El estado de la cuesti´on sobre la Historia Regum Britanniae

+PVWKVKXGN[ URGCMKPI C FGEKUKQP UVTCVGI[ TWNG KU ECNNGF TQDWUV KH KV KU PQV XGT[ UGPUK VKXG VQ VJG RTGXKQWUN[ FKUEWUUGF RTKQT WPEGTVCKPV[ QT FKUVQTVKQPU /QTG HQTOCNN[ NGV DG CP CTDKVTCT[ FGEKUKQP TWNG EQPUVTWEVGF WPFGT UQOG J[RQVJGVKECN OQFGN

¼ YJGTG

Ï KU VJG ENCUU VQ YJKEJ VJG QDUGTXCVKQP

ÜYKNN DG CUUKIPGF CPF KU C VTCKPKPI UCORNG UGV WUGF HQT VJG EQPUVTWEVKQP QH VJG FGEKUKQP TWNG .GV

¯FGPQVG CP CTDKVTCT[ CFOKUUKDNG FKUVQTVGF FCVC OQFGN HQT VJG FKUVQTVKQP V[RGU FKUEWUUGF KP 5GEVKQP YJGTG KU WUGF VQ EJCTCEVGTK\G VJG FKUVQTVKQP NGXGN .GV

£

¯ FGPQVG VJG UGV QH CFOKUUKDNG FKUVQTVGF FCVC OQFGNU 6JG ENCUUKſEC VKQP RGTHQTOCPEG QH VJG FGEKUKQP TWNGKP C UKVWCVKQP YJGTG FCVC CTG ſVVGF VQ VJG FKUVQTVGF OQFGN

¯

£

¯ YKNN DG EJCTCEVGTK\GF D[ VJG TKUM HWPEVKQPCN

¯

YJGTG FGPQVGU VJG GZRGEVCVKQP YKVJ TGURGEV VQ VJG RTQDCDKNKV[ FKUVTKDWVKQP QH EQTTGURQPFKPI VQ VJG FKUVQTVGF OQFGN

¯ £ ¯ .GV WU ECNN VJG HWPEVKQPCN · · ů¾Å ¯ ¯

VJGguaranteed (upper) risk=? HQT VJG FGEKUKQP TWNGKP VJG RTGUGPEG QH FKUVQT VKQPU ¯ £ ¯ +H YG MPQY VJG FKUVTKDWVKQP QH ¯QP £ ¯ YG ECP HWTVJGT FGſPG VJG HQNNQYKPI HWPEVKQPCN ¯

YJGTGFGPQVGU VJG GZRGEVCVKQP YKVJ TGURGEV VQ VJG FKUVTKDWVKQP QH QP

£ 9G ECNN VJGoverall risk #RRCTGPVN[ DQVJ

·

CPF ECP DG WUGF CU QRVKOCNKV[ ETKVGTKC KP UGCTEJKPI HQTrobust (with respect to distortions

£

) decision rules # FGEKUKQP TWNG

£

YKVJ VJG OKPKOCN XCNWG QH VJG IWCTCPVGGF TKUM HQT CNN CFOKUUKDNG FKUVQTVKQPU £ ´¡µ ·

KU TGHGTTGF VQ CU Cminimax decision rule # FGEKUKQP TWNG

YKVJ VJG OKPKOCN XCNWG QH VJG QXGTCNN TKUM HQT CNN CFOKUUKDNG FKUVQTVKQPU

´¡µ

KU TGHGTTGF VQ CU Cpredictive decision rule

6JG EQPUVTWEVKQP QH VJGUG TQDWUV FGEKUKQP TWNGU YKNN FGRGPF QP JQY VJG CFOKUUKDNG FKUVQTVKQPU

£

CTG FGſPGF CPF CNUQ HQT VJG ECUG QH VJG RTGFKEVKXG FGEKUKQP TWNG VJG FKUVTKDWVKQP QH VJG FKUVQTVKQP QP

£

+P VJG HQNNQYKPI VYQ UWDUGEVKQPU + UJQY VYQ GZCORNGU QH UWEJ TQDWUV FGEKUKQP TWNGU PCOGN[minimax decision ruleCPF Bayesian predictive decision rule TGURGEVKXGN[ $QVJ QH VJGO CUUWOG VJCV

VJG FKUVTKDWVKQPUCPFCTG MPQYP WR VQ UQOG URGEKſCDNG RCTCO GVGTU KP VJG HQTOU QH

£

CPF

VJG VTWG RCTCOGVGTU QH VJGUG FKUVTKDWVKQPU CPF NKG KP C PGKIJDQTJQQF QH VJG GUVKOCVGF QT J[RQVJGVKECN QPGU VJGTGHQTG

VJG RTKQT WPEGTVCKPV[ ECP DG OQFGNGF D[ FGſPKPI CPuncertainty neighborhood QH VJG OQFGN RCTCOGVGTU CPFQT RQUUKDN[ C FKUVTKDWVKQP QH OQFGN RCTCO GVGTUQP VJKU WPEGTVCKPV[ PGKIJDQTJQQF

9KVJ VJGUG CUUWORVKQPU VJG URGEKſE OKPKOCZ FGEKUKQP TWNG CPF RTGFKEVKXG FGEKUKQP TWNG ECP DG EQPUVTWEVGF CEEQTFKPIN[ VQ UCVKUH[ UQOG FGUKTGF TQDWUVPGUU RTQRGTVKGU

3.6.2

Minimax Classification Rule

.GV ¼ ¼

FGPQVG VJGuncertainty neighborhoodQH VJG VTWG OQFGN RCTCOGVGTU

KG ¼ ¼ YJGTG ¼

¼CTG OQFGN RCTCOGVGTU GUVKOCVGF HTQO VJG VTCKPKPI FCVC CPFECP DG XKGYGF CU C IGPGTKE RCTCOGVGT VQ EJCTCEVGTK\G VJG FGITGG QH VJG FKUVQTVKQP 6JGP YG JCXG £ £   ¼ ¼ YJGTG £

KU VJG UGV QH FKUVQTVGF OQFGNU CPF · · ´£ µ¾ ¯ ´£ ¼   ¼ µ ¾ªÏ   ¾ªÜ £

6Q EQPUVTWEV C OKPKOCZ FGEKUKQP TWNG YJKEJ OKPKOK\GU VJG CDQXG IWCTCPVGGF TKUM

·

CFQRVGF 1PG RQUUKDKNKV[ KU VQ WUG VJG WRRGT DQWPF QH · YJKEJ YG FGPQVG ·· ·· ·· Ͼª Ï ¾ªÜ ´£ µ¾¯´£¼ ¼µ £  

6Q UKORNKH[ QWT FKUEWUUKQP YG CUUWOG VJCV YG FQ PQV EQPUKFGT VJG WPEGTVCKPV[ QH VJGTGCHVGT CPF WUG

 ¼

CU VJG NCPIWCIG OQFGN YKVJ

¼DGKPI VJG UGV QH NCPIWCIG OQFGN RCTCOGVGTU GUVKOCVGF HTQO VJG VTCKPKPI VGZV FCVC $[ WUKPI VJG NQUU HWPEVKQP YG VJGP JCXG ·· ·· ¾ªÏ   ¼ ªÜ´µ £ ¯ ´£ ¼ µ £

# FGEKUKQP TWNG YJKEJ OKPKOK\GU VJG CDQXG ·· KU CU HQNNQYU ··  ¼ £ ¯ ´£ ¼ µ £

6JKU KU VJG UQECNNGFminimax decision ruleYJKEJ YCU ſTUV UVWFKGF D[ /GTJCX CPF .GG KP =? +V ECP DG UQNXGF KP VYQ UVGRU (KTUV YG GUVKOCVG VJG WPFGTN[KPI RCTCOG VGTU WUKPI VJG /. CRRTQCEJ YKVJKP GCEJ PGKIJDQTJQQF

´µ ¼ KG £¯´£ ´Ïµ ¼ µ £ YJGTG ´µ

¼ FGPQVGU RTGVTCKPGF OQFGN RCTCOGVGTU HQT YQTF

6JGP YG CRRN[ VJG RNWIKP /#2 FGEKUKQP TWNG YKVJ

TGRNCEKPI VJG QTKIKPCN

´µ

¼ 6JGTGHQTG EQPEGRVWCNN[ VJG OKPKOCZ FGEKUKQP TWNG FGUETKDGF KP 'S ECP DG XKGYGF CU C RTQEGFWTG YJKEJ OQFKſGU VJG RNWIKP /#2 FGEQFGT UJQYP KP 'S YKVJ CP GZVTC UVGR CU KP 'S VQ ſPF C OQFKſGF RQKPV GUVKOCVG KP VJG PGKIJDQTJQQF

¼ ´µ ¼

QH VJG QTKIKPCN ENCUUKſGT RCTCOGVGTU ¼

´µ ¼

6JG CDQXG TQDWUV OKPKOCZ ENCUUKſECVKQP TWNG OCMGU PQ CUUWORVKQP CDQWV VJG HQTO QH VJG FKUVQTVKQP *QYGXGT KVU GHſECE[ FQGU FGRGPF QP CP CRRTQRTKCVG URGEKſECVKQP QH VJG RCTCOGVGT WPEGTVCKPV[ PGKIJDQTJQQF

¼ ´µ ¼ +P VJG RCUV UGXGTCN [GCTU UQOG QVJGT URGEKſE VGEJPKSWGU JCXG CNUQ DGGP FGXGNQRGF VQ KORNGOGPV VJG CDQXG OKPKOCZ FGEKUKQP TWNG KP *//DCUGF #54 U[UVGOU = ? 6JG[ CTG UJQYP VQ DG GHHGEVKXG KP FGCNKPI YKVJ PQKU[ URGGEJ TGEQIPKVKQP CPF VJG OKUOCVEJ ECWUGF D[ FKHHGTGPV TGEQTFKPI EQPFKVKQPU

6JGTG CTG JQYGXGT QVJGT RQUUKDKNKVKGU VQ OQFGN VJG CFOKUUKDNG FKUVQTVKQPU (QT GZCORNG KH YG WUG £ ¼ YJGTG ¼

FGPQVGU C URGEKſE VTCPUHQTOCVKQP QH

¼YKVJ RCTCOGVGTU

+P VJKU YC[ VJG WPEGTVCKPV[ QHECP DG EJCTCEVGTK\GF D[ VJG WPEGTVCKPV[ QH 6JGP VJG minimax decision ruleYKVJ TGURGEV VQ VJG CDQXG

YKNN DG ··  ¼ ¼

6JG UQECNNGFmodel-space stochastic matchingOGVJQF FGUETKDGF KP = ? ECP DG VJGQTGVKECNN[ LWUVKſGF KP VJKU YC[