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°²©w§Ú­Ì²{¦b¥Î¬Y­Ó¸ÕÅç¨ÓÆ[¹î¤@­Ó¯S©wªºÀH¾÷²{¶H¡A§Ú­Ì¦ÛµM·|°Ý¦Û¤v¡G¯à¤£¯à§ä¨ì¤@­Ó¾A·íªº¶q¨Ó«×¶q¸Ó¸ÕÅç©Ò¯à´£¨Ñªº°T®§ (Information)¡H´«¤@­Ó¨¤«×¨Ó¬Ý¡A¦b¸ÕÅ礧«e¡A§Ú­ÌµLªk¹wª¾·|¥X²{¤°»òµ²ªG¡A¦]¦¹§Ú­Ì»¡¸ÕÅç¨ã¦³ÀH¾÷©Ê¡F¸ÕÅ礧«á§Ú­Ìª¾¹Dµ²ªG¤F¡AÀH¾÷©Ê´N®ø¥¢¤F¡A®ø¥¢¤FªºÀH¾÷©Ê¥i¥H¬Ý¦¨¬O§Ú­Ì©ÒÀò±oªº°T®§¡C©Ò¥H¡A¨Æ¹ê¤W¡A§Ú­Ìµ¥©ó¦b°Ý¡G¯à¤£¯à§ä¨ì¤@­Ó¾A·íªº¶q¨Ó«×¶q¸Ó¸ÕÅ窺ÀH¾÷©Ê¡H

°²©w A1, A2,¡K,An ¬O¬Y­Ó¸ÕÅ窺©Ò¦³¥i¯àµ²ªG¡C¦pªG P(Ai) > P(Aj)¡A«h¡uAj ¥X²{¡v¤ñ¡uAi ¥X²{¡v­n¨Ï§Ú­ÌÅå©_¡A´N¦p¹³µ}¦³ªºªÀ·|¨Æ¥ó¨ã¦³«D±`ªº·s»D»ù­È¤@¼Ë¡C¤]´N¬O»¡¡A¾÷²v¤£¦Pªºµ²ªG·|´£¨Ñ¤£¦Pªº°T®§¡C¦]¦¹¡A¥Î¨Ó¶q«×¸Ó¸ÕÅç©Ò¯à´£¨Ñªº°T®§¡]©Î¸ÕÅ窺ÀH¾÷©Ê¡^ªº¨º­Ó¶q¡A¥²¶·¬O¦U­Óµ²ªG©Ò´£¨Ñªº°T®§ªº¬YºØ¥­§¡­È¡C­º¥ý¡A§Ú­Ì¨Ó¬Ý¬Ý¦p¦ó«×¶q­Ó§O¨Æ¥ó©Ò¯à´£¨Ñªº°T®§¡C§Ú­Ì°²©w¦³¤@­Ó³o¼Ëªº¶q¡A¦Ó¥Î I(A) ¨Ó¥Nªí¨Æ¥ó A ©Ò¯à´£¨Ñªº°T®§¡A«h I(A) À³·íº¡¨¬¡G

(1) $I(A)\geq 0$¡F
(2) I(A) §¹¥þ¥Ñ A ªº¾÷²v¨M©w¡A´«¥y¸Ü»¡¡A $I(A) = \vartheta(P(A))$¡A$\vartheta$ ¬O­Ó©w¸q¦b 0 ¨ì 1 ¤§¶¡ªº¨ç¼Æ¡F
(3) ¦pªG $P(A)\geq P(B)$¡A«h $I(A)\leq I(B)$¡C

±ø¥ó(1)¶È¶Èªí¥Ü¨ú©w¤@­Ó¾A·íªº·ÇÂI¡F±ø¥ó(2)¬O±j½Õ¾÷²vªº¯SÂI¡A§Ú­Ì»¡¹L¨Æ¥óªºÀH¾÷³W«ß©Ê¬O¥Ñ¥¦ªº¾÷²v¨Ó¥Nªíªº¡A¦]¦¹©M¨Æ¥ó¦³Ãöªº­«­n¼Æ¶q¤]¸Ó¬O§¹¥þ¥Ñ¨Æ¥óªº¾÷²v¨Ó¨M©w¡C²{¦b§Ú­Ì¶i¤@¨B¦Ò¼{¨â­Ó¿W¥ß¨Æ¥ó A ©M B¡C$A\bigcap B$ ¥i¥H¸ÑÄÀ¬°¦b A ¥X²{ªº±¡ªp¤U¡AB ¤S¥X²{ªº¨Æ¥ó¡A¦ý¬O A ©M B ¬O¿W¥ßªº¡A¥Ñ A ¥X²{©Ò±o¨ìªº°T®§¡AÀ³¸ÓµLªkÀ°§U§Ú­Ì¹w´ú B ¥X²{ªº¥i¯à©Ê¡A¦]¦¹¤wª¾ A ¥X²{«á¡A¤Sª¾¹D B ¥X²{©Ò´£¨Ñªº°T®§¡AÀ³·í¬° A,B ¦U§O¥X²{©Ò±o°T®§¤§©M¡A¥ç§Y $I(A\bigcap B) = I(A)+I(B)$¡C¬ÛÀ³¦a¡A§Ú­Ì­n¨D $\vartheta$ º¡¨¬¡G

(4) $\vartheta(pq)=\vartheta(p)+\vartheta(q)$, $p,q \in[0,1]$

±N(1)¡B(2)¡B(3)¡A©M(4)ºî¦X°_¨Ó¡A´N¬O¤@­Ó©w¸q [0,1] ¦b¤W¡Bº¡¨¬(4)ªº»¼´î¨ç¼Æ $\vartheta$¡C³oºØ¨ç¼Æ«Ü¦h¡AÄ´¦p»¡¡A $\vartheta(f)=-\log_a t$¡A$t \in[0,1]$¡Aa ¬°¬Y­Ó¥¿¹ê¼Æ¡C³o¨à¨ú¤£¦Pªº a ¶È¶Èªí¥Ü¿ï¨ú¤£¦Pªº³æ¦ìªø«×¡C¦b¤U­±¡A§Ú­Ì¨ú $\vartheta(t)=-\log_2t$¡A¤]´N¬O¥O $I(A)=-\log_2P(A)$¡C

¦b¶i¤@¨B°Q½×¸ÕÅ窺°T®§¤§«e¡A§Ú­Ì¦^ÀY¬Ý¬Ý1.ªº¨Ò¤l¡C§Ú­Ì¦³ 2k ­Ó¿Oªw¡A¨ä¤¤¤@­ÓÃa¤F¡F¦bÀË´ú¤§«e¡A§Ú­Ì»{¬°¨C­Ó¿OÃaªº¾÷²v³£¬O¤@¼Ëªº¡A³£¬O 2-k¡C¥O An ¬°²Ä n ­Ó¿OªwÃa¤Fªº¨Æ¥ó¡A«h $I(A_n) = -\log_2 P(A)=-\log_2 2^{-k}=k$¡C³o®É¡A´ú¥X¥ô¦ó¤@­ÓÃa¿O©Ò¯à´£¨Ñªº°T®§¬Ò¬° k¡A¦]¦¹ k «×¶qµÛ´ú¥XÃa¿O¦ì¸m©ÒÀò±o¤§°T®§¡A¤]´N¬OÃa¿O¦ì¸mªºÀH¾÷©Ê¡C¤@¯ë¨Ó»¡¡A¦pªG¸ÕÅ礤ªº¨C­Óµ²ªG¨ã¦³¦P¼Ëªº¥i¯à©Ê¡]¥X²{ªº¾÷²v¤@¼Ë¡^¡A«h $-\log_2 \frac{1}{n} = \log_2 n$ ¥NªíµÛ¸Ó¸ÕÅç©Ò´£¨Ñªº°T®§¡A¨ä¤¤ n ¬O©Ò¦³¥i¯àµ²ªGªº­Ó¼Æ¡CÄ´¦p»¡¡A¥áÂY¤@ªT«D°¾­Ê»ÉªO¡AÆ[¹î¥¿­±¦¨¤Ï­±¥X²{©Ò±oªº°T®§¬° $-\log_2 \frac{1}{2} = \log_2 2=1$¡C¾ú¥v¤W¡A²Ä¤@­Ó¦Ò¼{¸ÕÅçæi¼Æªº¤H¬O¬ü°ê¹q°T¤uµ{®v Hartley¡]1928¦~¡^¡A¥L§â¸ÕÅçæi¼Æ©w¸q¬° $\log_2 n$¡C¥L¥u¦Ò¼{¨ì¸ÕÅ礤¥i¯à¥X²{ªºµ²ªGªº­Ó¼Æ¡A«o©¿²¤¤F¨C­Óµ²ªG¥X²{ªº¾÷²v¡C³o­Ó·§©À¦b 1947¡ã1948¦~¶¡¥ÑÅã¹A¤ó¤©¥H­×¥¿¡A¦Ó¦¨¤F¥Ø«e¼Æ¾Ç®a©M¤uµ{®v©Ò±Ä¥Îªº§Î¦¡¡C

°²©w A1,A2,¡K,An ¬O¬Y­Ó¸ÕÅ窺©Ò¦³¥i¯àµ²ªG¡C§Ú­Ìª¾¹D¨Æ¥ó Ai ©Ò¯à´£¨Ñªº°T®§¬O $-\log_2 P(A_i)$¡C¦]¦¹¡AÆ[¹î¤@¦¸¸ÕÅç©Ò±o¨ìªº°T®§¬O­ÓÀH¾÷ÅܼơC³o­ÓÀH¾÷Åܼƪº´Á±æ­È´N¬O¦h¦¸¿W¥ßÆ[¹î¸Ó¸ÕÅç©Ò±oªº°T®§ªº¥­§¡­È¡CÅã¹A¤ó§â³o¥­§¡­È¥s°µ¸ÕÅ窺æi¼Æ¡C§Î¦¡¤W»¡¡A¦pªG

\begin{displaymath}\alpha=\left<
\begin{array}{ccc}
A_1,&\cdots&,A_n\\
P(A_1),&\cdots&,P(A_n)
\end{array}\right>
\end{displaymath}

¬O¾÷²vªÅ¶¡ $(\Omega ,P)$ ªº¤@­Ó¸ÕÅç¡A«h £\ ªºæi¼Æ $H(\alpha)$ ¬O©w¸q¬°

\begin{displaymath}H(\alpha)=-\sum_{i=1}^nP(A_i)\log_2 P(A_i) \: .\end{displaymath}

¦pªG§Ú­Ì¥Î XB ªí¥Ü £[ ¤¤¨Æ¥ó B ªº«ü¥Ü¨ç¼Æ¡A«h $H(\alpha) = Ex$¡A¨ä¤¤ $x=-\sum_{i=1}^nX_{A_i}\cdot\log_2P(A_i)$¡C­n¬O¦b $H(\alpha)$ ªº©w¸q¤¤¡A¬Y­Ó¨Æ¥ó Ai ªº¾÷²v P(Ai)=0¡A«h¥O $P(A_i)\log_2P(A_i)=0$¡C³o¬O¦³¹D²zªº¡A¦]¬° $\lim_{t\rightarrow 0}t\log_2 t=0$¡A¥t¥~¡A¦pªG§Ú­Ì¥O $\eta(t)=-t\log_2 t$ ¥i¥H²³æªº¼g¦¨

\begin{displaymath}
H(\alpha)=\sum_{i=1}^n\eta(P(A_i)) \: .
\end{displaymath}

¦n¤F¡A¦b³o¨Çºò´êªº©â¶H°Q½×«á¡A§Ú­Ì¨Ó¦^ÀY¬Ý¬Ý¨º¦ê±m¿O§a¡I¨Ì·Ó¤W­zªº²Å¸¹¡A§Ú­Ì­n°Ýªº¬O¸ÕÅç

\begin{displaymath}
\alpha=\left<
\begin{array}{ccc}
A_1,&\cdots&,A_{2^k}\\
(\frac{1}{2})^k,&\cdots&,(\frac{1}{2})^k
\end{array}\right>\end{displaymath}

ªºæi¼Æ $H(\alpha)$¡C®Ú¾Ú­è¤~ªº©w¸q¡C

\begin{displaymath}H(\alpha)=-\sum_{i=1}^{2^k}\eta(P(A_i))=k ,\end{displaymath}

³o¥¿¬O§Ú­Ì³Ìªìªº·N«ä¡C

¸ÕÅ窺æi¼Æ¨ã¦³¤U­z©Ê½è¡G°²³] $\alpha=<A_1,A_2,\cdots,A_n>$, $\beta=<B_1,B_2,\cdots,B_n>$ ¬°¾÷²vªÅ¶¡ $(\Omega ,P)$ ªº¨â­Ó¸ÕÅç¡A«h

(i) $H(\alpha)\geq 0$; $H(\alpha)=0$ ªº¥R­n±ø¥ó¬O¬Y­Ó Ai ¬°¥²µM¨Æ¥ó¡A¦Ó¨ä¾lªº§¡¬°¤£¥i¯à¨Æ¥ó¡C
(ii) $H(\alpha) \leq \log_2 n$; $H(\alpha)=\log_2 n$ ªº¥R­n±ø¥ó¬O $P(A_1)=P(A_2)=\cdots=P(A_n)= \frac{1}{n}$¡C
(iii) ¦pªG £\ ©M £] ¬O¿W¥ßªº¡A«h $H(\alpha\vee\beta) = H(\alpha) + H(\beta)$

(i)ªºÃÒ©ú«Ü²³æ¡A(ii)ªºÃÒ©ú»P¾÷²vµLÃö¡A¦]¦¹¡A§Ú­Ì²¤±¼¥¦­Ì¡A¦Ó¨ÓÃÒ©ú(iii)¡C®Ú¾Ú©w¸q¡A $\alpha\vee\beta =$ $<A_1\bigcap B_1, \cdots, A_n \bigcap B_m>$¡C¥Ñ©ó £\ »P £] ¬O¿W¥ßªº¡A

\begin{displaymath}P(A_i\bigcap B_j)=P(A_i)P(B_j)\end{displaymath}

¦]¦¹

\begin{eqnarray*}
H(\alpha\vee\beta) &=& -\sum_{i,j}P(A_i\bigcap B_j)\log_2P(A_i...
...beta)\mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 1}}
\end{eqnarray*}


©Ê½è(i)¥i¸ÑÄÀ¬°¡G¦pªG¦b¬Y¸ÕÅ礤¡A·|¦³¤@­Ó¥²µM¨Æ¥ó²£¥Í¡A«hÆ[¹î³o­Ó¸ÕÅç¬O¤£·|´£¨Ñ¥ô¦ó°T®§ªº¡A¤]´N¬O»¡¡A³o­Ó°T®§¨S¦³ÀH¾÷©Ê¡C·íµÛ¸ÕÅ礤ªº¦U­Ó¨Æ¥ó¨ã¦³¦P¼Ëªº¾÷²v®É¡A§Ú­Ì§â¥¦¥s°µ«D°¾­Ê¸ÕÅç¡C©Ê½è(ii)§i¶D§Ú­Ì¡A¦b¨ã¦³ n ­Ó¨Æ¥óªº¸ÕÅ礤¡A«D°¾­Ê¸ÕÅ窺ÀH¾÷©Ê³Ì¤j¡A¨äæi¼Æ¬° $\log_2 n$¡C³o¬O¦X¥Gª½Ä±­n¨Dªº¡F¦]¬°¡A¦bÆ[¹î°¾­Ê¸ÕÅç®É¡A§Ú­Ì¬O¹w¥ý´Nª¾¹D¤F¬Y¨Ç¨Æ¥ó¤ñ¸û®e©öµo¥Í¡A¦Ó¥t¤@¨Ç¨Æ¥ó¬O¤ñ¸û¤£®e©öµo¥Í¡F³oºØ§t½kªº¹wª¾´N»¡©ú¤F°¾­Ê¸ÕÅ窺ÀH¾÷©Ê¤ñ¸û¤p¡C¨ä¹ê¡A¦pªG°¾­Ê¨ì¤F·¥ÂI¡A´N¨S¦³ÀH¾÷©Ê¤F¡A¦Ó³o¥¿¬O©Ê½è(i)©Ò­n´y­zªº¡C¨Ì·Ó«e¦Óªº»¡ªk¡A $\alpha\vee\beta$ «üªº¬O¦P®ÉÆ[¹î £\ ©M £] ¨â­Ó¸ÕÅç¡A¨Ì¦¹¡A©Ê½è(iii)¥i¥H¦p¤U±Ô­z¡G ¦pªG £\ ©M £] ¬O¿W¥ßªº¸ÕÅç¡A«h¦P®ÉÆ[¹î £\ ©M £] ©Ò±oªº°T®§¬°¤À§OÆ[¹î £\ ©M £] ©Ò±o°T®§ªº©M¡CÁ`µ²°_¨Ó¡A§Ú­Ì©Ò©w¸qªº¸ÕÅ窺æi¼Æªº½T¬O´y­z¤F§Ú­Ì©Ò¹w´Áªº¦U¶µÂ²³æ©Ê½è¡A³o¨Ç´Nª`©wµÛ¥¦·|¬O¤@­Ó­«­n¦Ó¦³¥Îªº·§©À¡C

¦b¤¶²Ðæi¼Æªº¨ä¥L©Ê½è¤§«e¡A§Ú­Ì¥ý½Í½Í±ø¥óæi¼Æ¡C°²³] $\alpha=<A_1,A_2,\cdots,A_n>$ ¬°¤@¸ÕÅç¡AB ¬°¤@¨Æ¥ó¡C«h

\begin{displaymath}\left<
\begin{array}{ccc}
A_1\bigcap B,&\cdots&,A_n\bigcap B\\
P(A_1\bigcap B),&\cdots&,P(A_n\bigcap B)
\end{array}\right>
\end{displaymath}

¤]¥i¥H¬Ý¦¨¤@­Ó¸ÕÅç¡C³o­Ó¸ÕÅç¬O¦b B ¤w¸gµo¥Íªº±¡ªp¤U¨ÓÆ[¹î £\ ªº¸ÕÅç¡C§Ú­Ì§â³o­Ó¸ÕÅç°O¬° $\alpha\vert B$¡C¸ÕÅç $\alpha\vert B$ ªºæi¼Æ $H(\alpha\vert B)$ ªí¥Ü¬Ý¦b¨Æ¥ó B ¤w¸gµo¥Íªº±¡ªp¤U¡A¸ÕÅç £\ ©Ò¯d¦sªºÀH¾÷©Ê¡C¨Ò¦p B=Ai¡A«h¦b Ai ¥X²{ªº±¡ªp¤U¡A£\ ¤w¤£¨ã¦³¥ô¦óÀH¾÷©Ê¡A¦]¦¹ $H(\alpha\vert A_i)=0$¡C¡]³oÂI¥i¥H«Ü®e©öªº±q©Ê½è(i)¾É¥X¡C¡^$H(\alpha\vert B)$ ºÙ¬°¸ÕÅç¡C¬Û¹ï©ó¨Æ¥ó B ªº±ø¥óæi¼Æ¡C°²³] $\beta=<B_1,\cdots,B_m>$ ¬°¥t¤@¸ÕÅç¡A¥O·í

\begin{displaymath}
x=\sum_jX_{B_j} H(\alpha\vert B_j) \: .
\end{displaymath}

Bj µo¥Í®É¡Ax ´N¬O $H(\alpha\vert B_j)$¡C§Ú­Ì§â x ªº´Á±æ­È°O¬° $H(\alpha\vert\beta)$¡C $H(\alpha\vert\beta)$ ¶q«×ªº¬O¦bÆ[¹î¤F¸ÕÅç £] ¤§«á¡A£\ ©Ò¯d¦s¤U¨ÓªºÀH¾÷©Ê¡C§Ú­Ì§â $H(\alpha\vert\beta)$ ¥s°µ £\ ¬Û¹ï©ó £] ªº±ø¥óæi¼Æ¡C ÅãµMªº¡A $H(\alpha\vert\alpha)=0$¡Aæi¼Æªº¥t¥~¨â­Ó­«­n©Ê½è¬O¡G°²³]£\,£] ¬O¨â­Ó¸ÕÅç¡A«h

(iv) $H(\alpha\vee\beta)=H(\beta)+H(\alpha\vert\beta)$
(v) $0\leq H(\alpha\vert\beta)\leq H(\alpha)$

©Ê½è(iii)¬O©Ê½è(iv)ªº¯S¨Ò¡A¦Ó©Ê½è(iv)ªºÃÒ©ú¤S¸ò©Ê½è(iii)ªº§¹¥þ¤@¼Ë¡A¥u­n§â $P(A_i\bigcap B_j) = P(A_i)P(B_j)$ ´«¦¨ $P(A_i\bigcap B_j)=P(B_j)P(A_i\vert B_j)$ ´N¦æ¤F¡C©Ê½è(v)ªºÃÒ©ú»P¾÷²v·§©ÀµLÃö¡A©Ò¥H¬Ù²¤¡C¤£¹L¡A§Ú­Ì­n´£¿ôŪªÌ¤@ÂI¡G¦bª½Ä±¤W¡A©Ê½è(v)¬O·¥¬°ÅãµMªº¡A¦]¬°¦bÆ[¹î £] ¤§«á¡A§Ú­Ì¦h¦h¤Ö¤Ö·|±o¨ì¨Ç°T®§¡A³o¨Ç°T®§¥u¥i¯à´î¤Ö £\ ªºÀH¾÷©Ê¡C¥t¥~¡A±q(iii)©M(iv)¥i¥H¬Ý¥X¡A¦pªG £\ ©M £] ¬O¿W¥ßªº¡A«h $H(\alpha\vert\beta)=H(\alpha)$¡C

±q¤W¬qªº°Q½×¥i¥H¬Ý¥X $H(\beta)-H(\beta\vert\alpha)$ ¶q«×ªº¬O¸ÕÅç £] ¦bÆ[¹î¤F¸ÕÅç £\ ¤§«á©Ò´î¤ÖªºÀH¾÷©Ê¡C¦]¦¹¡A§Ú­Ì¥i¥H§â $H(\beta)-H(\beta\vert\alpha)$ ¬Ý¦¨¬O £\ ´£¨Ñµ¹ £] ªº°T®§¡C§Ú­Ì§â $H(\beta)-H(\beta\vert\alpha)$ °O¬° $I(\alpha,\beta)$¡C¦³®É­Ô $I(\alpha,\beta)$ ¥s°µ £] ¦s©ó £\ ¤¤ªº°T®§¡C ¥Ñ©ó $H(\alpha\vee\beta) = H(\alpha)+H(\beta\vert\alpha)$ $=H(\beta)+H(\alpha\vert\beta)$¡A§Ú±o¨ì¤U­±ªºÃö«Y¦¡¡G

\begin{displaymath}
I(\alpha, \beta)=H(\beta)-H(\beta\vert\alpha)=H(\alpha)-H(\alpha\vert\beta)=I(\beta, \alpha)
\end{displaymath}

¦bÀ³¥Îªº®É­Ô¡A£] ¬O§Ú­Ì­n¬ã¨sªº¹ï¶H¡A£\ ¬O¬°¤F®ø°£ £] ªºÀH¾÷©Ê¦Ó¦Ò¼{ªº»²§U¸ÕÅç¡C³o¨Ç³£±N¦b¤U­±¸Ô²Ó°Q½×¡C

   

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