Lattice Definitions

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6JKU CRRTQZKOCVKQP KU RCTVKEWNCTN[ GCU[ VQ KORNGOGPV HQT CTDKVTCT[ NQUU HWPEVKQPU

*QYGXGT VJG WUG QH 0DGUV NKUVU OC[ KP UQOG ECUGU DG VQQ TGUVTKEVKXG CP CRRTQZKOC VKQP CPF UGCTEJ GTTQTU OC[ TGUWNV 6JGTGHQTG KV KU QH KPVGTGUV VQ KPETGCUG VJG UK\G QH VJGUG VYQ URCEGU VQ VJG TGEQIPKVKQP NCVVKEG KG VQ EQPUKFGT OQTG ECPFKFCVGU KP VJG UGCTEJ CPF VJG UWO

2.2.3 MBR Recognition with Lattices

+P VJG HQNNQYKPI YG RTGUGPV C OWNVKUVCEM RTGſZ VTGG^£ UGCTEJ CNIQTKVJO VJCV WUGU TGEQIPKVKQP NCVVKEGU CU VJG J[RQVJGUKU CPF GXKFGPEG URCEGU 6JG FGXGNQROGPV QH VJG CNIQTKVJO RTQEGGFU CU HQNNQYU 9G UVCTV D[ KPVTQFWEKPI UVCVKUVKECN SWCPVKVKGU FGTKXGF HTQO VJG NCVVKEG VJCV CTG PGGFGF D[ VJG UGCTEJ RTQEGFWTG 9G VJGP RTGUGPV C UKPING UVCEM^£UGCTEJ FKTGEVN[ QXGT VJG NCVVKEGU 6JKU UGCTEJ KU HWTVJGT TGſPGF D[ KPVTQFWE KPI C RTGſZ VTGG OWNVKUVCEM UVTCVGI[ (QT ENCTKV[ QH RTGUGPVCVKQP YG HQTOWNCVG VJG

£UGCTEJ HQT OKPKOK\CVKQP QH 9'4 6JKU KU TGCNK\GF CU C OKPKOWOTKUM RTQEGFWTG WPFGT C NQUU HWPEVKQP DCUGF QP .GXGPUJVGKP FKUVCPEG JGPEGHQTVJ TGHGTTGF VQ CU VJG .GXGPUJVGKP NQUU HWPEVKQP 9G GPF VJKU UGEVKQP D[ C FKUEWUUKQP QH VJG HGCUKDKNKV[ QH VJG^£UGCTEJ HQT QVJGT NQUU HWPEVKQPU

2.2.3.1 Lattice Definitions

# TGEQIPKVKQP NCVVKEG KU C EQORCEV TGRTGUGPVCVKQP HQT C NCTIG UGV QH YQTF UVTKPIU CPF VJGKT VKOG DQWPFCTKGU^£ +V KU CP CE[ENKE FKTGEVGF ITCRJ

KU VJG UGV QH PQFGU KU VJG UGV QH GFIGU KU VJG WPKSWG NCVVKEG UVCTV PQFG KU VJG WPKSWG NCVVKEG GPF PQFG CPF URGEKſGU NCVVKEG EQPPGEVKXKV[ 'CEJ PQFG KP VJG NCVVKEG KU NCDGNGF D[ C YQTF CPF C VKOG 'CEJ GFIG JCU C UVCTV PQFG CPF CP GPF PQFG 'FIGU CTG CUUQEKCVGF YKVJ VJG YQTFU CV VJGKT GPF PQFGU CPF YKVJ VJG

.CVVKEGU CTG IGPGTCVGF WUKPI YQTF UVTKPIU CPF VJGKT UVCVG NGXGN CNKIPOGPV YKVJ VJG CEQWUVKE HTCOGU *QY GXGT YG EQPUKFGT NCVVKEGU KP YJKEJ VJG UVCVG CNKIPOGPV KPHQTOCVKQP KU FKUECTFGF CPF QPN[ VJG YQTF VKOG DQWPFCTKGU CTG MGRV

VKOG KPVGTXCN HTQO VJGKT UVCTV PQFG VQ VJGKT GPF PQFG 6JG[ CTG CNUQ NCDGNGF D[ VJG LQKPV CEQWUVKE CPF NCPIWCIG OQFGN NQIRTQDCDKNKV[ VJCV VJGKT YQTF QEEWTU FWTKPI VJG CUUQEKCVGF KPVGTXCN 6JKU LQKPV NQIRTQDCDKNKV[ KU EQPFKVKQPGF QP VJG JKUVQT[ URGEKſGF D[ VJG UVCTV PQFG QH VJG GFIG (QT GZCORNG KP(KIWTG CP GFIG KFGPVKſGU VJG J[RQVJGUKU VJCV VJG YQTF 019 DGIKPU CV UGE CPF GPFU CV UGE 6JG PWODGT

QP VJKU GFIG KU NQIRTQDCDKNKV[ VJCV VJG YQTF 019 QEEWTU DGVYGGP UGE CPF UGE IKXGP VJCV VJG YQTF *'..1 KU RTGUGPV HTQO VJG UVCTV QH VJG CEQWUVKE FCVC WPVKN UGE

#pathQT complete path KU C UGSWGPEG QH EQPPGEVGF PQFGU CPF NKPMU HTQO VQ

VJTQWIJ VJG NCVVKEG # path segmentKU C UGSWGPEG QH EQPPGEVGF PQFGU HTQO CP KPVGTPCN NCVVKEG PQFG VQ CPQVJGT KPVGTPCN NCVVKEG PQFG OC[ DG CPF

OC[ DG #partial pathKU C UGSWGPEG QH EQPPGEVGF PQFGU CPF NKPMU HTQOVQ CP KPVGTPCN NCVVKEG PQFG KV OC[ DG C EQORNGVG RCVJ KHKU 6JG CEQWUVKE UGIOGPV EQTTGURQPFKPI VQ C RCVJ UGIOGPVHTQOVQUJCNN DG FGPQVGF .GVDG C RCTVKCN RCVJ HTQO NCVVKEG UVCTVVQDG C RCVJ UGIOGPV HTQOVQ

CPFDG C RCVJ UGIOGPV HTQOVQ VJG NCVVKEG GPF 6JG CEQWUVKE UGIOGPVU EQTTGURQPFKPI VQ VJGUG VJTGG RCVJ UGIOGPVU YKNN DG FGPQVGF

CPF

TGURGEVKXGN[ 6JG UWO QH NQIRTQDCDKNKVKGU QP VJG GFIGU CNQPI IKXGU

9G KPVTQFWEG VJGpartial path log-probability VJGlattice backward log-probability CPF VJGlattice total probabilityQH C RCTVKCN J[RQVJGUKU CU HQNNQYU 6JG RCTVKCN RCVJ NQIRTQDCDKNKV[ QHKU

6JG NCVVKEG DCEMYCTF NQIRTQDCDKNKV[ QHKU

YJGTG FGPQVGU VJG UGV QH CNN EQORNGVG RCVJU KP VJG NCVVKEG 6JG NCVVKEG VQVCN RTQDCDKNKV[ QHKU 5WDUVKVWVKPI VJG FGſPKVKQPU QH

An example lattice. The time marks correspond to the node times and the word ending times. The numbers on the edges are logarithms of conditional joint probabilities as described in the text. The partial path log-probability of a partial hypothesis is the log of the probability of its path; the partial path

(‘HELLO’,‘0.6’) in this lattice has value . The lattice backward log-probability of a partial hypothesis is the log of the sum of probabili-ties of all lattice paths from end node of to the lattice end node; for the partial path (‘HELLO’,‘0.6’) in this lattice these paths are indicated by dotted lines and the lattice backward log-probability of this is . The lattice total probability of a partial path is the exponentiated sum of its partial path log-probability and lattice backward log-probability; its value isfor

(‘HELLO’, ‘0.6’) in the lattice above.

6JG NCVVKEG VQVCN RTQDCDKNKV[

QH C RCTVKCN J[RQVJGUKU EQWNF VJGTGHQTG DG KPVGTRTGVGF CU VJG LQKPV RTQDCDKNKV[ QH QDUGTXKPI VJG CEQWUVKEU CPF CNN RQUUKDNG EQORNGVG J[RQVJGUGU VJCV JCXG VJG RTGſZ 6JGUG RTQDCDKNKVKGU CTG KNNWUVTCVGF KP (KIWTG

2.2.3.2 ^£Search Under General Loss Functions

6JG UGV QH CNN EQORNGVG RCVJU KP VJG NCVVKEG EQPUVKVWVGU VJG J[RQVJGUKU URCEG

QH 'SWCVKQP +V CNUQ HQTOU VJG GXKFGPEG URCEG 'SWCVKQP VJG CUUQEKCVGF LQKPV NQIRTQDCDKNKV[ECP DG EQORWVGF D[ CFFKPI VJG NQI RTQDCDKNKVKGU QP NCVVKEG GFIGU CNQPI 6JGTGHQTG QP VJG NCVVKEG YG YQWNF KORNGOGPV

Æ NCVVKEG UWEJ VJCV KVU GZRGEVGF NQUU

KU VJG NGCUV QH CNN EQORNGVG J[RQVJGUGU KP VJG NCVVKEG 6JKU UGCTEJ HQT^¼ECP DG GHHGE VKXGN[ KORNGOGPVGF CU CP^£CNIQTKVJO = ? YJKEJ RTQEGGFU D[ GZVGPFKPI RCTVKCN J[RQVJGUGU HQTYCTF VJTQWIJ VJG NCVVKEG

6YQ EQUV HWPEVKQPU CTG TGSWKTGF HQT VJG UGCTEJ 6JG ſTUV EQUV HWPEVKQP KU CUUQEKCVGF YKVJ GCEJ J[RQVJGUKU YJGVJGT RCTVKCN QT EQORNGVG +VU XCNWG KU C NQYGT DQWPF QP VJG GZRGEVGF NQUU 'SWCVKQP VJCV ECP DG QDVCKPGF D[ GZVGPFKPI VJG J[RQVJGUKU VJTQWIJ VJG NCVVKEG VQ EQORNGVKQP

6JG UGEQPF EQUV HWPEVKQP KU QPN[ CUUQEKCVGF YKVJ EQORNGVG J[RQVJGUGU +V KU CP QXGT GUVKOCVG QH VJG GZRGEVGF NQUU QH C EQORNGVG J[RQVJGUKU^¼

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*[RQVJGUGU CTG MGRV KP C RTKQTKV[ SWGWG YJKEJ KU UQTVGF D[ EQUV YKVJ VJG UOCNNGUV EQUV J[RQVJGUKU CV VJG VQR 9G UJCNN WUG VJG VGTO őUVCEMŒ VQ TGHGT VQ VJG SWGWG UKPEG KP URGGEJ TGEQIPKVKQP VJG^£CNIQTKVJOU JCXG JKUVQTKECNN[ DGGP RTGUGPVGF KP VGTOU QH UVCEMU = ? #V GXGT[ KVGTCVKQP VJG J[RQVJGUKU CV VJG VQR QH VJG UVCEM KU GZVGPFGF 9JGP VJGTG KU C EQORNGVG J[RQVJGUKU CV VJG VQR KVU UGEQPF EQUV KU EQORWVGF +H VJKU QXGTGUVKOCVGF EQUV KU UOCNNGT VJCP VJG WPFGT GUVKOCVGF EQUV QH VJG PGZV UVCEM J[

RQVJGUKU QT KH VJGTG KU PQ RCTVKCN J[RQVJGUKU NGHV KP VJG UVCEM VJG CNIQTKVJO VGTOKPCVGU 9G PQVG VJCV^£RTQEGFWTGU WUWCNN[ GORNQ[ CP GZCEV GZRGEVGF NQUU 'SWCVKQP HQT EQORNGVG J[RQVJGUGU JQYGXGT VJKU KU RTQJKDKVKXGN[ GZRGPUKXG VQ ſPF KP QWT ECUG VJGTGHQTG YG WUG VJG QXGTGUVKOCVG