ELEMENTS OF INFORMATIO INFOR MATION N THEOR THEORY Y Second Edition
THOMAS M. COVER JOY A. THOMAS
A JOHN WILEY & SONS, INC., INC., PUBLIC PUBLICATION ATION
ELEMENTS OF INFORMATION THEORY
ELEMENTS OF INFORMATIO INFOR MATION N THEOR THEORY Y Second Edition
THOMAS M. COVER JOY A. THOMAS
A JOHN WILEY & SONS, INC., INC., PUBLIC PUBLICATION ATION
Copyright
2006 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-P Cataloging-in-Publication ublication Data: Cover, T. M., 1938– Elements of information theory/by Thomas M. Cover, Joy A. Thomas.–2nd ed. p. cm. “A Wiley-Interscience publication.” Includes Inclu des bibli bibliograph ographical ical referen references ces and index index.. ISBN-13 978-0-471-24195-9 ISBN-10 0-471-24195-4 1. Information theory. I. Thomas, Joy A. II. Title. Q360.C68 2005 003 .54–dc22 2005047799
Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1
CONTENTS
Contents
v
Preface to the Second Edition
xv
Preface to the First Edition
xvii
Acknowledgments for the Second Edition
xxi
Acknowledgments for the First Edition 1
Introduction and Preview 1.1
2
xxiii 1
Preview of the Book
5
Entropy, Relative Entropy, and Mutual Information 2.1
Entropy
2.2 2. 2
Joi oint nt En Entr trop opy y an and d Co Cond ndit itiion onaal Ent ntro ropy py
2.3 2. 3
Rela Re lati tive ve En Entr trop opy y an and d Mu Mutu tual al In Info form rmat atio ion n
2.4 2. 4
Relati Rela tion onsh ship ip Be Betw twee een n En Entr trop opy y an and d Mu Mutu tual al Info In form rmat atio ion n 20
2.5 2. 5
Chain Chai n Ru Rule less fo forr En Entr trop opy, y, Re Rela lati tive ve En Entr trop opy, y, and an d Mu Mutu tual al In Infor forma mati tion on 22
2.6 2. 6
Jens Je nsen en’s ’s In Ineequ qual aliity an and d It Itss Co Cons nseequ quen ence cess
25
2.7 2. 7
Log Lo g Su Sum m In Ineq equa uali lity ty an and d It Itss Ap Appl plic icat atiion onss
30
2.8 2. 8
Data Da ta-P -Pro rocces esssin ing g In Ineq equa uali lity ty
2.9
Sufficient Statistics
2.10
Fano’s Inequality
Summary
41
Prob Pr oble lems ms
43
Hist Hi stori orica call No Note tess
13
13 16 19
34
35 37
54 v
vi
CONTENTS
3
Asymptotic Equipartition Property
57
3.1 3.1 Asym As ympt ptot otic ic Eq Equi uipa part rtit itio ion n Pr Prop oper erty ty Th Theo eore rem m 58 3.2 3. 2 Cons Co nseq eque uenc nces es of th thee AE AEP: P: Dat ataa Co Comp mpre ress ssio ion n 60 3.3 3. 3 High Hi gh-P -Pro roba babi bili lity ty Se Sets ts and th thee Typ ypic icaal Se Sett 62 Summ mmaary 64 Pro robl blem emss 64 Hist Hi stor oric ical al No Note tess 69
4
Entropy Rates of a Stochastic Process
71
4.1 4.2 4.3 4. 3
Markov Chains 71 Entropy Rate 74 Exam Ex ampl ple: e: En Entr trop opy y Ra Rate te of a Ra Rand ndom om Wal alk k on a Wei eigh ghte ted d Gr Grap aph h 78 4.4 Secon ond d Law of Thermodyn ynaamics 81 4.5 Functions of Markov Chains 84 Summ mmaary 87 Pro robl blem emss 88 Hist Hi stor oric ical al No Note tess 10 100 0
5
Data Compression 5.1 5.2 5.3 5.4 5.5 5. 5
Examples of Codes 103 Kraft Inequality 107 Optimal Codes 110 Bou Bo unds on the Optimal Co Cod de Length 112 Kraf Kr aftt In Ineq equa uali lity ty fo forr Un Uniq ique uely ly De Deco coda dabl blee Codes 115 5.6 Huffman Codes 118 5.7 Some Com omm ments on Huffman Codes 120 5.8 Optimality of Huffman Code dess 123 5.9 Shannon– Fano– Elias Coding 127 5.10 5. 10 Compe Com peti titi tive ve Op Opti tima mali lity ty of th thee Sh Shan annon non Code 130 5.11 5. 11 Gene Ge nera rati tion on of Di Disc scre rete te Di Dist stri ribut butio ions ns fro from m Fair Fa ir Co Coin inss 13 134 4 Sum umma mary ry 14 141 1 Pro robl blem emss 14 142 2 Hist Hi stor oric ical al No Note tess 15 157 7
103
CONTENTS
6
Gambling and Data Compression The Horse Race
6.2
Gamb mblling and Side Inf nfo ormation
6.3 6. 3
Depe De pend nden entt Hor orsse Ra Racces an and d En Entr trop opy y Ra Rate te
6.4
The Entropy of English
6.5
Data Co Com mpression and Gamb mblling
6.6 6. 6
Gamb Ga mbli ling ng Es Esttim imat atee of the En Entr trop opy y of En Engl glis ish h 175 17 5
Prob Pr oble lems ms
176 17 6
Hist Hi stor oric ical al No Note tess
7
159
6.1
Summ Su mmar ary y
159 164 166 16 6
168 171 173 17 3
182 18 2
Channel Capacity 7.1
183
Examples of Channe nell Capacity
184 18 4
7.1. 7. 1.1 1
Nois No isel eles esss Bi Bina nary ry Ch Chaann nnel el
7.1. 7. 1.2 2
Noisy Nois y Cha Channe nnell wi with th No Nono nove verl rlap appi ping ng Outp Ou tput utss 18 185 5
7.1.3
Noisy Typewriter
7.1. 7. 1.4 4
Bina Bi nary ry Sy Symm mmet etri ricc Ch Chan anne nell
7.1. 7. 1.5 5
Bina Bi nary ry Er Eras asur uree Ch Chan anne nell
184 18 4
186 18 6 187 18 7
188 18 8
7.2
Symmetric Channels
7.3
Properties of Ch Chaannel Capacity
7.4 7. 4
Prev Pr evie iew w of th thee Ch Chaann nnel el Co Codi ding ng Th Theo eore rem m
7.5
Definitions
7.6
Jointly Typ ypiical Sequences
195
7.7
Channel Coding Theorem
199
7.8
Zero-Error Codes
7.9 7. 9
Fano’s Fano ’s In Ineq equa uali lity ty an and d the the Co Conv nver erse se to th thee Cod Codin ing g Theo Th eore rem m 20 206 6
7.10 7. 10
Equalit Equal ity y in in the the Conv Conver erse se to th thee Chann Channel el Cod Codin ing g Theo Th eore rem m 20 208 8
7.11
Hamming Codes
7.12
Feedback Capacity
7.13 7. 13
Sour So urce– ce– Ch Chan anne nell Se Sepa para rati tion on Th Theo eore rem m
Summ Su mmar ary y
222 22 2
Prob Pr oble lems ms
223 22 3
Hist Hi stor oric ical al No Note tess
vii
189 191 191 19 1
192
240 24 0
205
210 216 218 21 8
viii
8
CONTENTS
Differential Entropy
243
8.1 8.2 8. 2 8.3 8. 3
Definitions 243 AEP AE P fo forr Co Cont ntin inuo uous us Ra Rand ndom om Var aria iabl bles es 24 245 5 Relat Rel ation ion of Di Diff ffer eren enti tial al En Entr tropy opy to Di Disc scre rete te Entr En trop opy y 24 247 7 8.4 8. 4 Join Jo intt an and d Co Cond ndit itio iona nall Di Difffe fere rent ntia iall En Entr trop opy y 24 249 9 8.5 8. 5 Rela Re lati tive ve En Entr trop opy y an and d Mu Mutu tual al In Info form rmaati tion on 25 250 0 8.6 8. 6 Prope Pr opert rtie iess of Di Diff ffer eren enti tial al En Entr trop opy, y, Rel Relat ativ ivee En Entr trop opy, y, and an d Mu Mutu tual al In Info form rmat atio ion n 25 252 2 Sum umma mary ry 25 256 6 Pro robl blem emss 25 256 6 Hist Hi stor oric ical al No Note tess 25 259 9
9
Gaussian Channel
261
9.1 9.1 9.2 9. 2
Gausssia Gaus ian n Ch Chan anne nel: l: Defi efini niti tion onss 26 263 3 Conv Co nver erse se to th thee Co Codi ding ng Th Theo eore rem m fo forr Ga Gaus ussi sian an Chan Ch anne nels ls 26 268 8 9.3 Bandlimited Channels 270 9.4 Parallel Gaussian Chann nneels 27 274 4 9.5 9. 5 Chan Ch anne nels ls wi with th Co Colo lore red d Ga Gaus ussi sian an Noi oisse 27 277 7 9.6 9. 6 Gaus Ga ussi sian an Ch Chan anne nels ls wi with th Fee eedb dbac ack k 28 280 0 Sum umma mary ry 28 289 9 Pro robl blem emss 29 290 0 Hist Hi stor oric ical al No Note tess 29 299 9
10 Rate Distortion Theory 10.1 10.2 10.3 10 .3
10.4 10.4 10.5 10 .5 10.6 10 .6 10.7 10 .7
Quantization 301 Definitions 303 Calc Ca lcul ulat atio ion n of th thee Ra Rate te Di Dist stor orti tion on Fu Func ncti tion on 30 307 7 10..3. 10 3.1 1 Bi Bina nary ry So Sour urce ce 30 307 7 10.3 10 .3..2 Ga Gaus usssia ian n So Sour urce ce 31 310 0 10.3.3 10. 3.3 Sim Simult ultane aneous ous Descr Descript iption ion of Indep Independ endent ent Gaussi Gau ssian an Ran Random dom Vari ariabl ables es 312 Conv Co nver erse se to th thee Ra Rate te Di Dist stor orti tion on Th Theo eore rem m 31 315 5 Achi Ac hiev evab abil ilit ity y of th thee Ra Rate te Di Dist stor orti tion on Fu Func ncti tion on 31 318 8 Stro St rongl ngly y Typi ypica call Se Sequ quen ence cess an and d Ra Rate te Di Dist stor orti tion on 325 Char Ch arac acte teri riza zati tion on of th thee Ra Rate te Di Dist stor orti tion on Fu Func ncti tion on 329
301
CONTENTS
ix
10.8 10. 8
Computat Comput atio ion n of Cha Channe nnell Cap Capac acit ity y and and th thee Rat Ratee Dist Di stor orti tion on Fu Func ncti tion on 332 Summ Su mmar ary y 33 335 5 Prob Pr oble lems ms 33 336 6 Hist Hi stor oric ical al No Note tess 34 345 5
11 Information Theory and Statistics
347
11.1 11.2 11..3 11 11..4 11 11.5 11 .5 11.6 11 .6 11..7 11 11. 1.8 8 11.9 11 .9 11.10 11. 10
Method of Types 347 Law of Large Numbers 355 Univ Un iver ersa sall So Sour urcce Co Codi ding ng 35 357 7 Larrge De La Devi viat atio ion n Th Theo eory ry 36 360 0 Exam Ex ampl ples es of Sa Sano nov’ v’ss Th Theeor oreem 36 364 4 Cond Co ndit itiion onaal Li Limi mitt Th Theo eore rem m 36 366 6 Hypo Hy potthe hessis Test stin ing g 37 375 5 Cheernoff– Stein Lemma 38 Ch 380 0 Cheern Ch rnof offf In Info form rmat atiion 38 384 4 Fish Fi sher er Info Inform rmat atio ion n and the the Cram Cramer–Rao e´ r–Rao Ineq In equa uali lity ty 39 392 2 Summ Su mmar ary y 39 397 7 Prob Pr oble lems ms 39 399 9 Hist Hi stor oric ical al No Note tess 40 408 8
12 Maximum Entropy
409
12.1 12.1 Maxi Ma ximu mum m En Entr trop opy y Di Dist stri ribu buti tion onss 40 409 9 12.2 Examples 411 12.3 12 .3 Anom An omal alou ouss Ma Maxi ximu mum m En Entr trop opy y Pr Prob oble lem m 41 413 3 12..4 12 Spec Sp ectr trum um Es Esti tima mattio ion n 41 415 5 12..5 12 Enttro En ropy py Ra Rattes of a Ga Gaus usssia ian n Pr Proc oces esss 41 416 6 12.6 12 .6 Burrg’ Bu g’ss Ma Maxi ximu mum m En Entr trop opy y Th Theo eore rem m 41 417 7 Summ Su mmar ary y 42 420 0 Prob Pr oble lems ms 42 421 1 Hist Hi stor oric ical al No Note tess 42 425 5
13 Universal Source Coding 13.1 13.1 13.2 13 .2 13. 3.3 3
Univer Univ ersa sall Co Code dess an and d Ch Chan anne nell Ca Capa paci city ty 428 428 Univ Un iver ersa sall Co Codi ding ng fo forr Bi Bina nary ry Se Sequ quen ence cess 43 433 3 Arithmetic Co Cod ding 436
427
x
CONTENTS
13.4
Lempel– Ziv Coding
440
13.4. 13. 4.1 1
13.5 13 .5
Slidin Slid ing g Win Window dow Le Lemp mpel– el– Zi Ziv v Algo Al gori rith thm m 44 441 1 13.4.2 13.4 .2 Tree ree-St -Struct ructured ured Lem Lempel– pel– Ziv Algo Al gori rith thms ms 44 442 2 Opti Op tima mali lity ty of Le Lemp mpel– el– Zi Ziv v Al Algo gori rith thms ms 13.5. 13. 5.1 1 13.5.2 13.5 .2
Sum umma mary ry
443 44 3
Slidin Slid ing g Win Window dow Le Lemp mpel– el– Zi Ziv v Algo Al gori rith thms ms 44 443 3 Optima Opt imalit lity y of Tree ree-St -Struc ructur tured ed Lempel– Lempel– Ziv Comp Co mpre ress ssio ion n 44 448 8
456 45 6
Pro robl blem emss 457 457 Hist Hi stor oric ical al No Note tess
461 46 1
14 Kolmogorov Complexity
463
14.1 14 .1
Mode Mo dels ls of Co Comp mput utaati tion on
14.2 14 .2
Kolmogo Kolm ogoro rov v Com Compl plex exit ity: y: De Defin finit itio ions ns and an d Ex Exam ampl ples es 46 466 6
14.3 14.3 14.4 14 .4
Kolmog Kolm ogor orov ov Co Comp mple lexi xity ty and and En Entr trop opy y 47 473 3 Kolm Ko lmog ogor orov ov Co Comp mple lexi xity ty of Inte Intege gers rs 475 475
14.5 14. 5
Algorithmi Algori thmical cally ly Rand Random om and Inc Incompr ompress essibl iblee Sequ Se quen ence cess 47 476 6
14.6 14.6 14.7 14 .7 14.8
Uni nive vers rsal al Pro roba babi bili lity ty 48 480 0 Kolm Ko lmog ogor orov ov com ompl pleexi xity ty 48 482 2 484
14.9 14.9 14.1 14 .10 0
Uni nive vers rsal al Gam ambl blin ing g 48 487 7 Occ ccam am’s ’s Ra Razo zorr 48 488 8
14.11 14. 11 14.1 14 .12 2
Kolmogorov Kolmogo rov Co Compl mplexi exity ty and and Unive Universa rsall Prob Pr obab abil ilit ity y 49 490 0 Kolm Ko lmogo ogoro rov v Su Suffi ffici cien entt St Stat atis isti ticc 49 496 6
14.1 14 .13 3
Mini Mi nimum mum De Desc scri ript ptio ion n Le Lengt ngth h Pr Prin inci cipl plee
Sum umma mary ry 50 501 1 Pro robl blem emss 503 503 Hist Hi stor oric ical al No Note tess
464 46 4
507 50 7
15 Network Information Theory 15.1 15 .1
500 50 0
Gaus Ga ussi sian an Mu Mult ltip iple le-U -Use serr Ch Chan anne nels ls
509 513 51 3
CONTENTS
15.1.1 15.1. 1 15.1.2 15.1 .2
15.2 15.2 15..3 15
15.4 15 .4
15.5 15. 5 15. 5.6 6
15.7 15.8 15 .8 15.9 15 .9
Single Sing le-U -Use serr Ga Gaus ussi sian an Ch Chan annel nel 513 Gaussi Gau ssian an Multi Multiple ple-Ac -Acces cesss Channe Channell with m Users 514 15.1. 15. 1.3 3 Ga Gaus ussi sian an Br Broa oadc dcas astt Cha Channe nnell 515 15.1 15 .1.4 .4 Ga Gaus ussi sian an Re Rela lay y Ch Chan anne nell 51 516 6 15.1. 15. 1.5 5 Ga Gaus ussi sian an In Inte terf rfer eren ence ce Cha Channe nnell 518 15.1. 15. 1.6 6 Ga Gaus ussi sian an Two wo-W -Way ay Cha Channe nnell 51 519 9 Join Jo intl tly y Typ ypic ical al Se Sequ quen ence cess 52 520 0 Mult Mu ltip iple le-A -Accce cess ss Ch Chan anne nell 52 524 4 15.3.1 15. 3.1 Ach Achiev ievabi abilit lity y of the Capac Capacity ity Regi Region on for the the Multip Mul tiplele-Acc Access ess Chan Channel nel 530 15.3. 15. 3.2 2 Com Comme ment ntss on the Capac Capacit ity y Region Region for the the Multip Mul tiplele-Acc Access ess Chan Channel nel 532 15.3. 15. 3.3 3 Con Conve vexi xity ty of the Capa Capaci city ty Regio Region n of the Multip Mul tiplele-Acc Access ess Chan Channel nel 534 15.3.4 15. 3.4 Conv Convers ersee for for the the Mult Multipl iple-A e-Acce ccess ss Chaann Ch nneel 53 538 8 15.3.5 m-Us -User er Mul Multip tiplele-Acc Access ess Chan Channel nelss 543 15.3. 15. 3.6 6 Ga Gaus ussi sian an Mu Mult ltip iple le-A -Acc cces esss Cha Chann nnel elss 54 544 4 Enco En codi ding ng of Co Corr rrel elat ated ed So Sour urce cess 54 549 9 15.4.1 15. 4.1 Ach Achiev ievabi abilit lity y of of the the Sle Slepia pian n – Wolf Theo Th eore rem m 55 551 1 15.4. 15. 4.2 2 Con Conve vers rsee for for the the Slepi Slepian– an– Wol olf f Theo Th eore rem m 55 555 5 15.4. 15. 4.3 3 Sl Slep epia ian n – Wol olff Theo Theore rem m for for Many Many Sour So urce cess 55 556 6 15.4.4 15. 4.4 Int Interp erpret retati ation on of Sle Slepia pian n – Wolf Codi ding ng 557 Dual Du alit ity y Be Betw twee een n Sl Slep epia ian n – Wol olff Enc Encod odin ing g and and Multip Mul tiplele-Acc Access ess Chan Channel nelss 558 Bro Br oadcast Ch Chaannel 56 560 0 15.6. 15. 6.1 1 De Defin finit itio ions ns fo forr a Br Broa oadc dcas astt Cha Chann nnel el 563 15.6. 15. 6.2 2 De Degra grade ded d Bro Broad adca cast st Ch Chan anne nels ls 56 564 4 15.6.3 15. 6.3 Capa Capacit city y Region Region for the the Degrad Degraded ed Broadc Broadcast ast Chaann Ch nneel 56 565 5 Relay Channel 571 Sour So urce ce Co Codi ding ng wi with th Si Side de In Info form rmat atio ion n 57 575 5 Rate Ra te Di Dist stor orti tion on wi with th Si Side de In Info form rmat atio ion n 58 580 0
xi
xii
CONTENTS
15.10 15.1 0 Ge Gene nera rall Mu Mult ltit iter ermi mina nall Ne Netw twor orks ks Sum umma mary ry 59 594 4 Pro robl blem emss 59 596 6 Hist Hi stor oric ical al No Note tess 60 609 9
587 58 7
16 Information Theory and Portfolio Theory
613
16.1 16.1 16.2 16 .2
The St The Stoc ock k Ma Mark rket et:: So Some me De Defin finit itio ions ns 61 613 3 Kuhn– Ku hn– Tuc ucke kerr Char Charac acte teri riza zati tion on of th thee LogLog-Op Opti tima mall Port Po rtfo foli lio o 61 617 7 16.3 16 .3 Asym As ympt ptot otic ic Op Opti tima mali lity ty of th thee Log Log-O -Opt ptim imal al Port Po rtfo foli lio o 61 619 9 16.4 16 .4 Side Si de In Info form rmat atio ion n an and d th thee Gr Grow owtth Ra Rate te 62 621 1 16.5 16 .5 Inve In vest stme ment nt in St Stat atio iona nary ry Ma Mark rket etss 62 623 3 16.6 16 .6 Compe Com peti titi tive ve Op Opti tima mali lity ty of th thee LogLog-Op Opti tima mall Port Po rtfo foli lio o 62 627 7 16.7 16 .7 Univ Un iver ersa sall Por orttfo follio ioss 62 629 9 16.7. 16. 7.1 1 Fi Fini nite te-H -Hor oriz izon on Un Univ iver ersa sall Po Port rtfol folio ioss 63 631 1 16.7. 16. 7.2 2 Ho Hori rizo zonn-Fr Free ee Un Univ iver ersa sall Po Port rtfo foli lios os 638 16.8 16 .8 Shan Sh anno non n – Mc McMi Mill llan– an– Br Brei eima man n Th Theo eore rem m (Gen (G ener eral al AE AEP) P) 64 644 4 Sum umma mary ry 65 650 0 Pro robl blem emss 65 652 2 Hist Hi stor oric ical al No Note tess 65 655 5
17 Inequalities in Information Theory 17.1 17.1 17.2 17 .2 17.3 17 .3 17.4 17 .4 17.5 17 .5 17.6 17 .6 17.7 17 .7 17.8 17 .8 17.9 17 .9
Basicc In Basi Ineq equa uali liti ties es of In Info form rmat atio ion n Th Theo eory ry 65 657 7 Difffe Di fere rent ntia iall En Enttro ropy py 66 660 0 Boun Bo unds ds on En Entr trop opy y an and d Re Rela lati tive ve En Entr trop opy y 66 663 3 Ineequ In quaali liti ties es fo forr Typ ypes es 66 665 5 Comb Co mbin inat ator oria iall Bo Boun unds ds on En Entr trop opy y 66 666 6 Enttro En ropy py Ra Rate tess of Su Subs bseets 66 667 7 Entr En trop opy y an and d Fi Fish sher er In Info form rmat atio ion n 67 671 1 Entrop Ent ropy y Po Powe werr In Ineq equa uali lity ty an and d Br Brunn– unn– Mi Minko nkows wski ki Ineq In equa uali lity ty 67 674 4 Ineq In equa uali liti ties es fo forr De Dete term rmin inan ants ts 67 679 9
657
CONTENTS
17.10 17.1 0 In Ineq equa uali liti ties es fo forr Rat Ratio ioss of of Det Deter ermi mina nant ntss Sum Su mmar ary y 686 Prob Pr oble lems ms 68 686 6 Hist Hi stor oric ical al No Note tess 68 687 7
xiii
683 68 3
Bibliography
689
List of Symbols
723
Index
727
