mathbf{V}}_p)\\ & & \cdots 2{\sin }^2\left(\frac{\theta }{2}\right)={\sin }^2\left(\frac{\theta }{2}\right)+{\sin }^2\left(\frac{\theta }{2}\right)=(1-{\cos }^2\left(\frac{\theta }{2}\right))+{\sin }^2\left(\frac{\theta }{2}\right)=1-({\cos }^2\left(\frac{\theta }{2}\right)-{\sin }^2\left(\frac{\theta }{2}\right))=1-\cos \left(\theta \right)\\ & & \cdots {\cos }^2\left(\frac{\theta }{2}\right)-{\sin }^2\left(\frac{\theta }{2}\right)=\cos (\frac{\theta }{2}+\frac{\theta }{2})=\cos \left(\theta \right)\\ & & \cdots 2\cos \left(\frac{\theta }{2}\right)\sin \left(\frac{\theta }{2}\right)=\sin (\frac{\theta }{2}+\frac{\theta }{2})=\sin \left(\theta \right)\\ & =& (0,(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}-\cos \left(\theta \right)(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+\cos \left(\theta \right){\mathbf{V}}_p+\sin \left(\theta \right)\mathbf{V}\times {\mathbf{V}}_p)\\ & =& (0,(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+\cos \left(\theta \right)({\mathbf{V}}_p-(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V})+\sin \left(\theta \right)\mathbf{V}\times {\mathbf{V}}_p)\end{array}$$ 位置ベクトル${\mathbf{V}}_p$を任意の回転軸$\mathbf{V}$周りで$\theta$だけ回転させる計算に対応する．

四元数の逆元

$\mathbf{q}\bar{\mathbf{q}}$, ${\mathbf{q}}^{-1}$

$$\begin{array}{rcl}\mathbf{q}\bar{\mathbf{q}}& =& (w,\mathbf{V})(w,-\mathbf{V})\\ & & \cdots \bar{\mathbf{q}}=w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}=(w,-\mathbf{V}),(\mathrm{実}\mathrm{部},\mathbf{i}\mathbf{j}\mathbf{k}\mathrm{部}),\mathbf{V}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\\ & =& (ww-\mathbf{V}\cdot (-\mathbf{V}),w(-\mathbf{V})+w\mathbf{V}+\mathbf{V}\times (-\mathbf{V}))\\ & & \cdots {\mathbf{q}}_1{\mathbf{q}}_2=(w_1,{\mathbf{V}}_1)(w_2,{\mathbf{V}}_2)=(w_1{w}_2-{\mathbf{V}}_1\cdot {\mathbf{V}}_2,w_1{\mathbf{V}}_2+w_2{\mathbf{V}}_1+{\mathbf{V}}_1\times {\mathbf{V}}_2)\\ & =& (w^2+|\mathbf{V}{|}^2,0)\cdots \mathbf{A}\cdot \mathbf{A}={\left|\mathbf{A}\right|}^2,\mathbf{A}\times \mathbf{A}=0\end{array}$$ $$\begin{array}{r}|\mathbf{q}{|}^2=\mathbf{q}\bar{\mathbf{q}}=\bar{\mathbf{q}}\mathbf{q}=w^2+|\mathbf{V}{|}^2\end{array}$$ $$\begin{array}{r}\\ \frac{\mathbf{q}\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}& =& \mathbf{q}\frac{\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}& =& \frac{\bar{\mathbf{q}}\mathbf{q}}{|\mathbf{q}{|}^2}& =& \frac{\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}\mathbf{q}=1\end{array}$$ $$\begin{array}{r}\\ {\mathbf{q}}^{-1}& =& \frac{\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}\cdots \mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}=\mathbf{E}(\mathrm{単}\mathrm{位}\mathrm{元})\mathrm{と}\mathrm{な}\mathrm{る}\mathbf{B}\mathrm{を}{\mathbf{A}}^{-1}(\mathrm{逆}\mathrm{元})\mathrm{と}\mathrm{す}\mathrm{る}\end{array}$$

$\mathbf{q}{\mathbf{q}}^{-1}$

$$\begin{array}{rcl}\mathbf{q}{\mathbf{q}}^{-1}& =& \mathbf{q}\frac{\bar{\mathbf{q}}}{|\mathbf{q}{|}^2}=\frac{1}{|\mathbf{q}{|}^2}\mathbf{q}\bar{\mathbf{q}}\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w,\mathbf{V})(w,-\mathbf{V})\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w^2+|\mathbf{V}{|}^2,w(-\mathbf{V})+w\mathbf{V}+\mathbf{V}\times \mathbf{V})\\ & =& \frac{1}{w^2+|\mathbf{V}{|}^2}(w^2+|\mathbf{V}{|}^2,0)\\ & =& (\frac{w^2+|\mathbf{V}{|}^2}{w^2+|\mathbf{V}{|}^2},\frac{w^2+|\mathbf{V}{|}^2}{w^2+|\mathbf{V}{|}^2})\\ & =& (1,0)(\mathrm{単}\mathrm{位}\mathrm{元})\end{array}$$

四元数(quaternion)の掛け算

四元数(quaternion)

$$\begin{array}{rcl}\mathbf{q}& =& w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\\ & =& (w,\mathbf{V})\cdots (\mathrm{実}\mathrm{部},\mathbf{i}\mathbf{j}\mathbf{k}\mathrm{部}),\mathbf{V}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\\ -\mathbf{q}& =& -(w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k})\\ & =& -w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}\\ & =& (-w,-\mathbf{V})\\ \bar{\mathbf{q}}& =& w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}\\ & =& (w,-\mathbf{V})\end{array}$$

${\mathbf{q}}_1{\mathbf{q}}_2$

$$\begin{array}{rcl}{\mathbf{q}}_1& =& w_1+x_1\mathbf{i}+y_1\mathbf{j}+z_1\mathbf{k}\\ & =& (w_1,{\mathbf{V}}_1)\\ {\mathbf{q}}_2& =& w_2+x_2\mathbf{i}+y_2\mathbf{j}+z_2\mathbf{k}\\ & =& (w_2,{\mathbf{V}}_2)\\ {\mathbf{q}}_1{\mathbf{q}}_2& =& (w_1,{\mathbf{V}}_{\mathbf{1}})(w_2,{\mathbf{V}}_{\mathbf{2}})\\ & =& (w_1+x_1\mathbf{i}+y_1\mathbf{j}+z_1\mathbf{k})(w_2+x_2\mathbf{i}+y_2\mathbf{j}+z_2\mathbf{k})\\ & =& w_1{w}_2+w_1{x}_2\mathbf{i}+w_1{y}_2\mathbf{j}+w_1{z}_2\mathbf{k}\\ & & +x_1\mathbf{i}{w}_2+x_1\mathbf{i}{x}_2\mathbf{i}+x_1\mathbf{i}{y}_2\mathbf{j}+x_1\mathbf{i}{z}_2\mathbf{k}\\ & & +y_1\mathbf{j}{w}_2+y_1\mathbf{j}{x}_2\mathbf{i}+y_1\mathbf{j}{y}_2\mathbf{j}+y_1\mathbf{j}{z}_2\mathbf{k}\\ & & +z_1\mathbf{k}{w}_2+z_1\mathbf{k}{x}_2\mathbf{i}+z_1\mathbf{k}{y}_2\mathbf{j}+z_1\mathbf{k}{z}_2\mathbf{k}\\ & =& w_1{w}_2+w_1{x}_2\mathbf{i}+w_1{y}_2\mathbf{j}+w_1{z}_2\mathbf{k}\\ & & +x_1{w}_2\mathbf{i}+x_1{x}_2{\mathbf{i}}^2+x_1{y}_2\mathbf{i}\mathbf{j}+x_1{z}_2\mathbf{i}\mathbf{k}\\ & & +y_1{w}_2\mathbf{j}+y_1{x}_2\mathbf{j}\mathbf{i}+y_1{y}_2{\mathbf{j}}^2+y_1{z}_2\mathbf{j}\mathbf{k}\\ & & +z_1{w}_2\mathbf{k}+z_1{x}_2\mathbf{k}\mathbf{i}+z_1{y}_2\mathbf{k}\mathbf{j}+z_1{z}_2{\mathbf{k}}^2\\ & =& w_1{w}_2+w_1{x}_2\mathbf{i}+w_1{y}_2\mathbf{j}+w_1{z}_2\mathbf{k}\\ & & +x_1{w}_2\mathbf{i}+x_1{x}_2(-1)+x_1{y}_2(\mathbf{k})+x_1{z}_2(\mathbf{-}\mathbf{j})\\ & & +y_1{w}_2\mathbf{j}+y_1{x}_2(-\mathbf{k})+y_1{y}_2(-1)+y_1{z}_2(\mathbf{i})\\ & & +z_1{w}_2\mathbf{k}+z_1{x}_2(\mathbf{j})+z_1{y}_2(-\mathbf{i})+z_1{z}_2(-1)\\ & & \cdots {\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1,\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k},\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j},\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i}\\ & =& (w_1{w}_2-x_1{x}_2-y_1{y}_2-z_1{z}_2)\\ & & +(w_1{x}_2+x_1{w}_2+y_1{z}_2-z_1{y}_2)\mathbf{i}\\ & & +(w_1{y}_2-x_1{z}_2+y_1{w}_2+z_1{x}_2)\mathbf{j}\\ & & +(w_1{z}_2+x_1{y}_2-y_1{x}_2+z_1{w}_2)\mathbf{k}\\ & =& (w_1{w}_2-x_1{x}_2-y_1{y}_2-z_1{z}_2)\\ & & +(w_1{x}_2+x_1{w}_2+y_1{z}_2-z_1{y}_2)\mathbf{i}\\ & & +(w_1{y}_2+y_1{w}_2-x_1{z}_2+z_1{x}_2)\mathbf{j}\\ & & +(w_1{z}_2+z_1{w}_2+x_1{y}_2-y_1{x}_2)\mathbf{k}\\ & =& (w_1{w}_2-{\mathbf{V}}_1\cdot {\mathbf{V}}_2,w_1{\mathbf{V}}_2+w_2{\mathbf{V}}_1+{\mathbf{V}}_1\times {\mathbf{V}}_2)\end{array}$$

位置ベクトルを任意の回転軸周りで回転させる計算

点$P$の位置ベクトル${\mathbf{V}}_{\mathbf{p}}$を任意の回転軸(単位ベクトル$\mathbf{V}$)周りで$\theta$だけ回転させる変換

$$\begin{array}{rcl}\mathrm{原}\mathrm{点}& & O\\ \overrightarrow{OP}& :& \mathrm{点}P\mathrm{の}\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}{\mathbf{V}}_p\\ \mathrm{回}\mathrm{転}\mathrm{軸}& :& \mathbf{V}(\mathrm{単}\mathrm{位}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\left|\mathbf{V}\right|=1)\\ \mathrm{回}\mathrm{転}\mathrm{量}& :& \theta \\ \overrightarrow{O{P}^{\prime }}& :& \mathrm{点}P\mathrm{を}\mathbf{V}\mathrm{周}\mathrm{り}\mathrm{に}\theta \mathrm{だ}\mathrm{け}\mathrm{回}\mathrm{転}\mathrm{後}\mathrm{の}\mathrm{点}{P}^{\prime }\mathrm{の}\mathrm{位}\mathrm{置}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\\ O^{\prime }& :& \mathrm{点}P\mathrm{を}\mathbf{V}\mathrm{周}\mathrm{り}\mathrm{に}\mathrm{回}\mathrm{し}\mathrm{た}\mathrm{時}\mathrm{に}\mathrm{で}\mathrm{き}\mathrm{る}\mathrm{面}(\mathrm{点}{P}^{\prime }\mathrm{も}\mathrm{こ}\mathrm{の}\mathrm{面}\mathrm{に}\mathrm{乗}\mathrm{る})\mathrm{と}\mathbf{V}\mathrm{の}\mathrm{延}\mathrm{長}\mathrm{線}\mathrm{の}\mathrm{交}\mathrm{点}\\ & & (\mathrm{点}P\mathrm{か}\mathrm{ら}\mathbf{V}\mathrm{の}\mathrm{延}\mathrm{長}\mathrm{線}\mathrm{に}\mathrm{垂}\mathrm{線}\mathrm{を}\mathrm{下}\mathrm{ろ}\mathrm{し}\mathrm{た}\mathrm{時}\mathrm{の}\mathrm{交}\mathrm{点})\end{array}$$ $$\begin{array}{rcl}\overrightarrow{O{O}^{\prime }}& =& (\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}\cdots (\left|\mathbf{V}\right|\left|{\mathbf{V}}_p\right|\cos \left(\varphi \right))\mathbf{V}=(\left|{\mathbf{V}}_p\right|\cos \left(\varphi \right))\mathbf{V}\\ & & \varphi \mathrm{は}\mathbf{V}\mathrm{と}{\mathbf{V}}_p(=\overrightarrow{OP})\mathrm{の}\mathrm{な}\mathrm{す}\mathrm{角}．\mathrm{ま}\mathrm{た}\overrightarrow{O{O}^{\prime }}\mathrm{と}\overrightarrow{O^{\prime }P}\mathrm{は}\mathrm{直}\mathrm{角}\mathrm{で}\mathrm{あ}\mathrm{る}．\\ \overrightarrow{O^{\prime }P}& =& \overrightarrow{OP}-\overrightarrow{O{O}^{\prime }}\cdots (\overrightarrow{OP}=\overrightarrow{O{O}^{\prime }}+\overrightarrow{O^{\prime }P})\\ (\left|\overrightarrow{O^{\prime }{P}^{\prime }}\right|\cos \left(\theta \right))\frac{\overrightarrow{O^{\prime }P}}{\left|\overrightarrow{O^{\prime }P}\right|}& =& \cos \left(\theta \right)\overrightarrow{O^{\prime }P}\cdots \overrightarrow{O^{\prime }{P}^{\prime }}\mathrm{の}\overrightarrow{O^{\prime }P}\mathrm{方}\mathrm{向}\mathrm{成}\mathrm{分}\\ & =& \cos \left(\theta \right)(\overrightarrow{OP}-\overrightarrow{O{O}^{\prime }})\\ & =& \cos \left(\theta \right)({\mathbf{V}}_p-(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V})\\ \mathbf{V}\times {\mathbf{V}}_p& :& \mathbf{V}\mathrm{と}{\mathbf{V}}_p\mathrm{に}\mathrm{直}\mathrm{角}\mathrm{の}\mathrm{向}\mathrm{き}\mathrm{で}\mathrm{あ}\mathrm{り}，\mathrm{外}\mathrm{積}\mathrm{の}\mathrm{イ}\mathrm{メ}\mathrm{ー}\mathrm{ジ}\mathrm{で}\mathrm{は}O\mathrm{か}\mathrm{ら}\mathrm{伸}\mathrm{び}\mathrm{る}\mathrm{わ}\mathrm{け}\mathrm{だ}\mathrm{が}\\ & & \mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{平}\mathrm{行}\mathrm{移}\mathrm{動}\mathrm{し}\mathrm{て}\mathrm{も}\mathrm{同}\mathrm{じ}\mathrm{で}\mathrm{あ}\mathrm{る}\mathrm{の}\mathrm{で}{O}^{\prime }\mathrm{に}\mathrm{移}\mathrm{動}\mathrm{さ}\mathrm{せ}\mathrm{て}\mathrm{考}\mathrm{え}\mathrm{る}．\overrightarrow{O^{\prime }P}\mathrm{と}\mathrm{も}\mathrm{直}\mathrm{角}．\\ & & \mathrm{大}\mathrm{き}\mathrm{さ}\mathrm{は}\left|\mathbf{V}\right|\left|{\mathbf{V}}_p\right|\sin \left(\varphi \right)=\left|{\mathbf{V}}_p\right|\sin \left(\varphi \right)=\left|\overrightarrow{O^{\prime }P}\right|=\left|\overrightarrow{O^{\prime }{P}^{\prime }}\right|\mathrm{と}\mathrm{な}\mathrm{る}，\\ (\left|\overrightarrow{O^{\prime }{P}^{\prime }}\right|\cos (\frac{\pi }{2}-\theta ))\frac{\mathbf{V}\times {\mathbf{V}}_p}{|\mathbf{V}\times {\mathbf{V}}_p|}& =& (\left|\overrightarrow{O^{\prime }{P}^{\prime }}\right|\sin \left(\theta \right))\frac{\mathbf{V}\times {\mathbf{V}}_p}{\left|\overrightarrow{O^{\prime }{P}^{\prime }}\right|}\cdots \overrightarrow{O^{\prime }{P}^{\prime }}\mathrm{の}\mathbf{V}\times {\mathbf{V}}_p\mathrm{方}\mathrm{向}\mathrm{成}\mathrm{分}\\ \\ & & \cdots \cos (\frac{\pi }{2}-\theta )=\cos \left(\frac{\pi }{2}\right)\cos \left(\theta \right)-\sin \left(\frac{\pi }{2}\right)\sin \left(\theta \right)=0\cos \left(\theta \right)-1\sin \left(\theta \right)=\sin \left(\theta \right)\\ & =& \sin \left(\theta \right)\mathbf{V}\times {\mathbf{V}}_p\\ \overrightarrow{O{P}^{\prime }}& =& \overrightarrow{O{O}^{\prime }}+\overrightarrow{O^{\prime }{P}^{\prime }}\\ & =& \overrightarrow{O{O}^{\prime }}+\cos \left(\theta \right)\overrightarrow{O^{\prime }P}+\sin \left(\theta \right)\mathbf{V}\times {\mathbf{V}}_p\\ & =& (\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V}+\cos \left(\theta \right)({\mathbf{V}}_p-(\mathbf{V}\cdot {\mathbf{V}}_p)\mathbf{V})+\sin \left(\theta \right)\mathbf{V}\times {\mathbf{V}}_p\end{array}$$ これは四元数において，元と逆元を左右から作用させる際の特殊な形と同様の計算となる．

1/6, 1/12, 1/30 公式

$\frac{1}{6}$公式

$x\mathrm{軸}\mathrm{と}\mathrm{二}\mathrm{次}\mathrm{凾}\mathrm{数}(x-\alpha )(x-\beta )\mathrm{で}\mathrm{囲}\mathrm{ま}\mathrm{れ}\mathrm{る}\mathrm{面}\mathrm{積}(\mathrm{直}\mathrm{線}\mathrm{と}\mathrm{二}\mathrm{次}\mathrm{凾}\mathrm{数}\mathrm{で}\mathrm{も}\mathrm{引}\mathrm{き}\mathrm{算}\mathrm{す}\mathrm{れ}\mathrm{ば}\mathrm{同}\mathrm{様})$は 二次凾数の解(x軸との交点)を$x=\alpha ,\beta$とした，$p=1,q=1$の第一種オイラー積分である．
(X軸の下側にできる面積は負値で表される) $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha )(x-\beta )\mathrm{d}x& =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^1(-1(\beta -x){)}^1\mathrm{d}x\\ & =& -\frac{1!1!}{(1+1+1)!}(\beta -\alpha {)}^{(1+1+1)}\cdots {\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\\ & =& -\frac{1!1!}{3!}(\beta -\alpha {)}^3\\ & =& -\frac{1}{6}(\beta -\alpha {)}^3\end{array}$$ ( 積分として解いた記事 )

$\frac{1}{12}$公式

x軸とそれと接する三次凾数$(x-\alpha )(x-\beta {)}^2$で囲まれる面積は三次凾数の解を$x=\alpha ,\beta$(後者を重複側とする)とした， $p=1,q=2$の第一種オイラー積分である． $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha )(x-\beta {)}^2\mathrm{d}x& =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^1(-1(\beta -x){)}^2\mathrm{d}x\\ & =& \frac{1!2!}{(1+2+1)!}(\beta -\alpha {)}^{(1+2+1)}\cdots {\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{1!2!}{4!}(\beta -\alpha {)}^4\\ & =& \frac{2}{24}(\beta -\alpha {)}^4\\ & =& \frac{1}{12}(\beta -\alpha {)}^4\end{array}$$

$\frac{1}{30}$公式

x軸とそれと接する四次凾数$(x-\alpha {)}^2(x-\beta {)}^2$で囲まれる面積は四次凾数の解を$x=\alpha ,\beta$(両方を重複とする)とした， $p=2,q=2$の第一種オイラー積分である． $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^2(x-\beta {)}^2\mathrm{d}x& =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^2(-1(\beta -x){)}^2\mathrm{d}x\\ & =& \frac{2!2!}{(2+2+1)!}(\beta -\alpha {)}^{(2+2+1)}\cdots {\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{2!2!}{5!}(\beta -\alpha {)}^5\\ & =& \frac{4}{120}(\beta -\alpha {)}^5\\ & =& \frac{1}{30}(\beta -\alpha {)}^5\end{array}$$

$\frac{1}{3}$公式(?)

x軸と接する二次凾数$(x-\alpha {)}^2$があり，それらと$x=\beta$で囲まれる面積は二次凾数の解を$x=\alpha$(重複)とした， $p=2,q=0$の第一種オイラー積分である． $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^2(x-\beta {)}^0\mathrm{d}x& =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^2(-1(\beta -x){)}^0\mathrm{d}x\\ & =& \frac{2!0!}{(2+0+1)!}(\beta -\alpha {)}^{(2+0+1)}\cdots {\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{2!0!}{3!}(\beta -\alpha {)}^3\\ & =& \frac{2}{6}(\beta -\alpha {)}^3\\ & =& \frac{1}{3}(\beta -\alpha {)}^3\end{array}$$ これは$(x-\alpha {)}^2$の$\alpha$から$\beta$までの積分結果と同様になっている． $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^2\mathrm{d}x& =& {[\frac{1}{3}(x-\alpha {)}^3]}_{\alpha }^{\beta }\\ & =& [\frac{1}{3}(\beta -\alpha {)}^3-\frac{1}{3}(\alpha -\alpha {)}^3]\\ & =& [\frac{1}{3}(\beta -\alpha {)}^3-0]\\ & =& \frac{1}{3}(\beta -\alpha {)}^3\end{array}$$

第一種オイラー積分 / ベータ凾数

第一種オイラー積分 / ベータ凾数

$p,q\ge 0$の整数の元で以下の式を第一種オイラー積分という． $$\begin{array}{r}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x\end{array}$$ 部分積分を一回適用すると以下のように$p+1$,$q-1$の第一種オイラー積分がでてくる． $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x& =& {\int }_{\alpha }^{\beta }\left[\frac{\mathrm{d}}{\mathrm{d}x}\{\frac{1}{p+1}(x-\alpha {)}^{(p+1)}\}\right](\beta -x{)}^q\mathrm{d}x\cdots \frac{\mathrm{d}}{\mathrm{d}x}\{\frac{1}{p+1}(x-\alpha {)}^{(p+1)}\}=(x-\alpha {)}^p\\ & =& {[\frac{1}{p+1}(x-\alpha {)}^{(p+1)}(\beta -x{)}^q]}_{\alpha }^{\beta }-{\int }_{\alpha }^{\beta }\frac{1}{p+1}(x-\alpha {)}^{(p+1)}\{q(\beta -x{)}^{(q-1)}(-1)\}\mathrm{d}x\cdots \int f^{\prime }g\mathrm{d}x=fg-\int f{g}^{\prime }\mathrm{d}x\\ & =& 0-{\int }_{\alpha }^{\beta }\frac{-q}{p+1}(x-\alpha {)}^{(p+1)}(\beta -x{)}^{(q-1)}\mathrm{d}x\cdots (\alpha -\alpha )=0,(\beta -\beta )=0\\ & =& \frac{q}{p+1}{\int }_{\alpha }^{\beta }(x-\alpha {)}^{(p+1)}(\beta -x{)}^{(q-1)}\mathrm{d}x\end{array}$$ 第一種オイラー積分の乗数$p,q$に着目し元の式を$I_{p,q}$,部分積分によって出てきた積分を$I_{p+1,q-1}$と表すと以下の様になる． $$\begin{array}{rcl}{I}_{p,q}& =& \frac{q}{p+1}{I}_{p+1,q-1}\cdots I_{a,b}={\int }_{\alpha }^{\beta }(x-\alpha {)}^a(\beta -x{)}^b\mathrm{d}x\\ & =& \frac{q}{p+1}\frac{q-1}{p+2}{I}_{p+2,q-2}\\ & =& \frac{q}{p+1}\frac{q-1}{p+2}\cdots \frac{2}{p+(q-1)}\frac{1}{p+q}{I}_{p+q,0}\\ I_{p+q,0}& =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^{(p+q)}(\beta -x{)}^0\mathrm{d}x\\ & =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^{(p+q)}\mathrm{d}x\\ & =& {[\frac{1}{p+q+1}(x-\alpha {)}^{\{(p+q)+1\}}]}_{\alpha }^{\beta }\\ & =& \frac{1}{p+q+1}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{(\beta -\alpha {)}^{(p+q+1)}}{p+q+1}\\ I_{p,q}& =& \frac{q}{p+1}\frac{q-1}{p+2}\cdots \frac{2}{p+q-1}\frac{1}{p+q}\frac{(\beta -\alpha {)}^{(p+q+1)}}{p+q+1}\\ & =& \frac{q(q-1)\cdots 21}{(p+1)(p+2)\cdots (p+q-1)(p+q)(p+q+1)}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{q!}{\frac{(p+q+1)!}{p!}}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\end{array}$$ 特に$\alpha =0,\beta =1$の時 $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x& =& {\int }_0^1(x-0{)}^p(1-x{)}^q\mathrm{d}x\\ & =& {\int }_0^1{x}^p(1-x{)}^q\mathrm{d}x\\ & =& \frac{p!q!}{(p+q+1)!}(1-0{)}^{(p+q+1)}\cdots {\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{p!q!}{(p+q+1)!}\end{array}$$ 特に$\alpha =0,\beta =1,p=m-1,q=n-1$の時 $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x& =& {\int }_0^1{x}^{(m-1)}(1-x{)}^{(n-1)}\mathrm{d}x\\ & =& \frac{(m-1)!(n-1)!}{((m-1)+(n-1)+1)!}\cdots {\int }_0^1{x}^p(1-x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}\\ & =& \frac{(m-1)!(n-1)!}{(m+n-1)!}\\ & =& \frac{\mathrm{\Gamma }(m)\mathrm{\Gamma }(n)}{\mathrm{\Gamma }(m+n)}\cdots \mathrm{\Gamma }(a)=(a-1)!\\ & =& B(m,n)\cdots \mathrm{ベ}\mathrm{ー}\mathrm{タ}\mathrm{凾}\mathrm{数}\end{array}$$ 逆を言えば上記積分の値を1にしたい時はベータ凾数で割れば(逆数を掛ければ)よい(確率密度凾数との関係性)． $$\begin{array}{rcl}\frac{1}{B(m,n)}{\int }_0^1{x}^{(m-1)}(1-x{)}^{(n-1)}\mathrm{d}x& =& 1\\ f(x;m,n)& =& \frac{x^{(m-1)}(1-x{)}^{(n-1)}}{B(m,n)}\cdots \mathrm{ベ}\mathrm{ー}\mathrm{タ}\mathrm{分}\mathrm{布}\end{array}$$

n回の事象が発生するまでの間隔の確率密度分布 (ガンマ分布)

$n$回の事象が発生するまでの間隔(時間)の確率$Y_n$について考える． $$\begin{array}{rcl}{F}_n(t)& =& P(Y_n\le t)\cdots F_n\mathrm{を}{Y}_n\mathrm{の}\mathrm{累}\mathrm{積}\mathrm{分}\mathrm{布}\mathrm{凾}\mathrm{数}(cumulativedistributionfunction,CDF)\mathrm{と}\mathrm{す}\mathrm{る}．(\mathrm{少}\mathrm{な}\mathrm{く}\mathrm{と}\mathrm{も}t\mathrm{ま}\mathrm{で}\mathrm{に}n\mathrm{回}\mathrm{の}\mathrm{事}\mathrm{象}\mathrm{が}\mathrm{発}\mathrm{生}\mathrm{し}\mathrm{て}\mathrm{い}\mathrm{る})\\ & =& P(X_t\ge n)\cdots n\mathrm{回}\mathrm{以}\mathrm{上}\mathrm{の}\mathrm{事}\mathrm{象}\mathrm{が}\mathrm{発}\mathrm{生}\mathrm{す}\mathrm{る}\mathrm{の}\mathrm{が}\mathrm{時}\mathrm{刻}t\mathrm{以}\mathrm{降}\mathrm{に}\mathrm{な}\mathrm{る}\mathrm{確}\mathrm{率}\mathrm{と}\mathrm{も}\mathrm{言}\mathrm{え}\mathrm{る}\\ & =& \sum _{x=n}^{\mathrm{\infty }}{P}_o(x;\lambda t)\cdots \mathrm{事}\mathrm{象}\mathrm{の}\mathrm{発}\mathrm{生}\mathrm{回}\mathrm{数}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{を}\lambda t\mathrm{を}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{と}\mathrm{す}\mathrm{る}\mathrm{ポ}\mathrm{ア}\mathrm{ソ}\mathrm{ン}\mathrm{分}\mathrm{布}\mathrm{と}\mathrm{す}\mathrm{る}．\mathrm{ポ}\mathrm{ア}\mathrm{ソ}\mathrm{ン}\mathrm{分}\mathrm{布}:P_o(x;\lambda )={\mathrm{e}}^{-\lambda }\frac{{\left(\lambda \right)}^x}{x!}\\ & =& \sum _{x=n}^{\mathrm{\infty }}{\mathrm{e}}^{-\lambda t}\frac{{\left(\lambda t\right)}^x}{x!}\cdots \mathrm{ポ}\mathrm{ア}\mathrm{ソ}\mathrm{ン}\mathrm{分}\mathrm{布}\mathrm{は}\mathrm{期}\mathrm{間}\mathrm{内}\mathrm{で}\mathrm{何}\mathrm{回}\mathrm{起}\mathrm{こ}\mathrm{る}\mathrm{か}？\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}(\mathrm{指}\mathrm{数}\mathrm{分}\mathrm{布}\mathrm{は}\mathrm{次}\mathrm{の}\mathrm{発}\mathrm{生}\mathrm{ま}\mathrm{で}\mathrm{の}\mathrm{期}\mathrm{間}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布})\\ & =& 1-\sum _{x=0}^{n-1}{\mathrm{e}}^{-\lambda t}\frac{{\left(\lambda t\right)}^x}{x!}\cdots \mathrm{時}\mathrm{刻}t\mathrm{ま}\mathrm{で}\mathrm{に}n\mathrm{回}\mathrm{未}\mathrm{満}\mathrm{の}\mathrm{事}\mathrm{象}\mathrm{し}\mathrm{か}\mathrm{発}\mathrm{生}\mathrm{し}\mathrm{な}\mathrm{い}\mathrm{場}\mathrm{合}\mathrm{の}\mathrm{余}\mathrm{事}\mathrm{象}\\ & =& 1-{\mathrm{e}}^{-\lambda t}\{\frac{{\left(\lambda t\right)}^0}{0!}+\frac{{\left(\lambda t\right)}^1}{1!}+\frac{{\left(\lambda t\right)}^2}{2!}+\cdots +\frac{{\left(\lambda t\right)}^{(n-2)}}{(n-2)!}+\frac{{\left(\lambda t\right)}^{(n-1)}}{(n-1)!}\}\\ & =& 1-{\mathrm{e}}^{-\lambda t}A\cdots A=\{\frac{{\left(\lambda t\right)}^0}{0!}+\frac{{\left(\lambda t\right)}^1}{1!}+\frac{{\left(\lambda t\right)}^2}{2!}+\cdots +\frac{{\left(\lambda t\right)}^{(n-2)}}{(n-2)!}+\frac{{\left(\lambda t\right)}^{(n-1)}}{(n-1)!}\}\\ Y_n=\frac{\partial F_n(t)}{\partial t}& =& 0-[({\mathrm{e}}^{-\lambda t}{)}^{\prime }A+{\mathrm{e}}^{-\lambda t}(A{)}^{\prime }]\cdots (fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }\cdots \mathrm{累}\mathrm{積}\mathrm{分}\mathrm{布}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{微}\mathrm{分}\mathrm{は}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}(probabilitydensityfunction、PDF)\\ ({\mathrm{e}}^{-\lambda t}{)}^{\prime }A& =& {\mathrm{e}}^{-\lambda t}(-\lambda )A\\ & =& {\mathrm{e}}^{-\lambda t}(-\lambda )\{\frac{{\left(\lambda t\right)}^0}{0!}+\frac{{\left(\lambda t\right)}^1}{1!}+\frac{{\left(\lambda t\right)}^2}{2!}+\cdots +\frac{{\left(\lambda t\right)}^{(n-2)}}{(n-2)!}+\frac{{\left(\lambda t\right)}^{(n-1)}}{(n-1)!}\}\\ & =& {\mathrm{e}}^{-\lambda t}\{-\frac{\lambda }{0!}-\frac{{\lambda }^{2}t}{1!}-\frac{{\lambda }^3{t}^2}{2!}-\cdots -\frac{{\lambda }^{(n-1)}{t}^{(n-2)}}{(n-2)!}-\frac{{\lambda }^n{t}^{(n-1)}}{(n-1)!}\}\\ \\ {\mathrm{e}}^{-\lambda t}(A{)}^{\prime }& =& {\mathrm{e}}^{-\lambda t}{\left(\{\frac{{\left(\lambda t\right)}^0}{0!}+\frac{{\left(\lambda t\right)}^1}{1!}+\frac{{\left(\lambda t\right)}^2}{2!}+\cdots +\frac{{\left(\lambda t\right)}^{(n-2)}}{(n-2)!}+\frac{{\left(\lambda t\right)}^{(n-1)}}{(n-1)!}\}\right)}^{\prime }\\ & =& {\mathrm{e}}^{-\lambda t}\{{\left(\frac{1}{0!}\right)}^{\prime }+{\left(\frac{\lambda t}{1!}\right)}^{\prime }+{\left(\frac{{\lambda }^2{t}^2}{2!}\right)}^{\prime }+\cdots +{\left(\frac{{\lambda }^{(n-2)}{t}^{(n-2)}}{(n-2)!}\right)}^{\prime }+{\left(\frac{{\lambda }^{(n-1)}{t}^{(n-1)}}{(n-1)!}\right)}^{\prime }\}\\ & =& {\mathrm{e}}^{-\lambda t}\{0+\frac{\lambda }{1!}+\frac{2{\lambda }^{2}t}{2!}+\cdots +\frac{(n-2){\lambda }^{(n-2)}{t}^{(n-3)}}{(n-2)!}+\frac{(n-1){\lambda }^{(n-1)}{t}^{(n-2)}}{(n-1)!}\}\\ & =& {\mathrm{e}}^{-\lambda t}\{\frac{\lambda }{1!}+\frac{{\lambda }^{2}t}{1!}+\cdots +\frac{{\lambda }^{(n-2)}{t}^{(n-3)}}{(n-3)!}+\frac{{\lambda }^{(n-1)}{t}^{(n-2)}}{(n-2)!}\}\cdots \frac{n}{n!}=\frac{n}{n\times (n-1)\times \cdots \times 1}=\frac{1}{(n-1)\times \cdots \times 1}=\frac{1}{(n-1)!}\\ ({\mathrm{e}}^{-\lambda t}{)}^{\prime }A+{\mathrm{e}}^{-\lambda t}(A{)}^{\prime }& =& {\mathrm{e}}^{-\lambda t}\{-\frac{{\lambda }^1}{0!}-\frac{{\lambda }^{2}t}{1!}-\frac{{\lambda }^3{t}^2}{2!}-\cdots -\frac{{\lambda }^{(n-1)}{t}^{(n-2)}}{(n-2)!}-\frac{{\lambda }^n{t}^{(n-1)}}{(n-1)!}\}+{\mathrm{e}}^{-\lambda t}\{\frac{\lambda }{1!}+\frac{{\lambda }^{2}t}{1!}+\cdots +\frac{{\lambda }^{(n-2)}{t}^{(n-3)}}{(n-3)!}+\frac{{\lambda }^{(n-1)}{t}^{(n-2)}}{(n-2)!}\}\\ & =& {\mathrm{e}}^{-\lambda t}\{(-\frac{\lambda }{0!}+\frac{\lambda }{1!})+(-\frac{{\lambda }^{2}t}{1!}+\frac{{\lambda }^{2}t}{1!})+\cdots +(-\frac{{\lambda }^{(n-1)}{t}^{(n-2)}}{(n-2)!}+\frac{{\lambda }^{(n-1)}{t}^{(n-2)}}{(n-2)!})-\frac{{\lambda }^n{t}^{(n-1)}}{(n-1)!}\}\\ & =& {\mathrm{e}}^{-\lambda t}\{-\frac{{\lambda }^n{t}^{(n-1)}}{(n-1)!}\}\\ Y_n=\frac{\partial F_n(t)}{\partial t}& =& -[({\mathrm{e}}^{-\lambda t}{)}^{\prime }A+{\mathrm{e}}^{-\lambda t}(A{)}^{\prime }]\\ & =& -\left[{\mathrm{e}}^{-\lambda t}\{-\frac{{\lambda }^n{t}^{(n-1)}}{(n-1)!}\}\right]\\ & =& {\mathrm{e}}^{-\lambda t}\frac{{\lambda }^n{t}^{(n-1)}}{(n-1)!}\\ & =& {\mathrm{e}}^{-\beta t}\frac{{\beta }^{\alpha }{t}^{(\alpha -1)}}{(\alpha -1)!}\cdots \alpha =n,\beta =\lambda \mathrm{と}\mathrm{す}\mathrm{る}.\\ & =& \frac{{\mathrm{e}}^{-\beta t}}{(\alpha -1)!}{\beta }^{\alpha }{t}^{(\alpha -1)}\\ & =& \frac{{\mathrm{e}}^{-\beta t}}{\mathrm{\Gamma }(\alpha )}{\beta }^{\alpha }{t}^{(\alpha -1)}\cdots \mathrm{ガ}\mathrm{ン}\mathrm{マ}\mathrm{分}\mathrm{布},\mathrm{\Gamma }(n)=(n-1)!\end{array}$$ $\alpha =1$の時，指数分布となる(事象と事象の間隔の確率分布)． $$\begin{array}{rcl}\frac{{\mathrm{e}}^{-\beta t}}{\mathrm{\Gamma }(\alpha )}{\beta }^{\alpha }{t}^{(\alpha -1)}& =& \frac{{\mathrm{e}}^{-\beta t}}{\mathrm{\Gamma }(1)}{\beta }^1{t}^{(1-1)}\cdots \alpha =1\\ & =& \frac{{\mathrm{e}}^{-\beta t}}{1}\beta t^{\left(0\right)}\cdots \mathrm{\Gamma }(n)=(n-1)!,\mathrm{\Gamma }(1)=(1-1)!=0!=1\\ & =& {\mathrm{e}}^{-\beta t}\beta \cdots t^0=1\\ & =& \lambda {\mathrm{e}}^{-\lambda t}\cdots \beta =\lambda \end{array}$$

事象の発生する回数についての確率密度分布 (ポアソン分布)

単位時間当たりの事象の発生確率を$\lambda$とし，t時間の間にk回事象が発生する確率を考える． tまでをn等分した時間を$\mathrm{\Delta }t=\frac{t}{n}$として二項分布により以下のように表すことができる． $$\begin{array}{rcl}p(k;\lambda ,\mathrm{\Delta }t)& =& {}_n{C}_k{\left(\lambda \mathrm{\Delta }t\right)}^k{(1-\left(\lambda \mathrm{\Delta }t\right))}^{(n-k)}\end{array}$$ 二項分布からポアソン分布を導いた時と同様に$\lambda$を$\lambda \mathrm{\Delta }t$として計算を進めると以下のようになる． $$\begin{array}{rcl}p(k;\lambda ,\mathrm{\Delta }t)& =& {}_n{C}_k{\left(\lambda \mathrm{\Delta }t\right)}^k{(1-\left(\lambda \mathrm{\Delta }t\right))}^{(n-k)}\\ & =& \frac{n!}{k!(n-k)!}{\left(\lambda \mathrm{\Delta }t\right)}^k{(1-\left(\lambda \mathrm{\Delta }t\right))}^{(n-k)}\\ & =& \frac{1}{k!}\{n(n-1)\cdots (n-k+1)\}{\left(\lambda \mathrm{\Delta }t\right)}^k{(1-\left(\lambda \mathrm{\Delta }t\right))}^{(n-k)}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})n^k\}{\left(\lambda \mathrm{\Delta }t\right)}^k{(1-\left(\lambda \mathrm{\Delta }t\right))}^{(n-k)}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(n\lambda \mathrm{\Delta }t\right)}^k{(1-\left(\lambda \mathrm{\Delta }t\right))}^{(n-k)}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(n\lambda \frac{t}{n}\right)}^k{(1-\left(\lambda \frac{t}{n}\right))}^n{(1-\left(\lambda \frac{t}{n}\right))}^{-k}\cdots \mathrm{\Delta }t=\frac{t}{n}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(\lambda t\right)}^k{(1-\left(\frac{\lambda t}{n}\right))}^n{(1-\left(\frac{\lambda t}{n}\right))}^{-k}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(\lambda t\right)}^k{(1+\frac{1}{\frac{-n}{\lambda t}})}^n{(1-\left(\frac{\lambda t}{n}\right))}^{-k}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(\lambda t\right)}^k{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}\frac{-\lambda t}{n}}\right\}}^n{(1-\left(\frac{\lambda t}{n}\right))}^{-k}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(\lambda t\right)}^k{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}}\right\}}^{\frac{-\lambda t}{n}n}{(1-\left(\frac{\lambda t}{n}\right))}^{-k}\\ & =& \frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(\lambda t\right)}^k{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}}\right\}}^{-\lambda t}{(1-\left(\frac{\lambda t}{n}\right))}^{-k}\end{array}$$ ここでnを$\mathrm{\infty }$に極限をとるとnを分母にもつ分数は0へ収束することや，収束数列によるネイピア数の定義により以下のようになる． $$\begin{array}{rcl}& & \underset{n\to \mathrm{\infty }}{lim}\frac{1}{k!}\{1(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})\}{\left(\lambda t\right)}^k{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}}\right\}}^{-\lambda t}{(1-\left(\frac{\lambda t}{n}\right))}^{-k}\\ & =& \frac{1}{k!}\{1(1-0)\cdots (1-0)\}{\left(\lambda t\right)}^k{\mathrm{e}}^{-\lambda t}{(1-0)}^{-k}\\ & =& \frac{1}{k!}{\left(\lambda t\right)}^k{\mathrm{e}}^{-\lambda t}\\ & =& \frac{{\left(\lambda t\right)}^k}{k!}{\mathrm{e}}^{-\lambda t}\end{array}$$

事象の発生する間隔についての確率密度分布 (指数分布)

ある事象が起こってから次の事象が起きるまでの時間をTとする． 事象が起こってからt時間までに次の事象が起きる確率は，tまでをn等分した時間を$\mathrm{\Delta }t=\frac{t}{n}$として， “事象が$(k-1)$回の$\mathrm{\Delta }t$の間に発生しなかったのち次の$k$回目の$\mathrm{\Delta }t$の間に発生する確率”の$k=1$から$n$までの和によって以下のように表すことができる (独立同分布の和によって決まる過程，単位時間当たりの事象の発生確率(個々の分布)を$\lambda$とする)． $$\begin{array}{rcl}P(T\le t)& =& \lambda \mathrm{\Delta }t+(1-\lambda \mathrm{\Delta }t)\mathrm{\Delta }t+(1-\lambda \mathrm{\Delta }t{)}^2\mathrm{\Delta }t+\cdots +(1-\lambda \mathrm{\Delta }t{)}^{(n-1)}\mathrm{\Delta }t\\ & =& \sum _{k=1}^n(\lambda \mathrm{\Delta }t)(1-\lambda \mathrm{\Delta }t{)}^{k-1}\\ \lambda \mathrm{\Delta }t& & \mathrm{\Delta }t\mathrm{の}\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\\ (1-\lambda \mathrm{\Delta }t)\lambda \mathrm{\Delta }t& & \mathrm{最}\mathrm{初}\mathrm{の}\mathrm{\Delta }t\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{せ}\mathrm{ず}，\mathrm{次}\mathrm{の}\mathrm{\Delta }t\mathrm{の}\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\\ (1-\lambda \mathrm{\Delta }t{)}^2\lambda \mathrm{\Delta }t& & 2\mathrm{回}\mathrm{の}\mathrm{\Delta }t\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{せ}\mathrm{ず}，\mathrm{次}\mathrm{の}\mathrm{\Delta }t\mathrm{の}\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\\ (1-\lambda \mathrm{\Delta }t{)}^{n-1}\lambda \mathrm{\Delta }t& & n-1\mathrm{回}\mathrm{の}\mathrm{\Delta }t\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{せ}\mathrm{ず}，\mathrm{次}\mathrm{の}\mathrm{\Delta }t\mathrm{の}\mathrm{間}\mathrm{に}\mathrm{発}\mathrm{生}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\end{array}$$ 初項を$\lambda \mathrm{\Delta }t$，公比を$(1-\lambda \mathrm{\Delta }t)$とする等比級数として上式は以下のようになる． $$\begin{array}{rcl}P(T\le t)& =& \lambda \mathrm{\Delta }t+(1-\lambda \mathrm{\Delta }t)\mathrm{\Delta }t+(1-\lambda \mathrm{\Delta }t{)}^2\mathrm{\Delta }t+\cdots +(1-\lambda \mathrm{\Delta }t{)}^{(n-1)}\mathrm{\Delta }t\\ & =& \lambda \mathrm{\Delta }t\frac{1-(1-\lambda \mathrm{\Delta }t{)}^n}{1-(1-\lambda \mathrm{\Delta }t)}\cdots \sum _{k=1}^{n}a{r}^{(k-1)}=\frac{a(1-r^n)}{1-r}\\ & =& 1-{(1-\lambda \mathrm{\Delta }t)}^n\\ & =& 1-{(1-\lambda \frac{t}{n})}^n\cdots \mathrm{\Delta }t=\frac{t}{n}\\ & =& 1-{(1+\frac{1}{-\frac{n}{\lambda t}})}^n\\ & =& 1-{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}\frac{\lambda t}{-n}}\right\}}^n\\ & =& 1-{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}}\right\}}^{\frac{\lambda t}{-n}n}\\ & =& 1-{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}}\right\}}^{-\lambda t}\end{array}$$ ここでnを$\mathrm{\infty }$に極限をとると収束数列によるネイピア数の定義より以下のようになる． $$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}1-{\left\{{(1+\frac{1}{\frac{-n}{\lambda t}})}^{\frac{-n}{\lambda t}}\right\}}^{-\lambda t}& =& 1-{\mathrm{e}}^{-\lambda t}\cdots \underset{n\to +\mathrm{\infty }}{lim}{(1+\frac{1}{n})}^n=\mathrm{e}\end{array}$$ 上式は時刻tまでの累積分布凾数$F(t)$となる． 累積分布凾数を微分することで確率密度凾数を得ることができる． $$\begin{array}{rcl}F(t)& =& 1-{\mathrm{e}}^{-\lambda t}\\ f(t)& =& \frac{\mathrm{d}F(t)}{\mathrm{d}t}\\ & =& 0-(-\lambda {\mathrm{e}}^{-\lambda t})\\ & =& \lambda {\mathrm{e}}^{-\lambda t}\cdots \mathrm{指}\mathrm{数}\mathrm{分}\mathrm{布}\end{array}$$

等比級数 (等比数列の和)

初項を$a$，公比を$r$とする数列の和，級数は以下のようになる． $$\begin{array}{rclrclr}{S}_n& =& \sum _{k=1}^{n}a{r}^{(k-1)}& =& a+& ar+a{r}^2+a{r}^3+\cdots +a{r}^{n-1}& \\ r{S}_n& =& r\sum _{k=1}^{n}a{r}^{(k-1)}& =& & ar+a{r}^2+a{r}^3+\cdots +a{r}^{n-1}& +a{r}^n\\ S_n-r{S}_n& =& a-a{r}^n\\ (1-r)S_n& =& a(1-r^n)\\ S_n& =& \frac{a(1-r^n)}{1-r}\end{array}$$

フィボナッチ数列(Fibonacci numbers)の行列表現と一般項

フィボナッチ数列(Fibonacci numbers)の行列表現

フィボナッチ数列 $$\begin{array}{r}{a}_{n+1}=a_n+a_{n-1}\cdots \mathrm{た}\mathrm{だ}\mathrm{し}{a}_1=1,a_2=1\end{array}$$ これを行列で表現すると以下となる． $$\begin{array}{rcl}\left(\begin{array}{c}{a}_{n+1}\\ a_n\end{array}\right)& =& \left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\left(\begin{array}{c}{a}_n\\ a_{n-1}\end{array}\right)\\ {\overrightarrow{a}}_{(n+1),n}& =& \mathrm{A}{\overrightarrow{a}}_{n,(n-1)}\end{array}$$

フィボナッチ数列の一般項

${\overrightarrow{a}}_{n,(n-1)}$を$\mathrm{A}$の固有ベクトル${\overrightarrow{x}}_{1,2}$を用い表せれば， 上式は対応する固有値${\lambda }_{1,2}$を用いて以下のように表すことができる． $$\begin{array}{rcl}{\overrightarrow{a}}_{(n+1),n}& =& \mathrm{A}{\overrightarrow{a}}_{n,(n-1)}\\ & =& \mathrm{A}(C_1{\overrightarrow{x}}_1+C_2{\overrightarrow{x}}_2)\cdots C_1,C_2\mathrm{は}\mathrm{定}\mathrm{数}\\ & =& C_1\mathrm{A}{\overrightarrow{x}}_1+C_2\mathrm{A}{\overrightarrow{x}}_2\\ & =& C_1{\lambda }_1{\overrightarrow{x}}_1+C_2{\lambda }_2{\overrightarrow{x}}_2\cdots \mathrm{A}\overrightarrow{x}=\lambda \overrightarrow{x}\cdots \mathrm{固}\mathrm{有}\mathrm{値}\mathrm{・}\mathrm{固}\mathrm{有}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{の}\mathrm{定}\mathrm{義}\end{array}$$ またこれを繰り返すことで一般項として以下のように表せることになる． $$\begin{array}{rcl}{\overrightarrow{a}}_{(n+1),n}& =& {\mathrm{A}}^n{\overrightarrow{a}}_{init}\cdots {\overrightarrow{a}}_{init}\mathrm{は}\mathrm{固}\mathrm{有}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{の}\mathrm{和}\mathrm{で}\mathrm{表}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{初}\mathrm{期}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{定}\mathrm{数}{C}_1,C_2\mathrm{を}\mathrm{含}\mathrm{む}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\\ & =& {\mathrm{A}}^n(C_{{\overrightarrow{a}}_{init}1}{\overrightarrow{x}}_1+C_{{\overrightarrow{a}}_{init}2}{\overrightarrow{x}}_2)\\ & =& C_{{\overrightarrow{a}}_{init}1}{\mathrm{A}}^n{\overrightarrow{x}}_1+C_{{\overrightarrow{a}}_{init}2}{\mathrm{A}}^n{\overrightarrow{x}}_2\\ & =& C_{{\overrightarrow{a}}_{init}1}{\lambda }_1^n{\overrightarrow{x}}_1+C_{{\overrightarrow{a}}_{init}2}{\lambda }_2^n{\overrightarrow{x}}_2\end{array}$$ そこでまず$\mathrm{A}$の固有値を求める． $$\begin{array}{rcl}|\mathrm{A}-\lambda \mathrm{E}|& =& 0\\ |\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]-\lambda \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]|& =& \\ \left|\begin{array}{cc}1-\lambda & 1\\ 1& -\lambda \end{array}\right|& =& \\ (1-\lambda )(-\lambda )-1& =& \\ {\lambda }^2-\lambda -1& =& \\ \lambda & =& \frac{-(-1)\pm \sqrt{(-1{)}^2-4(1)(-1)}}{2(1)}\\ & =& \frac{1\pm \sqrt{5}}{2}\\ {\lambda }_1,{\lambda }_2& =& \frac{1+\sqrt{5}}{2},\frac{1-\sqrt{5}}{2}\cdots \mathrm{こ}\mathrm{こ}\mathrm{で}\mathrm{は}{\lambda }_1>{\lambda }_2\mathrm{と}\mathrm{す}\mathrm{る}．\end{array}$$ ${\lambda }_1$に対応する固有値ベクトル$\overrightarrow{x_1}$を求める． $$\begin{array}{rcl}(\mathrm{A}-{\lambda }_1\mathrm{E})\overrightarrow{x_1}& =& \overrightarrow{0}\\ \left[\begin{array}{cc}1-{\lambda }_1& 1\\ 1& -{\lambda }_1\end{array}\right]\left(\begin{array}{c}{x}_{11}\\ x_{12}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right)\\ \left[\begin{array}{cc}1-\frac{1+\sqrt{5}}{2}& 1\\ 1& -\frac{1+\sqrt{5}}{2}\end{array}\right]\left(\begin{array}{c}{x}_{11}\\ x_{12}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right)\\ \left[\begin{array}{cc}\frac{1-\sqrt{5}}{2}& 1\\ 1& -\frac{1+\sqrt{5}}{2}\end{array}\right]\left(\begin{array}{c}{x}_{11}\\ x_{12}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right)\end{array}$$ $$\{\begin{array}{rlr}\frac{1-\sqrt{5}}{2}{x}_{11}+x_{12}& =& 0\\ x_{11}-\frac{1+\sqrt{5}}{2}{x}_{12}& =& 0\end{array}$$ $$\begin{array}{rcl}{x}_{11}& =& \frac{1+\sqrt{5}}{2}{x}_{12}\\ {\overrightarrow{x}}_1& =& t\left(\begin{array}{c}\frac{1+\sqrt{5}}{2}\\ 1\end{array}\right)\cdots t:\mathrm{任}\mathrm{意}\mathrm{の}\mathrm{実}\mathrm{数}\end{array}$$ 同様に${\lambda }_2$に対応する固有値ベクトル$\overrightarrow{x_2}$を求める． $$\begin{array}{rcl}(\mathrm{A}-{\lambda }_2\mathrm{E})\overrightarrow{x_2}& =& \overrightarrow{0}\\ \left[\begin{array}{cc}1-{\lambda }_2& 1\\ 1& -{\lambda }_2\end{array}\right]\left(\begin{array}{c}{x}_{21}\\ x_{22}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right)\\ \left[\begin{array}{cc}1-\frac{1-\sqrt{5}}{2}& 1\\ 1& -\frac{1-\sqrt{5}}{2}\end{array}\right]\left(\begin{array}{c}{x}_{21}\\ x_{22}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right)\\ \left[\begin{array}{cc}\frac{1+\sqrt{5}}{2}& 1\\ 1& -\frac{1-\sqrt{5}}{2}\end{array}\right]\left(\begin{array}{c}{x}_{21}\\ x_{22}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right)\end{array}$$ $$\{\begin{array}{rlr}\frac{1+\sqrt{5}}{2}{x}_{21}+x_{22}& =& 0\\ x_{21}-\frac{1-\sqrt{5}}{2}{x}_{22}& =& 0\end{array}$$ $$\begin{array}{rcl}{x}_{21}& =& \frac{1-\sqrt{5}}{2}{x}_{22}\\ {\overrightarrow{x}}_2& =& t\left(\begin{array}{c}\frac{1-\sqrt{5}}{2}\\ 1\end{array}\right)\cdots t:\mathrm{任}\mathrm{意}\mathrm{の}\mathrm{実}\mathrm{数}\end{array}$$ よって前述の表現は以下のように書き換えられる． $$\begin{array}{rcl}\left(\begin{array}{c}{a}_{n+1}\\ a_n\end{array}\right)& =& C_{{\overrightarrow{a}}_{init}1}{\lambda }_1^n{\overrightarrow{x}}_1+C_{{\overrightarrow{a}}_{init}2}{\lambda }_2^n{\overrightarrow{x}}_2\\ & =& C_{{\overrightarrow{a}}_{init}1}{\left(\frac{1+\sqrt{5}}{2}\right)}^n\left(\begin{array}{c}\frac{1+\sqrt{5}}{2}\\ 1\end{array}\right)+C_{{\overrightarrow{a}}_{init}2}{\left(\frac{1-\sqrt{5}}{2}\right)}^n\left(\begin{array}{c}\frac{1-\sqrt{5}}{2}\\ 1\end{array}\right)\end{array}$$ $n=1$の時，$a_n=a_1=1,a_{n+1}=a_{1+1}=a_2=1$なので，ここから$C_{{\overrightarrow{a}}_{init}1},C_{{\overrightarrow{a}}_{init}2}$を求める． $$\begin{array}{rcl}{C}_{{\overrightarrow{a}}_{init}1}{\left(\frac{1+\sqrt{5}}{2}\right)}^1\left(\begin{array}{c}\frac{1+\sqrt{5}}{2}\\ 1\end{array}\right)+C_{{\overrightarrow{a}}_{init}2}{\left(\frac{1-\sqrt{5}}{2}\right)}^1\left(\begin{array}{c}\frac{1-\sqrt{5}}{2}\\ 1\end{array}\right)& =& \left(\begin{array}{c}1\\ 1\end{array}\right)\end{array}$$ 簡単のために以下のようにひとまず置き直す． $$\begin{array}{rcl}{C}_1& :& C_{{\overrightarrow{a}}_{init}1}\\ C_2& :& C_{{\overrightarrow{a}}_{init}2}\\ {\alpha }_1& :& \frac{1+\sqrt{5}}{2}\\ {\alpha }_2& :& \frac{1-\sqrt{5}}{2}\end{array}$$ 上記で書き直す． $$\begin{array}{r}{C}_1{\left({\lambda }_1\right)}^1\left(\begin{array}{c}{\alpha }_1\\ 1\end{array}\right)+C_2{\left({\lambda }_2\right)}^1\left(\begin{array}{c}{\alpha }_2\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right)\end{array}$$ 行ごとに書き出すと以下のような連立方程式となる． $$\{\begin{array}{rlr}{C}_1{\lambda }_1{\alpha }_1+C_2{\lambda }_2{\alpha }_2& =& 1\\ C_1{\lambda }_1+C_2{\lambda }_2& =& 1\end{array}$$ $C_1$について解く． $$\begin{array}{rcl}{C}_1{\lambda }_1{\alpha }_2+C_2{\lambda }_2{\alpha }_2& =& {\alpha }_2\cdots \mathrm{連}\mathrm{立}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{二}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{に}{\alpha }_2\mathrm{を}\mathrm{掛}\mathrm{け}\mathrm{る}\\ (C_1{\lambda }_1{\alpha }_1+C_2{\lambda }_2{\alpha }_2)-(C_1{\lambda }_1{\alpha }_2+C_2{\lambda }_2{\alpha }_2)& =& \left(1\right)-\left({\alpha }_2\right)\cdots \mathrm{連}\mathrm{立}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{一}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{よ}\mathrm{り}\mathrm{上}\mathrm{式}\mathrm{を}\mathrm{引}\mathrm{く}\\ C_1{\lambda }_1{\alpha }_1-C_1{\lambda }_1{\alpha }_2& =& 1-{\alpha }_2\\ C_1{\lambda }_1({\alpha }_1-{\alpha }_2)& =& 1-{\alpha }_2\\ C_1{\lambda }_1({\alpha }_1-{\alpha }_2)& =& {\alpha }_1\cdots 1-{\alpha }_2=1-\frac{1-\sqrt{5}}{2}=\frac{1+\sqrt{5}}{2}={\alpha }_1\\ C_1& =& \frac{{\alpha }_1}{{\lambda }_1({\alpha }_1-{\alpha }_2)}=\frac{\frac{1+\sqrt{5}}{2}}{\frac{1+\sqrt{5}}{2}(\frac{1+\sqrt{5}}{2}-\frac{1-\sqrt{5}}{2})}\\ & =& \frac{1}{\frac{1+\sqrt{5}}{2}-\frac{1-\sqrt{5}}{2}}=\frac{1}{\frac{2\sqrt{5}}{2}}=\frac{1}{\sqrt{5}}\end{array}$$ $C_2$について解く． $$\begin{array}{rcl}{C}_1{\lambda }_1{\alpha }_1+C_2{\lambda }_2{\alpha }_1& =& {\alpha }_1\cdots \mathrm{連}\mathrm{立}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{二}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{に}{\alpha }_1\mathrm{を}\mathrm{掛}\mathrm{け}\mathrm{る}\\ (C_1{\lambda }_1{\alpha }_1+C_2{\lambda }_2{\alpha }_2)-(C_1{\lambda }_1{\alpha }_1+C_2{\lambda }_2{\alpha }_1)& =& \left(1\right)-\left({\alpha }_1\right)\cdots \mathrm{連}\mathrm{立}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{一}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{よ}\mathrm{り}\mathrm{上}\mathrm{式}\mathrm{を}\mathrm{引}\mathrm{く}\\ C_2{\lambda }_2{\alpha }_2-C_2{\lambda }_2{\alpha }_1& =& 1-{\alpha }_1\\ C_2{\lambda }_2({\alpha }_2-{\alpha }_1)& =& 1-{\alpha }_1\\ C_2{\lambda }_2({\alpha }_2-{\alpha }_1)& =& {\alpha }_2\cdots 1-{\alpha }_1=1-\frac{1+\sqrt{5}}{2}=\frac{1-\sqrt{5}}{2}={\alpha }_2\\ C_2& =& \frac{{\alpha }_2}{{\lambda }_2({\alpha }_2-{\alpha }_1)}=\frac{\frac{1-\sqrt{5}}{2}}{\frac{1-\sqrt{5}}{2}(\frac{1-\sqrt{5}}{2}-\frac{1+\sqrt{5}}{2})}\\ & =& \frac{1}{\frac{1-\sqrt{5}}{2}-\frac{1+\sqrt{5}}{2}}=\frac{1}{-\frac{2\sqrt{5}}{2}}=-\frac{1}{\sqrt{5}}\end{array}$$ 以上よりフィボナッチ数列の一般項$a_n$は以下のように表すことができた． $$\begin{array}{rcl}{a}_n& =& C_1{\lambda }_1^n+C_2{\lambda }_2^n\\ & =& \frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^n-\frac{1}{\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^n\end{array}$$ $a_{n+1}$側の式は以下のようになり, $n+1$を改めて$n$と置き直せば上式と同じになる． $$\begin{array}{rcl}{a}_{n+1}& =& C_1{\lambda }_1^n{\alpha }_1+C_2{\lambda }_2^n{\alpha }_2\\ & =& \frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^n\frac{1+\sqrt{5}}{2}-\frac{1}{\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^n\frac{1-\sqrt{5}}{2}\\ & =& \frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^{n+1}-\frac{1}{\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^{n+1}\end{array}$$

指数分布族モデル / 共役な事前分布 / 予測分布

指数分布族モデル

$$\begin{array}{rcl}q(x;\theta )& =& v(x)\exp (f(\theta )\cdot g(x))\\ {\int }_{X}q(x;\theta )\mathrm{d}x& =& {\int }_{X}v(x)\exp (f(\theta )\cdot g(x))\mathrm{d}x\\ & =& 1\cdots \mathrm{確}\mathrm{率}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{こ}\mathrm{の}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{よ}\mathrm{う}\mathrm{に}v,f,g\mathrm{を}\mathrm{用}\mathrm{意}\mathrm{す}\mathrm{る}\end{array}$$

共役な事前分布

$$\begin{array}{rcl}\pi (\theta ;\varphi )& =& \frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\\ {\int }_X\pi (\theta ;\varphi )\mathrm{d}x& =& {\int }_X\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\mathrm{d}x\\ & =& 1\cdots \mathrm{確}\mathrm{率}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{こ}\mathrm{の}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{よ}\mathrm{う}\mathrm{に}z,f,\varphi \mathrm{を}\mathrm{用}\mathrm{意}\mathrm{す}\mathrm{る}\\ z(\varphi )& =& {\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )\mathrm{d}\theta \cdots \mathrm{む}\mathrm{し}\mathrm{ろ}z\mathrm{は}\mathrm{上}\mathrm{記}\mathrm{条}\mathrm{件}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{た}\mathrm{め}\mathrm{に}\mathrm{用}\mathrm{意}\mathrm{さ}\mathrm{れ}\mathrm{る}\end{array}$$

予測分布

$$\begin{array}{rcl}q(X_{n+1}|X^n)& =& \frac{1}{Z_n(\beta )}\int q(X_{n+1}|\theta {)}^{\beta }\prod _{i=1}^n\{q(X_i|\theta {)}^{\beta }\}\pi (\theta )\mathrm{d}\theta \\ & =& \frac{1}{Z_n(\beta )}{\int }_{\mathrm{\Theta }}\pi (\theta )q(X_{n+1}|\theta {)}^{\beta }{\displaystyle \prod _{i=1}^n{q(X_i|\theta )}^{\beta }\mathrm{d}\theta }\\ & =& \frac{1}{Z_n(\beta )}{\int }_{\mathrm{\Theta }}\pi (\theta ){\displaystyle \prod _{i=1}^{n+1}q(X_i|\theta {)}^{\beta }\mathrm{d}\theta }\dots {\displaystyle \prod _{i=1}^{n+1}q(X_i|\theta {)}^{\beta }=q(X_{n+1}|\theta {)}^{\beta }{\displaystyle \prod _{i=1}^{n}q(X_i|\theta {)}^{\beta }}}\\ & =& \frac{1}{Z_n(\beta )}{Z}_{n+1}(\beta )\dots Z_{n+1}(\beta )={\int }_{\mathrm{\Theta }}\pi (\theta ){\displaystyle \prod _{i=1}^{n+1}q(X_i|\theta {)}^{\beta }\mathrm{d}\theta }\\ & =& \frac{Z_{n+1}(\beta )}{Z_n(\beta )}\end{array}$$

---

分配凾数

$$\begin{array}{rcl}{Z}_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta ;\varphi )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta ;\varphi )\prod _{i=1}^n{[v(X_i)\exp (f(\theta )\cdot g(X_i))]}^{\beta }\mathrm{d}\theta \cdots q(X_i;\theta )=v(X_i)\exp (f(\theta )\cdot g(X_i))\\ & =& {\int }_{\mathrm{\Theta }}[\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )]\prod _{i=1}^n{[v(X_i)\exp (f(\theta )\cdot g(X_i))]}^{\beta }\mathrm{d}\theta \cdots \pi (\theta ;\varphi )=\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\\ & =& \frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )\prod _{i=1}^n{[v(X_i)\exp (f(\theta )\cdot g(X_i))]}^{\beta }\mathrm{d}\theta \cdots \pi (\theta ;\varphi )=\frac{1}{z(\varphi )}\exp (f(\theta )\cdot \varphi )\\ & =& \frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )\prod _{i=1}^n[v(X_i{)}^{\beta }\exp {(f(\theta )\cdot g(X_i))}^{\beta }]\mathrm{d}\theta \cdots (ab{)}^c=a^c{b}^c\\ & =& \frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )[\prod _{i=1}^{n}v(X_i{)}^{\beta }][\prod _{i=1}^n\exp {(f(\theta )\cdot g(X_i))}^{\beta }]\mathrm{d}\theta \cdots \prod _{i=1}^{n}AB=\prod _{i=1}^{n}A\prod _{i=1}^{n}B\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )[\prod _{i=1}^n\exp \left(\beta \{f\left(\theta \right)\cdot g\left(X_i\right)\}\right)]\mathrm{d}\theta \cdots \exp {\left(A\right)}^B=\exp \left(AB\right)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi )[\prod _{i=1}^n\exp (f\left(\theta \right)\cdot \beta g\left(X_i\right))]\mathrm{d}\theta \cdots c(A\cdot B)=A\cdot cB(c:\mathrm{定}\mathrm{数})\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi +\sum _{i=1}^{n}f\left(\theta \right)\cdot \beta g\left(X_i\right))\mathrm{d}\theta \cdots \exp \left(A\right)\exp \left(B\right)=\exp (A+B)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \varphi +f\left(\theta \right)\cdot \sum _{i=1}^n\beta g\left(X_i\right))\mathrm{d}\theta \cdots A\cdot B+A\cdot C=A\cdot (B+C)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot \{\varphi +\sum _{i=1}^n\beta g\left(X_i\right)\})\mathrm{d}\theta \cdots A\cdot B+A\cdot C=A\cdot (B+C)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}{\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \cdots {\hat{\varphi }}_{\beta }=\varphi +\sum _{i=1}^n\beta g\left(X_i\right)\\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}z({\hat{\varphi }}_{\beta })\cdots z({\hat{\varphi }}_{\beta })={\int }_{\mathrm{\Theta }}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& [\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\end{array}$$

自由エネルギー

$$\begin{array}{rcl}{F}_n(\beta )& =& -\frac{1}{\beta }\log Z_n(\beta )\\ & =& -\frac{1}{\beta }\log \left([\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\log \left([\prod _{i=1}^{n}v(X_i{)}^{\beta }]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\log \left([v(X_1{)}^{\beta }v(X_2{)}^{\beta }\cdots v(X_n{)}^{\beta }]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\log \left({[v(X_1)v(X_2)\cdots v(X_n)]}^{\beta }\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\frac{1}{\beta }\beta \log \left([v(X_1)v(X_2)\cdots v(X_n)]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\log \left([v(X_1)v(X_2)\cdots v(X_n)]\right)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\\ & =& -\sum _{i=1}^n\log v(X_i)-\frac{1}{\beta }\log \left(\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\right)\end{array}$$

事後分布

$$\begin{array}{rcl}q(\theta ;X^n)& =& \frac{1}{Z_n(\beta )}\pi (\theta ;\varphi )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\\ & =& \frac{1}{Z_n(\beta )}[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\\ & =& \frac{1}{[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}}[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{1}{z(\varphi )}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\cdots Z_n(\beta )=[\prod _{i=1}^{n}v(X_i{)}^{\beta }]\frac{z({\hat{\varphi }}_{\beta })}{z(\varphi )}\\ & =& \frac{1}{z({\hat{\varphi }}_{\beta })}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\\ & =& \pi (\theta ;{\hat{\varphi }}_{\beta })\end{array}$$

---

上記事後分布で予測分布を書き直す

$$\begin{array}{rcl}q(x;X^n)& =& {\int }_{\mathrm{\Theta }}q(x;\theta {)}^{\beta }q(\theta ;X^n)\mathrm{d}\theta \cdots X^n\mathrm{か}\mathrm{ら}\theta \mathrm{を}，\theta \mathrm{か}\mathrm{ら}x\mathrm{の}\mathrm{推}\mathrm{測}\mathrm{確}\mathrm{率}(\mathrm{分}\mathrm{布})\mathrm{を}\mathrm{求}\mathrm{め}\mathrm{る}\\ & =& {\int }_{\mathrm{\Theta }}q(x;\theta {)}^{\beta }\pi (\theta ;{\hat{\varphi }}_{\beta })\mathrm{d}\theta \cdots {\hat{\varphi }}_{\beta }\mathrm{か}\mathrm{ら}\theta \mathrm{を}，\theta \mathrm{か}\mathrm{ら}x\mathrm{の}\mathrm{推}\mathrm{測}\mathrm{確}\mathrm{率}(\mathrm{分}\mathrm{布})\mathrm{を}\mathrm{求}\mathrm{め}\mathrm{る}\\ & =& {\int }_{\mathrm{\Theta }}q(x;\theta {)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}{(v(x)\exp (f(\theta )\cdot g(x)))}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\int \exp {(f(\theta )\cdot g(x))}^{\beta }\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\int \exp (f(\theta )\cdot \beta g(x))\exp (f(\theta )\cdot {\hat{\varphi }}_{\beta })\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}\int \exp (f(\theta )\cdot ({\hat{\varphi }}_{\beta }+\beta g(x)))\mathrm{d}\theta \\ & =& v(x{)}^{\beta }\frac{1}{z({\hat{\varphi }}_{\beta })}z({\hat{\varphi }}_{\beta }+\beta g(x))\\ & =& v(x{)}^{\beta }\frac{z({\hat{\varphi }}_{\beta }+\beta g(x))}{z({\hat{\varphi }}_{\beta })}\\ & =& v(x{)}^{\beta }\frac{z(\varphi +\beta g(X_1)+\cdots +\beta g(X_n)+\beta g(x))}{z(\varphi +\beta g(X_1)+\cdots +\beta g(X_n))}\end{array}$$

尤度凾数の期待値の平均情報量(離散/標本からの)

尤度凾数の期待値の平均情報量(離散/標本からの)

$$\begin{array}{rcl}{T}_n& =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;\theta )]\right)\cdots \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))]\right)\cdots q(X;\theta )=q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\\ & =& -\frac{1}{n}\sum _{i=1}^n\log (q(X;{\theta }_0){\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))])\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -\frac{1}{n}\sum _{i=1}^n\{\log (q(X;{\theta }_0))+\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\}\cdots \log \left(AB\right)=\log \left(A\right)+\log \left(B\right)\\ & =& -\frac{1}{n}\{\sum _{i=1}^n\log (q(X;{\theta }_0))+\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\}\cdots \sum (A+B)=\sum A+\sum B\\ & =& -\frac{1}{n}\sum _{i=1}^n\log (q(X;{\theta }_0))-\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\\ & =& L_n({\theta }_0)-\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\cdots -\frac{1}{n}\sum _{i=1}^n\log (q(X;{\theta }_0))=L_n({\theta }_0)\\ & =& L_n({\theta }_0)+T_n^{(0)}\cdots -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)=T_n^{(0)}\end{array}$$

---

$$\begin{array}{rcl}{T}_n^{(0)}& =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-\log \frac{q(X_i;{\theta }_0)}{q(X_i;\theta )})]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (\log \frac{q(X_i;\theta )}{q(X_i;{\theta }_0)})]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}\left[\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left(\frac{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}{q(X_i;{\theta }_0)}\right)\\ & =& \frac{1}{n}\sum _{i=1}^n-\log \left(\frac{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}{q(X_i;{\theta }_0)}\right)\\ & =& \frac{1}{n}\sum _{i=1}^n\log \left(\frac{q(X_i;{\theta }_0)}{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}\right)\end{array}$$

尤度凾数の期待値の平均情報量(連続)

尤度凾数の期待値の平均情報量(連続)

$$\begin{array}{rcl}{G}_n& =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;\theta )]\right)]\cdots \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}\\ & =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots q(X;\theta )=q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\\ & =& -{\mathrm{E}}_X[\log (q(X;{\theta }_0){\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))])]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -{\mathrm{E}}_X[\log (q(X;{\theta }_0))+\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots \log \left(AB\right)=\log \left(A\right)+\log \left(B\right)\\ & =& -{\mathrm{E}}_X[\log (q(X;{\theta }_0))]-{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots \mathrm{E}[X+Y]=\mathrm{E}\left[X\right]+\mathrm{E}\left[Y\right]\\ & =& L({\theta }_0)-{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots -{\mathrm{E}}_X[\log (q(X;{\theta }_0))]=L({\theta }_0)\\ & =& L({\theta }_0)+G_n^{(0)}\cdots G_n^{(0)}=-{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]:\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}\end{array}$$

---

$$\begin{array}{rcl}{G}_n^{(0)}& =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\\ & =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-\log \frac{q(X_i;{\theta }_0)}{q(X_i;\theta )})]\right)]\\ & =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (\log \frac{q(X_i;\theta )}{q(X_i;{\theta }_0)})]\right)]\\ & =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}\left[\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right]\right)]\\ & =& -{\mathrm{E}}_X[\log \left(\frac{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}{q(X_i;{\theta }_0)}\right)]\\ & =& {\mathrm{E}}_X[-\log \left(\frac{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}{q(X_i;{\theta }_0)}\right)]\\ & =& {\mathrm{E}}_X[\log \left(\frac{q(X_i;{\theta }_0)}{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}\right)]\end{array}$$

自由エネルギー

自由エネルギー

$$\begin{array}{rcl}{F}_n(\beta )& =& -\frac{1}{\beta }\log Z_n(\beta )\cdots Z_n(\beta ):\mathrm{分}\mathrm{配}\mathrm{凾}\mathrm{数}\\ & =& -\frac{1}{\beta }\log (Z_n^{(0)}(\beta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta })\cdots Z_n(\beta )=Z_n^{(0)}(\beta )\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }\\ & =& -\frac{1}{\beta }(\log (\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta })+\log (Z_n^{(0)}(\beta )))\cdots \log \left(AB\right)=\log \left(A\right)+\log \left(B\right)\\ & =& -\frac{1}{\beta }\log (\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta })-\frac{1}{\beta }\log (Z_n^{(0)}(\beta ))\\ & =& -\frac{1}{\beta }\log (\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta })+F_n^{(0)}(\beta )\cdots F_n^{(0)}(\beta )=-\frac{1}{\beta }\log (Z_n^{(0)}(\beta )):\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{自}\mathrm{由}\mathrm{エ}\mathrm{ネ}\mathrm{ル}\mathrm{ギ}\mathrm{ー}\\ & =& -\frac{1}{\beta }\log {(\prod _{i=1}^{n}q(X_i;\theta ))}^{\beta }+F_n^{(0)}(\beta )\cdots \prod A^B={(\prod A)}^B\\ & =& -\frac{1}{\beta }\beta \log (\prod _{i=1}^{n}q(X_i;\theta ))+F_n^{(0)}(\beta )\cdots \log A^B=B\log A\\ & =& -\log (\prod _{i=1}^{n}q(X_i;\theta ))+F_n^{(0)}(\beta )\\ & =& -\sum _{i=1}^n\log q(X_i;\theta )+F_n^{(0)}(\beta )\cdots \log \left(\prod _{i=1}^n{A}_i\right)=\sum _{i=1}^n(\log A_i)\\ & =& -n\frac{1}{n}\sum _{i=1}^n\log q(X_i;\theta )+F_n^{(0)}(\beta )\cdots n\frac{1}{n}=1\\ & =& n\{-\frac{1}{n}\sum _{i=1}^n\log q(X_i;\theta )\}+F_n^{(0)}(\beta )\\ & =& n{L}_n({\theta }_0)+F_n^{(0)}(\beta )\cdots -\frac{1}{n}\sum _{i=1}^n\log q(X_i;\theta )=L_n({\theta }_0)\\ \end{array}$$

事後分布

最適パラメータにおける条件付き分布

$$\begin{array}{rcl}{Z}_n(\beta )& =& Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))\\ & =& Z_n^{(0)}(\beta )\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }\\ \prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }& =& \exp (-n\beta L_n({\theta }_0))\cdots \mathrm{最}\mathrm{適}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{条}\mathrm{件}\mathrm{付}\mathrm{き}\mathrm{分}\mathrm{布}\end{array}$$

---

$$\begin{array}{rcl}q(\theta ;X^n)& =& \frac{1}{Z_n(\beta )}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\\ & =& \frac{1}{Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\cdots Z_n(\beta )=Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\frac{\exp (-n\beta L_n(\theta ))}{\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\cdots \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }=\exp (-n\beta L_n(\theta ))\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\frac{\exp (-n\beta (L_n({\theta }_0)+K_n(\theta )))}{\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\cdots L_n(\theta )=L_n({\theta }_0)+K_n(\theta )\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\frac{\exp (-n\beta L_n({\theta }_0))\exp (-n\beta K_n(\theta ))}{\exp (-n\beta L_n({\theta }_0))}\pi (\theta )\cdots \exp (A+B)=\exp (A)\exp (B)\\ & =& \frac{1}{Z_n^{(0)}(\beta )}\exp (-n\beta K_n(\theta ))\pi \left(\theta \right)\end{array}$$

正規化された分配凾数(2)

正規化された分配凾数(2)

$$\begin{array}{rcl}{Z}_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^n{\{q(X_i;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\}}^{\beta }\mathrm{d}\theta \cdots q(X_i;\theta {)}^{\beta }={\{q(X_i;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\}}^{\beta }\\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^n\exp {(-f(X_i,{\theta }_0,\theta ))}^{\beta }\mathrm{d}\theta \\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^n\exp (-\beta f(X_i,{\theta }_0,\theta ))\mathrm{d}\theta \\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-\beta \sum _{i=1}^{n}f(X_i,{\theta }_0,\theta ))\mathrm{d}\theta \cdots \prod _{i=1}^n\exp \left(A_i\right)=\exp \left(\sum _{i=1}^n{A}_i\right)\\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-\beta n{K}_n)\mathrm{d}\theta \cdots \sum _{i=1}^{n}f(X_i,{\theta }_0,\theta )=n{K}_n(\theta )\\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n)\mathrm{d}\theta \\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta })Z_n^{(0)}(\beta )\cdots Z_n^{(0)}(\beta )={\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n(\theta ))\mathrm{d}\theta :\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{分}\mathrm{配}\mathrm{凾}\mathrm{数}\\ & =& Z_n^{(0)}(\beta )\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }\end{array}$$

正規化された分配凾数

正規化された分配凾数

$$\begin{array}{rcl}\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }& =& \exp (\log \left({\displaystyle \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }}\right))\cdots A=\exp (\log \left(A\right))\\ & =& \exp (\sum _{i=1}^n\log (q(X_i;\theta {)}^{\beta }))\cdots \log \left(\prod _{i=1}^n{A}_i\right)=\sum _{i=1}^n\log \left(A_i\right)\\ & =& \exp (\beta \sum _{i=1}^n\log (q(X_i;\theta )))\\ & =& \exp (-n\beta (-\frac{1}{n}\sum _{i=1}^n\log \left(q(X_i;\theta )\right)))\cdots (-n)(-\frac{1}{n})=1\\ & =& \exp (-n\beta L_n(\theta ))\cdots L_n(\theta )=-\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;\theta ))}\end{array}$$

---

$$\begin{array}{rcl}{Z}_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta L_n(\theta ))\mathrm{d}\theta \cdots \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }=\exp (-n\beta L_n(\theta ))\\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta (L_n({\theta }_0)+K_n(\theta )))\mathrm{d}\theta \cdots L_n(\theta )=L_n({\theta }_0)+K_n(\theta )\\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta L_n({\theta }_0)-n\beta K_n(\theta ))\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta L_n({\theta }_0))\exp (-n\beta K_n(\theta ))\mathrm{d}\theta \\ & =& \exp (-n\beta L_n({\theta }_0)){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n(\theta ))\mathrm{d}\theta \\ & =& \exp (-n\beta L_n({\theta }_0))Z_n^{(0)}(\beta )\cdots Z_n^{(0)}(\beta )={\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n(\theta ))\mathrm{d}\theta :\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{分}\mathrm{配}\mathrm{凾}\mathrm{数}\\ & =& Z_n^{(0)}(\beta )\exp (-n\beta L_n({\theta }_0))\end{array}$$

事後分布/分配凾数

分配凾数

$$\begin{array}{rclrc}{x}^n& =& (x_1,x_2,\dots ,x_n)& \dots & n\mathrm{個}\mathrm{の}\mathrm{標}\mathrm{本}\\ p(x)& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}\\ p(x^n)& =& {\displaystyle \prod _{i=1}^{n}p(x_i)}\\ & =& p(x_1)p(x_2)\dots p(x_n)\\ X^n& =& (X_1,X_2,\dots ,X_n)& \dots & x^n\mathrm{を}\mathrm{実}\mathrm{現}\mathrm{値}\mathrm{と}\mathrm{考}\mathrm{え}\mathrm{る}n\mathrm{個}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}\\ & & & & (x^n\mathrm{を}p(x)\mathrm{上}\mathrm{の}\mathrm{分}\mathrm{布}\mathrm{か}\mathrm{ら}\mathrm{得}\mathrm{ら}\mathrm{れ}\mathrm{た}\mathrm{も}\mathrm{の}\mathrm{と}\mathrm{考}\mathrm{え}\mathrm{る})\end{array}$$

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$$\begin{array}{rcl}q(x;\theta )& & \mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\theta \mathrm{を}\mathrm{用}\mathrm{い}\mathrm{た}\mathrm{推}\mathrm{定}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}(\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル})\\ \pi (\theta )& & \mathrm{事}\mathrm{前}\mathrm{分}\mathrm{布}\\ \beta & & \mathrm{逆}\mathrm{温}\mathrm{度}0<\beta <\mathrm{\infty }\\ q(\theta ;X^n)& =& \frac{1}{Z_n(\beta )}\pi (\theta ){\displaystyle \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\dots \mathrm{逆}\mathrm{温}\mathrm{度}\beta \mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\theta \mathrm{の}\mathrm{事}\mathrm{後}\mathrm{分}\mathrm{布}．}\\ Z_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta ){\displaystyle \prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \dots \mathrm{分}\mathrm{配}\mathrm{凾}\mathrm{数}(\beta =1\mathrm{の}\mathrm{時}，\mathrm{周}\mathrm{辺}\mathrm{尤}\mathrm{度})}\\ & & \mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\theta \mathrm{の}\mathrm{逆}\mathrm{温}\mathrm{度}\beta \mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{事}\mathrm{後}\mathrm{分}\mathrm{布}\mathrm{の}\mathrm{総}\mathrm{和}\mathrm{を}1\mathrm{と}\mathrm{す}\mathrm{る}\mathrm{た}\mathrm{め}\mathrm{の}\mathrm{定}\mathrm{数}\end{array}$$

不偏推定量は一意性を持たない

不偏推定量は一意性を持たない

$$\begin{array}{rcl}\mathrm{E}\left[\overline{X}\right]& =& {\mu }_{\overline{X}}\\ \mathrm{E}\left[Y\right]& =& {\mu }_Y\\ & =& 0\end{array}$$ の時， $$\begin{array}{rcl}\mathrm{E}[\overline{X}+Y]& =& \mathrm{E}\left[\overline{X}\right]+\mathrm{E}\left[Y\right]\\ & =& {\mu }_{\overline{X}}+{\mu }_Y\\ & =& {\mu }_{\overline{X}}+0\\ & =& {\mu }_{\overline{X}}\end{array}$$ であり，$\mathrm{E}[\overline{X}+Y]\mathrm{と}\mathrm{E}\left[\overline{X}\right]$で不偏推定量の値に違いがない．

尤度凾数の平均情報量(離散/標本からの)

尤度凾数の平均情報量(離散/標本からの)

$$\begin{array}{rcl}p(x)& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}\\ \theta & & \mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\\ q(x;\theta )& & \theta \mathrm{を}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{と}\mathrm{し}\mathrm{た}\mathrm{推}\mathrm{測}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}\mathrm{モ}\mathrm{デ}\mathrm{ル}(\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル})(\theta \mathrm{を}\mathrm{固}\mathrm{定}\mathrm{し}x\mathrm{を}\mathrm{動}\mathrm{か}\mathrm{す}\mathrm{イ}\mathrm{メ}\mathrm{ー}\mathrm{ジ})\\ {\mathrm{\Theta }}_0& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}p(x)\mathrm{に}\mathrm{対}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル}q(x;\theta )\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{最}\mathrm{適}\mathrm{な}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{の}\mathrm{集}\mathrm{合}\\ {\theta }_0& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}p(x)\mathrm{に}\mathrm{対}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル}q(x;\theta )\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{最}\mathrm{適}\mathrm{な}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}({\theta }_0\in {\mathrm{\Theta }}_0)\\ q(x;\theta )& & \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}(x\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\theta \mathrm{の}\mathrm{尤}\mathrm{も}\mathrm{ら}\mathrm{し}\mathrm{さ}(\mathrm{式}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{は}\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル}\mathrm{と}\mathrm{同}\mathrm{じ}．x\mathrm{を}\mathrm{固}\mathrm{定}\mathrm{し}\theta \mathrm{を}\mathrm{動}\mathrm{か}\mathrm{す}\mathrm{イ}\mathrm{メ}\mathrm{ー}\mathrm{ジ}))\\ \log q(x;\theta )& & \mathrm{対}\mathrm{数}\mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\\ -\log q(x;\theta )& & \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{選}\mathrm{択}\mathrm{情}\mathrm{報}\mathrm{量}(\mathrm{逆}\mathrm{数}\mathrm{の}\mathrm{対}\mathrm{数})\end{array}$$

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$$\begin{array}{rcl}{L}_n(\theta )& =& \mathrm{E}[\log \left(\frac{1}{q(X_i;\theta )}\right)]\cdots \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}(\mathrm{選}\mathrm{択}\mathrm{情}\mathrm{報}\mathrm{量}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値})\\ & =& \mathrm{E}[\log (q(X_i;\theta {)}^{-1})]\cdots \frac{1}{A}=A^{-1}\\ & =& \mathrm{E}[-\log (q(X_i;\theta ))]\cdots \log A^B=B\log A\\ & =& -\mathrm{E}[\log (q(X_i;\theta ))]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;\theta ))\cdots n\mathrm{個}\mathrm{の}\mathrm{標}\mathrm{本}，\mathrm{離}\mathrm{散}}\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;\theta )\frac{q(X_i;{\theta }_0)}{q(X_i;{\theta }_0)})\cdots {\theta }_0:\mathrm{最}\mathrm{適}\mathrm{な}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}}\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;{\theta }_0)\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)})}\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\{\log (q(X_i;{\theta }_0))+\log \left(\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right)\}\cdots \log AB=\log A+\log B}\\ & =& -\frac{1}{n}\left\{{\displaystyle \sum _{i=1}^n\log (q(X_i;{\theta }_0))+{\displaystyle \sum _{i=1}^n\log \left(\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right)}}\right\}\cdots \sum (A+B)=\sum A+\sum B\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;{\theta }_0))-\frac{1}{n}{\displaystyle \sum _{i=1}^n\log \left(\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right)}}\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;{\theta }_0))+\frac{1}{n}{\displaystyle \sum _{i=1}^n\log \left({\left(\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right)}^{-1}\right)\cdots -\log x=\log x^{-1}}}\\ & =& -\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;{\theta }_0))+\frac{1}{n}{\displaystyle \sum _{i=1}^n\log \left(\frac{q(X_i;{\theta }_0)}{q(X_i;\theta )}\right)\cdots {\left(\frac{A}{B}\right)}^{-1}=\frac{1}{\frac{A}{B}}=\frac{B}{A}}}\\ & =& L_n({\theta }_0)+K_n(\theta )\\ & & \cdots L_n({\theta }_0)=-\frac{1}{n}{\displaystyle \sum _{i=1}^n\log (q(X_i;{\theta }_0)),K_n=\frac{1}{n}{\displaystyle \sum _{i=1}^n\log \left(\frac{q(X_i;{\theta }_0)}{q(X_i;\theta )}\right)}}\end{array}$$

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$$\begin{array}{rcl}f(X_i,{\theta }_0,\theta )& =& \log \frac{q(X_i;{\theta }_0)}{q(X_i;\theta )}\cdots \mathrm{尤}\mathrm{度}\mathrm{の}\mathrm{比}\mathrm{の}\mathrm{対}\mathrm{数}(\mathrm{対}\mathrm{数}\mathrm{の}\mathrm{比}\mathrm{は}，\mathrm{分}\mathrm{母}\mathrm{分}\mathrm{子}\mathrm{の}\mathrm{対}\mathrm{数}\mathrm{の}\mathrm{差}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{尤}\mathrm{度}\mathrm{の}\mathrm{差}\mathrm{で}\mathrm{も}\mathrm{あ}\mathrm{る})\\ K_n(\theta )& =& \frac{1}{n}\sum _{i=1}^{n}f(X_i,{\theta }_0,\theta )\cdots \mathrm{標}\mathrm{本}\mathrm{か}\mathrm{ら}\mathrm{の}KL\mathrm{ダ}\mathrm{イ}\mathrm{バ}\mathrm{ー}\mathrm{ジ}\mathrm{ェ}\mathrm{ン}\mathrm{ス}\\ n{K}_n(\theta )& =& \sum _{i=1}^{n}f(X_i,{\theta }_0,\theta )\\ L_n(\theta )& =& L_n({\theta }_0)+K_n(\theta )\\ K_n(\theta )& >& 0\cdots KL\mathrm{ダ}\mathrm{イ}\mathrm{バ}\mathrm{ー}\mathrm{ジ}\mathrm{ェ}\mathrm{ン}\mathrm{ス}\mathrm{の}\mathrm{下}\mathrm{限}\mathrm{よ}\mathrm{り}\\ K_n(\theta )=0& ⟺& \theta \in {\mathrm{\Theta }}_0\\ \exp (f(X_i,{\theta }_0,\theta ))& =& \frac{q(X_i;{\theta }_0)}{q(X_i;\theta )}\cdots A=\log (B),\exp (A)=B\\ q(X_i;\theta )& =& q(X_i;{\theta }_0)\frac{1}{\exp (f(X_i,{\theta }_0,\theta ))}\\ & =& q(X_i;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\end{array}$$

尤度凾数の平均情報量(連続)

尤度凾数の平均情報量(連続)

$$\begin{array}{rcl}p(x)& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}\\ \theta & & \mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\\ q(x;\theta )& & \theta \mathrm{を}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{と}\mathrm{し}\mathrm{た}\mathrm{推}\mathrm{測}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}\mathrm{モ}\mathrm{デ}\mathrm{ル}(\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル})(\theta \mathrm{を}\mathrm{固}\mathrm{定}\mathrm{し}x\mathrm{を}\mathrm{動}\mathrm{か}\mathrm{す}\mathrm{イ}\mathrm{メ}\mathrm{ー}\mathrm{ジ})\\ {\mathrm{\Theta }}_0& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}p(x)\mathrm{に}\mathrm{対}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル}q(x;\theta )\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{最}\mathrm{適}\mathrm{な}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\mathrm{の}\mathrm{集}\mathrm{合}\\ {\theta }_0& & \mathrm{真}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{分}\mathrm{布}p(x)\mathrm{に}\mathrm{対}\mathrm{す}\mathrm{る}\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル}q(x;\theta )\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\mathrm{最}\mathrm{適}\mathrm{な}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}({\theta }_0\in {\mathrm{\Theta }}_0)\\ q(x;\theta )& & \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}(x\mathrm{に}\mathrm{お}\mathrm{け}\mathrm{る}\theta \mathrm{の}\mathrm{尤}\mathrm{も}\mathrm{ら}\mathrm{し}\mathrm{さ}(\mathrm{式}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{は}\mathrm{確}\mathrm{率}\mathrm{モ}\mathrm{デ}\mathrm{ル}\mathrm{と}\mathrm{同}\mathrm{じ}．x\mathrm{を}\mathrm{固}\mathrm{定}\mathrm{し}\theta \mathrm{を}\mathrm{動}\mathrm{か}\mathrm{す}\mathrm{イ}\mathrm{メ}\mathrm{ー}\mathrm{ジ}))\\ \log q(x;\theta )& & \mathrm{対}\mathrm{数}\mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\\ -\log q(x;\theta )& & \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{選}\mathrm{択}\mathrm{情}\mathrm{報}\mathrm{量}(\mathrm{逆}\mathrm{数}\mathrm{の}\mathrm{対}\mathrm{数})\end{array}$$

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$$\begin{array}{rcl}L(\theta )& =& {\mathrm{E}}_X[\log \left(\frac{1}{q(x;\theta )}\right)]\cdots \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}(\mathrm{選}\mathrm{択}\mathrm{情}\mathrm{報}\mathrm{量}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値})\\ & =& {\mathrm{E}}_X[\log (q(x;\theta {)}^{-1})]\cdots \frac{1}{A}=A^{-1}\\ & =& {\mathrm{E}}_X[-\log (q(x;\theta ))]\cdots \log A^B=B\log A\\ & =& -{\mathrm{E}}_X[\log q(x;\theta )]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -\int q(x;{\theta }_0)\log (q(x;\theta ))\mathrm{d}x\cdots {\theta }_0:\mathrm{最}\mathrm{適}\mathrm{な}\mathrm{パ}\mathrm{ラ}\mathrm{メ}\mathrm{ー}\mathrm{タ}\\ & =& -\int q(x;{\theta }_0)\log (q(x;\theta )\frac{q(x;{\theta }_0)}{q(x;{\theta }_0)})\mathrm{d}x\\ & =& -\int q(x;{\theta }_0)\log (q(x;{\theta }_0)\frac{q(x;\theta )}{q(x;{\theta }_0)})\mathrm{d}x\\ & =& -\int q(x;{\theta }_0)\{\log (q(x;{\theta }_0))+\log \left(\frac{q(x;\theta )}{q(x;{\theta }_0)}\right)\}\mathrm{d}x\cdots \log AB=\log A+\log B\\ & =& -\int q(x;{\theta }_0)\log (q(x;{\theta }_0))\mathrm{d}x-\int q(x;{\theta }_0)\log \left(\frac{q(x;\theta )}{q(x;{\theta }_0)}\right)\mathrm{d}x\cdots {\int }_{X}AB\mathrm{d}x={\int }_{X}A\mathrm{d}x+{\int }_{X}B\mathrm{d}x\\ & =& -\int q(x;{\theta }_0)\log (q(x;{\theta }_0))\mathrm{d}x+\int q(x;{\theta }_0)\log \left({\left(\frac{q(x;\theta )}{q(x;{\theta }_0)}\right)}^{-1}\right)\mathrm{d}x\cdots -\log x=\log x^{-1}\\ & =& -\int q(x;{\theta }_0)\log (q(x;{\theta }_0))\mathrm{d}x+\int q(x;{\theta }_0)\log \left(\frac{q(x;{\theta }_0)}{q(x;\theta )}\right)\mathrm{d}x\cdots {\left(\frac{A}{B}\right)}^{-1}=\frac{1}{\frac{A}{B}}=\frac{B}{A}\\ & =& -{\mathrm{E}}_X[\log q(x;{\theta }_0)]+{\mathrm{E}}_X[\log \left(\frac{q(x;{\theta }_0)}{q(x;\theta )}\right)]\\ & =& L({\theta }_0)+K(\theta )\\ & & \cdots L({\theta }_0)=-{\mathrm{E}}_X[\log q(x;{\theta }_0)],K(\theta )={\mathrm{E}}_X[\log \frac{q(x;{\theta }_0)}{q(x;\theta )}]\end{array}$$

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$$\begin{array}{rcl}f({\theta }_0,\theta ;x)& =& \log \frac{q(x;{\theta }_0)}{q(x;\theta )}\cdots \mathrm{尤}\mathrm{度}\mathrm{の}\mathrm{比}\mathrm{の}\mathrm{対}\mathrm{数}(\mathrm{対}\mathrm{数}\mathrm{の}\mathrm{比}\mathrm{は}，\mathrm{分}\mathrm{母}\mathrm{分}\mathrm{子}\mathrm{の}\mathrm{対}\mathrm{数}\mathrm{の}\mathrm{差}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{尤}\mathrm{度}\mathrm{の}\mathrm{差}\mathrm{で}\mathrm{も}\mathrm{あ}\mathrm{る})\\ K(\theta )& =& {\mathrm{E}}_X[f({\theta }_0,\theta ;x)]\cdots KL\mathrm{ダ}\mathrm{イ}\mathrm{バ}\mathrm{ー}\mathrm{ジ}\mathrm{ェ}\mathrm{ン}\mathrm{ス}\\ L(\theta )& =& L({\theta }_0)+K(\theta )\\ K(\theta )& >& 0\cdots KL\mathrm{ダ}\mathrm{イ}\mathrm{バ}\mathrm{ー}\mathrm{ジ}\mathrm{ェ}\mathrm{ン}\mathrm{ス}\mathrm{の}\mathrm{下}\mathrm{限}\mathrm{よ}\mathrm{り}\\ K(\theta )=0& ⟺& \theta \in {\mathrm{\Theta }}_0\\ \exp (f({\theta }_0,\theta ;x))& =& \frac{q(x;{\theta }_0)}{q(x;\theta )}\cdots A=\log (B),\exp (A)=B\\ q(x;\theta )& =& q(x;{\theta }_0)\frac{1}{\exp (f({\theta }_0,\theta ;x))}\\ & =& q(x;{\theta }_0)\exp (-f({\theta }_0,\theta ;x))\end{array}$$