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Spike Density Estimation

- Histogram Method

- Kernel Method

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$BE}7WE*?dB,$NM}O@$HZ$K;H$&$3$H$r?d?J$7$?!%%U%#%C%7%c!<$O%G!<%?$+$iF@$i$l$kE}7WNL$HE}7W%b%G%k$NJl?t$H$rL@3N$K6hJL$7!$E}7WNL!J?dDjNL!K$NK>$^$7$$@-Dj$O!$Jl?t$,Dj?t$G$"$k$3$H$G$"$k!%(B $B7WB,4|4VCf$KJl?t$,JQF0$7$?$j!$;n9T$4$H$KJl?t$,JQF0$9$k$3$H$OA[Dj$7$J$$!%?@7P2J3X$K$*$1$k$$F10l

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$\hat{\theta}$ $B$r%Q%i%a!<%?(B$\theta$ $B$N?dDjNL$H$9$k!%Nc$r$"$2$F$_$h$&!%(B$X_{i}$ $B$,J?6Q(B$\theta$ $B$N;X?tJ,I[$K=>$&Mp?t$H$9$k$H!$(B$\theta$ $B$N?dDjNL(B $\hat{\theta}$ $B$NNc$H$7$F$^$:9M$($i$l$k$N$,!$J?6Q(B $\hat {\theta}_{n}=\frac{1}{n}\sum X_{i}$ $B!$$,$"$k!%?dDjNL$OB>$K$b9M$($i$l$k!%Nc$($P=i$a$N%5%s%W%k$@$1$r;H$&(B $\hat{\theta}_{1}=X_{i}$ $B$b(B$\theta$ $B$N?dDjNL$G$"$k!%$=$l$G$OK>$^$7$$?dDjNL$H$O$I$N$h$&$J@-h8m:9$O$=$N0l$D$N4p=`$rM?$($k$H9M$($i$l$k!%(B

$$=E[(\hat{\theta}-\theta)^{2}]%$$ (4.1)

$BFs>hB;<:4X?t!"B;<:4X?t$r??$NJ,I[$G4|BTCM$r$H$C$?4X?t$r%j%9%/4X?t$H$$$&!%695A$N0UL#$G$O!$%j%9%/4X?t$N:G>.2=$,E}7W3X$d3X=,M}O@$N%F!<%^$G$"$k!%?dDjNL$N4|BTCM$r(B $E\hat{\theta}=\bar{\theta}$ $B$H$9$k$H(B,

$$=E[(\hat{\theta}-\bar{\theta}+\bar{\theta}-\theta )^{2}]$$ MSE

$$=E[(\hat{\theta}-\bar{\theta})^{2}+2(\hat{\theta}-\bar{\theta})(\bar {\theta}-\theta)+(\bar{\theta}-\theta)^{2}]$$

$$=E[(\hat{\theta}-\bar{\theta})^{2}]+(\bar{\theta}-\theta)^{2} %$$ (4.2)

$BBh0l9`(B $E[(\hat{\theta}-\bar{\theta})^{2}]$ $B$O?dDjJ,;6!%BhFs9`$O%P%$%"%9$NFs>h$G$"$k!%=>$C$F%P%$%"%9$,$J$/!$J,;6$,>.$5$$?dDjNL$,K>$^$7$$?dDjNL$G$"$k!%(B $B3V$?$j$N$"$k?dDjNL$,>.$5$$J,;6$r;}$A!$$=$N(BMSE$B$,2DG=$JITJP?dDjNL$N(BMSE$B$h$j$b>.$5$/$J$k$3$H$O$"$j$&$k!%$3$l$,K>$^$7$$?dDjNL$G$"$k$N$O4V0c$$$J$$$,!$Nr;KE*$K$OITJP@-!$M-8z@-$OJL!9$K6cL#$5$l$F$-$?!%$^$:IaJW@-$N$"$k?dDjNL$r8+$D$1$F!$.$N$b$N$rA*$V$H$$$&

$BLdBj(B 18   Gauss$BJ,I[$K=>$&3NN(JQ?t$KBP$9$kE}7WNL(B $S_{n-1},S_{n},S_{n+1}$ $B$N(BMSE$B$r5a$a$h!%(B

$B$?$@$7(B $S_{n-1}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar {X})^{2}$ $B!$(B $S_{n}^{2}=\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}$ , $S_{n+1}^{2}=\frac{1}{n+1}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}$ .

$B$I$l$,0lHV>.$5$$$+!%(B

$B2rK!(B 19   MSE $(S_{n+1})<$ MSE $(S_{n})\leq$ MSE$(S_{n-1})$

$$(S_{n-1}) =E(S_{n-1}-\sigma^{2})^{2}=\frac{2}{n-1}\sigma^{4}$$

$$(S_{n}) =E(S_{n}-\sigma^{2})^{2}=\frac{2n-1}{n^{2}}\sigma^{4}$$

$$(S_{n+1}) =E(S_{n+1}-\sigma^{2})^{2}=\frac{2}{n+1}\sigma^{2}%$$

$B:GL`?dDj(B

$BL`EY(B(likelihood)$B$r:GBg$K$9$k%Q%i%a!<%?$r:GL`?dDjNL$H$$$&!%(B

$$l(\theta)=\log p\left( X\vert\theta\right) =\sum\log p\left( X_{i}% \vert\theta\right)$$

$BNc(B    $B%]%"%=%sJ,I[$N>l9g(B

$$l(\theta) =\sum_{i=1}^{n}\log p\left( X_{i}\vert\theta,\Delta\right)$$

$$=\sum_{i=1}^{n}\log\frac{\left( \theta\Delta\right) ^{X_{i}}}{X_{i}% !}e^{-\theta\Delta}$$

$$=\sum_{i=1}^{n}[X_{i}\log\left( \theta\Delta\right) -\theta\Delta-\log X_{i}!]$$

$B:GL`?dDj$O(B

$$\frac{\partial l(\theta)}{\partial\theta} =\sum_{i=1}^{n}[\frac{X_{i}% }{\theta}-\Delta]$$

$$=\frac{\sum_{i=1}^{n}X_{i}}{\theta}-n\Delta$$

$\frac{\partial l(\theta)}{\partial\theta}=0$ $B$h$j(B

$$\hat{\theta}_{n}=\frac{\sum_{i=1}^{n}X_{i}}{n\Delta}=\frac{\bar{X}}{\Delta}%$$

$B$?$@$7J?6QCM(B $\bar{X}=\frac{1}{n}\sum X_{i}$

$B:GL`?dDjCM$N4|BTCM$O(B

$$E\hat{\theta}_{n}=\frac{\sum_{i=1}^{n}EX_{i}}{n\Delta}=\theta$$

$BJ,;6$O(B

$$E(\hat{\theta}_{n}-E\hat{\theta}_{n})^{2} =E\left[ \frac{\sum_{i=1}% ^{n}X_{i}}{n\Delta}-\theta\right] ^{2}$$

$$=\frac{1}{(n\Delta)^{2}}\sum_{i=1}^{n}E(X_{i}-\Delta\theta)^{2}$$

$$=\frac{\theta}{n\Delta}%$$

$B:G8e$NEy<0$K$O%]%"%=%sJ,I[$N@- $E(X_{i}-\Delta\theta)^{2}=EX_{i}=\Delta\theta$ $B$rMQ$$$?!%(B

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$BH/2PN((B$\lambda$ $B$NDj>o%]%"%=%s2aDx$K=>$&%9%Q%$%/;~7ONs$rF@$?$H$7$h$&!%%9%Q%$%/;~7ONs$r$4$/C;$$(B$\Delta$ $BIC$N(B$n$ $B8D$N%S%s$K6h@Z$j!$(B$t$ $BHVL\$N%S%s$K%9%Q%$%/$,B8:_$9$l$P(B$X_{t}=0$ $B!$$7$J$1$l$P(B$X_{t}=1$ $B$H$7$F!$(B $X=\left( X_{1},X_{2},\ldots,X_{n}\right)$ $B$r%Y%k%L!<%$2aDx$H8+$J$9!%(B $B$3$N$H$-E}7WNL(B $T=\sum_{t=1}^{n}X_{i}$ $B$O%Y%k%L!<%$2aDx$N(B$\lambda$ $B$K4X$9$k==J,E}7WNL$G$"$k!%(B

$$P\left( X\vert\lambda,\Delta\right) =% {\displaystyle\prod\limits_{t=1}^{n}} P\left( X_{i}\vert\lambda,\Delta\right)$$

$$=% {\displaystyle\prod\limits_{t=1}^{n}} (\lambda\Delta)^{X_{i}}\left( 1-\lambda\Delta\right) ^{1-X_{i}}$$

$$=(\lambda\Delta)^{X_{1}+X_{2}+\cdots+X_{n}}(1-\lambda\Delta)^{n-(X_{1}% +X_{2}+\cdots+X_{n})}%$$

$\Delta$ $B$O7h$^$C$F$$$k$H$9$l$P!$(B$\lambda$ $B$NCM$r?dDj$9$k$N$K!$$3$3$N%G!<%?(B $\left( X_{1},X_{2},\ldots,X_{n}\right)$ $B$OI,MW$G$J$/!$E}7WNL(B $T=\sum_{t=1}^{n}X_{i}$ $B$rMQ$$$l$P==J,$G$"$k$3$H$,$o$+$k!%(B

$B:#EY$ODj>o%]%"%=%s2aDx$r$d$dBg$-$a$N(B$n$ $B8D$N4QB,6h4V(B$\Delta$ $B$G6h@Z$C$?$H$7$h$&!%(B $B$3$N$H$-3F6h4VFb$N%9%Q%$%/$N8D?t(B $X=\left( X_{1},X_{2},\ldots,X_{n}\right)$ $B$OJ?6Q(B $\lambda \Delta$ $B$N%]%"%=%sJ,I[$K=>$&!%$3$N$H$-(B $T=\sum_{t=1}^{n}X_{i}$ $B$O(B$\lambda$ $B$K4X$9$k==J,E}7WNL$G$"$k!%(B

$$P\left( X\vert\lambda,\Delta\right) =% {\displaystyle\prod\limits_{i=1}^{n}} P\left( X_{i}\vert\lambda,\Delta\right)$$

$$=% {\displaystyle\prod\limits_{t=1}^{n}} \frac{\left( \lambda\Delta\right) ^{X_{i}}}{X_{i}!}e^{-\lambda\Delta}$$

$$=\frac{1}{X_{1}!X_{1}!\cdots X_{n}!}\cdot\left( \lambda\Delta\right) ^{X_{1}+X_{2}+\cdots+X_{n}}e^{-n\lambda\Delta}%$$

$B:GL`?dDj$r9M$($F$_$h$&!%0x;R(B $\frac{1}{X_{1}!X_{1}!\cdots X_{n}!}$ $B$O(B$\lambda$ $B$K4X78$,$J$$!%(B $\lambda$ $B$K4X78$9$kBhFs0x;R$O(B $\left( \lambda\Delta\right) ^{T}e^{-n\lambda\Delta }$ $B$H(B$T$ $B$N4X?t$H$7$F=q$1$k$+$i!$8D!9$N%G!<%?$rCN$kI,MW$O$J$$$3$H$,$o$+$k!%(B

$B0lHL$KL)EYJ,I[$N%b%G%k(B $f\left( x\vert\theta\right)$ $B$,(B

$$f\left( x\vert\theta\right) =h\left( x\right) g\left( T\left( x\right) \vert\theta\right)$$ (4.3)

$B$N7A$K0x;RJ,2r$G$-$k>l9g!$$+$D$=$N>l9g$K8B$j!$E}7WNL(B$T$ $B$O%b%G%k%Q%i%a!<%?(B$\theta$ $B$N==J,E}7WNL$G$"$k!%$3$l$r%U%#%C%7%c!$B!%L)EYJ,I[$N%b%G%k(B $f\left( x\vert\theta\right)$ $B$r(B$\theta$ $B$K0MB8$9$k0x;R$H0MB8$7$J$$0x;R$KJ,2r$G$-$F(B, $B0MB8$9$k0x;R$,E}7WNL(B $T\left( x\right)$ $B$N4X?t$GI=$9$3$H$,$G$-$k$H$9$l$P!$(B$\theta$ $B$N?dDj$K$O(B $T\left( x\right)$ $B$rMQ$$$l$P==J,$G$"$k!%%Y%k%L!<%$2aDx$NNc$G$O(B $h\left( x\right) =1$ $B!$(B $g\left( T\left( x\right) \vert\theta\right) =\theta^{T}(1-\theta)^{n-T}$ $B$H8+$F!$(B $T=\sum_{t=1}^{n}X_{i}$ $B$,==J,E}7WNL$G$"$k$3$H$,$o$+$k!%$^$?%]%"%=%sJ,I[$O(B $h\left( x\right) =\frac{1}{X_{1}!X_{1}!\cdots X_{n}!}$ $B!$(B $g\left( T\left( x\right) \vert\theta\right) =\left( \lambda\Delta\right) ^{T}e^{-n\lambda\Delta}$ $B$H$J$j!$(B $T=\sum_{t=1}^{n}X_{i}$ $B$,==J,E}7WNL$G$"$k$3$H$,$o$+$k!%(B

$BDj5A(B 20   $B%G!<%?(B $X=\left( X_{1},X_{2},\ldots,X_{n}\right)$ $B$,(B $T\left( X\right)$ $B$,M?$($i$l$?2<$G(B$\theta$ $B$HFHN)$J;~(B, $B4X?t(B $T\left( X\right)$ $B$r%b%G%k(B $f\left( x\vert\theta\right)$ $B$N==J,E}7WNL$H$$$&!%(B

$$P\left( X=x\vert T\left( X\right) =t,\theta\right) =P\left( X=x\vert T\left( X\right) =t\right)$$

$B%U%#%C%7%c!<>pJsNL(B

$B%9%3%"(B$B$H8F$P$l$kE}7WNL$rF3F~$9$k(B

$$V\left( X;\theta\right) =\frac{\partial}{\partial\theta}\log f\le... ...ert\theta\right) }\frac{\partial f\left( X\vert\theta\right) }{\partial\theta}$$ (4.4)

$B%9%3%"$N4|BTCM$O(B

$$E[V] =\int V\left( x;\theta\right) f\left( X\vert\theta\right) dx$$

$$=\int\frac{\partial f\left( x\vert\theta\right) }{\partial\theta}dx$$

$$=\frac{\partial}{\partial\theta}\int f\left( x\vert\theta\right) dx$$

$$=0$$

$B%9%3%"$NJ,;6$O(B

$$E[V^{2}]=E\left[ \frac{\partial}{\partial\theta}\log f\left( X\vert\theta \right) \right] ^{2}%$$

$B%9%3%"$NJ,;6$r%U%#%C%7%c!<>pJsNL(B$B$H$h$S!$(B $J\left( \theta\right) =E[V^{2}]$ $B$GI=$9!%(B

$B3NN(JQ?t$K4X$9$k%3!<%7! )

$$\vert E [X Y]\vert^2 \leq E X^2 E Y^2$$ (4.5)

$B$rMQ$$$l$P

$$E[V^{2}]\cdot E[(\hat{\theta}-\bar{\theta})^{2}]\geq\left\vert E\left[ V(\hat{\theta}-\bar{\theta})\right] \right\vert ^{2}$$

$B$3$3$G:8JU$NBh0l0x;R$O%U%#%C%7%c!<>pJsNL(B $E[V^{2}]=J\left( \theta\right)$ $B!%(B $BBhFs0x;R$O?dDjNL$NJ,;6!$(B $var\hat{\theta}=E[(\hat{\theta}-\bar{\theta})^{2}]$ $B$G$"$k!%1&JU$K$D$$$F$O$OFs>h$NCf?H$,(B

$$EV\hat{\theta} =% {\displaystyle\int} V(\hat{\theta}-\bar{\theta})f\left( x\right) dx$$

$$=\int\frac{1}{f\left( x\vert\theta\right) }\frac{\partial f\left(... ...ht) }{\partial\theta}(\hat{\theta}-\bar{\theta})f\left( x\vert\theta\right) dx$$

$$=\int(\hat{\theta}-\bar{\theta})\frac{\partial f\left( x\vert\theta\right) }{\partial\theta}dx$$

$$=\int\hat{\theta}\frac{\partial f\left( x\vert\theta\right) }{\pa... ...\bar{\theta}\frac{\partial}{\partial\theta}\int f\left( x\vert\theta\right) dx$$

$$=\frac{\partial}{\partial\theta}\int\hat{\theta}f\left( x\vert\theta\right) dx$$

$$=\frac{\partial}{\partial\theta}E\hat{\theta}%$$

$B$H$J$k!%$f$($KE}7WNL$NJ,;6$N2<8B$O%U%#%C%7%c!<>pJsNL$N5U?t$GM?$($i$l$k!%(B

$$var\hat{\theta}\geq\frac{\left( \frac{\partial E\hat{\theta}}{\partial\theta }\right) ^{2}}{J\left( \theta\right) }%$$ (4.6)

$B$3$NITEy<0$r%/%i%^!$B$H$$$&!%%U%#%C%7%c!<>pJsNL$OE}7WNL$r$"$kFCDj$N%b%G%k(B $f\left( x\vert\theta\right)$ $B$N%Q%i%a!<%?(B$\theta$ $B$N?dDjCM$KMQ$$$k>l9g$NJ,;6$N2<8B$rM?$($k!%ITJP?dDjNL(B $E\hat{\theta}=\theta$ $B$N>l9g(B

$$var\hat{\theta}\geq\frac{1}{J\left( \theta\right) }%$$ (4.7)

$B%P%$%"%9$,$"$k>l9g$O(B $E\hat{\theta}=\theta+b\left( \theta\right)$

$$var\hat{\theta}\geq\frac{1+b^{\prime}\left( \theta\right) }{J\left( \theta\right) }%$$ (4.8)

$B%U%#%C%7%c!<>pJsNL$O

$$J\left( \theta\right) =E\left[ \left( \frac{\partial}{\partial\th... ...ac{\partial^{2}% }{\partial\theta^{2}}\log f\left( x\vert\theta\right) \right]$$ (4.9)

$BLdBj(B 21   $B3NN(JQ?t$K4X$9$k%3!<%7! )$B$rF3$-$J$5$$!%(B

$BLdBj(B 22   $B%U%#%C%7%c!<>pJsNL$NEy<0$r>ZL@$7$J$5$$!%(B

$B2rK!(B 23   $B1&JU$NCf?H$K$D$$$F(B

$$-\frac{\partial^{2}}{\partial\theta^{2}}\log f\left( x;\theta\rig... ...me}}}{f^{2}}=-\frac{f^{^{\prime\prime}}}{f}+\frac{(f^{^{\prime}})^{2}}{f^{2}}$$

$B$N4X78$,$"$k$+$i!$4|BTCM$r$H$C$F(B

$$E\left[ -\frac{f^{^{\prime\prime}}}{f}+\left( \frac{f^{^{\prime}}... ...}}{f}\right] ^{2}=E\left[ \frac{\partial}{\partial\theta}\log f\right] ^{2}%$$

$B%]%"%=%sJ,I[$N%U%#%C%7%c!<>pJsNL!%(B

$BDj>o%]%"%=%s%9%Q%$%/;~7ONs$r(B$n$ $B8D$N4QB,6h4V(B$\Delta$ $B$G6h@Z$C$?$H$7$h$&!%$3$N$H$-3F6h4VFb$N%9%Q%$%/$N8D?t(B $X=\left( X_{1},X_{2},\ldots,X_{n}\right)$ $B$OJ?6Q(B $\lambda \Delta$ $B$N%]%"%=%sJ,I[(B

$$\Pr\left( X=\left( X_{1},X_{2},\ldots,X_{n}\right) \vert\lambda,\... ...^{n}} \frac{\left( \lambda\Delta\right) ^{X_{i}}}{X_{i}!}e^{-\lambda\Delta}%$$

$B$K=>$&!%%U%#%C%7%c!<>pJsNL$O(B

$$J\left( \lambda\right) =-E\left[ \frac{\partial^{2}}{\partial \lambda^{2}}\log f\left( X;\lambda,\Delta\right) \right]$$

$$=-E\left[ \frac{\partial^{2}}{\partial\lambda^{2}}\sum_{i=1}^{n}\left( X_{i}\log\lambda\Delta-\lambda\Delta-\log X_{i}!\right) \right]$$

$$=-E\left[ \sum_{i=1}^{n}\frac{\partial}{\partial\lambda}\left( X_{i}% \frac{1}{\lambda}-\Delta\right) \right]$$

$$=\frac{\sum_{i=1}^{n}EX_{i}}{\lambda^{2}}$$

$$=\frac{n\lambda\Delta}{\lambda^{2}}=\frac{n\Delta}{\lambda}%$$

$B=>$C$FJ,;6$N2<8B$O(B $\lambda/n\Delta$ $B!%(B

$BLdBj(B 24   $B%]%"%=%sJ,I[$N(B$\lambda$ $B$K4X$9$k%U%#%C%7%c!<>pJsNL$r8x<0(B $J\left( \theta\right) =E\left[ \left( \frac{\partial }{\partial\theta}\log f\left( x\vert\theta\right) \right) ^{2}\right]$ $B$rMQ$$$FF3=P$;$h!%(B

$BLdBj(B 25  

$B;X?tJ,I[$N%U%#%C%7%c!<>pJsNL$r5a$a$h!%(B

$B2rK!(B 26   $BJ?6Q(B$1/\lambda$ $B$N;X?tJ,I[(B $f\left( x;\lambda\right) =\lambda e^{-\lambda x}$ $B$N(B$\lambda$ $B$K4X$9$k%U%#%C%7%c!<>pJsNL$O(B

$$J\left( \lambda\right) =-E\left[ \frac{\partial^{2}}{\partial\lam... ...al\lambda}\left( \frac{1}{\lambda}-x\right) \right] =\frac {1}{\lambda^{2}}%$$

$B$f$($KJ,;6$N2<8B$O(B $\lambda^{2}$ $B!%(B

$B%Y%$%:?dDj!J$O$d$o$+$j!K(B

Bayes$B$NDjM}(B

$$p\left( \theta\vert x,w\right) =\frac{p(x\vert\theta)\pi(\theta\vert w)}{p(x\vert w)}%$$

$p(x\vert\theta)$ $B$rL`EY(B(likelihood)$B!$(B $\pi(\theta)$ $B$r;vA0J,I[(B(prior distribution)$B!$(B $p\left( \theta\vert x\right)$ $B$r;v8eJ,I[(B(posterior distribution)$B$H$$$&!%(B

$BJ,Jl$O@55,2=9`$GJ,;R$NAmOB$r$H$C$F(B

$$p(x\vert w)=\int p\left( x,\theta\vert w\right) d\theta$$

$B$GM?$($i$l$k!%<~JUL`EY4X?t!&%(%S%G%s%9$H$$$&!%(B

$B;v8eJ,I[$K$h$k?dDjCM(B

$B;v8eB;<:4X?t(B(posterior loss function, posterior expected loss) $B$rDj5A$7!$;v8eJ,I[$K$h$k4|BTCM(B: $B;v8e%j%9%/4X?t(B(posterior risk function)    

$$L^{\text{Bayes}}(\delta,\theta;X,w)\equiv E^{\theta\vert X,w}[L(\... ...a)]=\int L\left( \delta,\theta\right) p\left( \theta\vert X,w\right) d\theta$$

$B$r:G>.2=$9$k%Q%i%a!<%?$r?dDjNL$H$9$k(B

$$\hat{\theta}_{\text{Bayes}}(X,w)=\arg\underset{\theta}{\min}L^{\text{Bayes}% }(\delta,\theta;X,w)$$

$BB;<:4X?t$H$7$F;Y;}4X?t(B(indicator function)$L_{0}$

$$L_{0}(\delta,\theta)=1_{\delta,\theta}$$

$B$rA*$V$H(BMAP$B?dDjCM(B(maximum a posteriori estimate)$B$,F@$i$l$k!%(B

$$\hat{\theta}_{\text{MAP}}(X,w)=\arg\underset{\theta}{\max}p\left( \theta\vert X,w\right)$$

$BB;<:4X?t$H$7$F@dBP8m:9(B$L_{1}$

$$L_{1}(\delta,\theta)=\vert\delta-\theta\vert$$

$B$rA*$V$H;v8eCf1{CM?dDj(B(posterior median)$B$,F@$i$l$k!%(B

$BB;<:4X?t$H$7$FFs>h8m:9(B$L_{2}$

$$L_{2}(\delta,\theta)=(\delta-\theta)^{2}%$$

$B$rMQ$$$k$H;v8eJ?6Q(B(posterior mean)

$$\hat{\theta}_{\text{PM}}(X,w)=\int\theta p\left( \theta\vert X,w\right) d\theta$$

$B$,F@$i$l$k!%;v8eJ?6Q$rFC$K%Y%$%:?dDjCM(B(Bayes estimate)$B$H8F$V$3$H$,$"$k!%(B

$BLdBj(B 27   $B;v8eB;<:(B $L^{Post}(\delta,\theta)$ $B:G>.2=$N86M}$+$i(BMAP$B?dDjCM!$;v8eJ?6QCM$r5a$a$h!%(B

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$B%Y%$%:?dDjNL$OD6%Q%i%a!<%?$K0MB8$9$k!%D6%Q%i%a!<%?$r:GL`?dDj$G5a$a%Y%$%:?dDjNL$KMQ$$$k$3$H$r7P83%Y%$%:K!$H8F$V(B(empirical Bayes estimator)$B!%(B

$B<~JUL`EY(B

$$p(x\vert w)=\int p\left( x,\theta\vert w\right) d\theta=\int p(x\vert\theta)\pi (\theta\vert w)d\theta$$

$B<~JUL`EY$N:GL`?dDjCM(B($B<~JUL`EY:GBg2=(B)

$$\hat{w}=\arg\underset{w}{\max}p(x\vert w)$$

$B7P83%Y%$%:?dDjNL$O:GE,2=$7$?D6%Q%i%a!<%?(B$\hat{w}$ $B$rMQ$$$F(B

$$\hat{\theta}_{\text{EB}}(X)=\hat{\theta}_{\text{Bayes}}(X,\hat{w})$$

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