Information Theory, Inference, and Learning ... - Inference Group

More documents

Recommendations

Info

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.204 12 — Hash Codes: Codes for Efficient Information Retrievalon empirical experience of playing the game ‘guess that tune’, it seems tome that whereas many four-note sequences are shared in common betweenmelodies, the number of collisions between five-note sequences is rather smaller– most famous five-note sequences are unique.Solution to exercise 12.11 (p.201). (a) Let the DNA-binding protein recognizea sequence of length L nucleotides. That is, it binds preferentially to thatDNA sequence, and not to any other pieces of DNA in the whole genome. (Inreality, the recognized sequence may contain some wildcard characters, e.g.,the * in TATAA*A, which denotes ‘any of A, C, G and T’; so, to be precise, we areassuming that the recognized sequence contains L non-wildcard characters.)Assuming the rest of the genome is ‘random’, i.e., that the sequence consistsof random nucleotides A, C, G and T with equal probability – which isobviously untrue, but it shouldn’t make too much difference to our calculation– the chance that there is no other occurrence of the target sequence in thewhole genome, of length N nucleotides, is roughlywhich is close to one only ifthat is,(1 − (1/4) L ) N ≃ exp(−N(1/4) L ), (12.9)N4 −L ≪ 1, (12.10)L > log N/ log 4. (12.11)Using N = 3 × 10 9 , we require the recognized sequence to be longer thanL min = 16 nucleotides.What size of protein does this imply?• A weak lower bound can be obtained by assuming that the informationcontent of the protein sequence itself is greater than the informationcontent of the nucleotide sequence the protein prefers to bind to (whichwe have argued above must be at least 32 bits). This gives a minimumprotein length of 32/ log 2 (20) ≃ 7 amino acids.In guess that tune, one playerchooses a melody, and sings agradually-increasing number of itsnotes, while the other participantstry to guess the whole melody.The Parsons code is a related hashfunction for melodies: each pair ofconsecutive notes is coded as U(‘up’) if the second note is higherthan the first, R (‘repeat’) if thepitches are equal, and D (‘down’)otherwise. You can find out howwell this hash function works athttp://musipedia.org/.• Thinking realistically, the recognition of the DNA sequence by the proteinpresumably involves the protein coming into contact with all sixteennucleotides in the target sequence. If the protein is a monomer, it mustbe big enough that it can simultaneously make contact with sixteen nucleotidesof DNA. One helical turn of DNA containing ten nucleotideshas a length of 3.4 nm, so a contiguous sequence of sixteen nucleotideshas a length of 5.4 nm. The diameter of the protein must therefore beabout 5.4 nm or greater. Egg-white lysozyme is a small globular proteinwith a length of 129 amino acids and a diameter of about 4 nm. Assumingthat volume is proportional to sequence length and that volumescales as the cube of the diameter, a protein of diameter 5.4 nm musthave a sequence of length 2.5 × 129 ≃ 324 amino acids.(b) If, however, a target sequence consists of a twice-repeated sub-sequence, wecan get by with a much smaller protein that recognizes only the sub-sequence,and that binds to the DNA strongly only if it can form a dimer, both halvesof which are bound to the recognized sequence. Halving the diameter of theprotein, we now only need a protein whose length is greater than 324/8 = 40amino acids. A protein of length smaller than this cannot by itself serve asa regulatory protein specific to one gene, because it’s simply too small to beable to make a sufficiently specific match – its available surface does not haveenough information content.
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.About Chapter 13In Chapters 8–11, we established Shannon’s noisy-channel coding theoremfor a general channel with any input and output alphabets. A great deal ofattention in coding theory focuses on the special case of channels with binaryinputs. Constraining ourselves to these channels simplifies matters, and leadsus into an exceptionally rich world, which we will only taste in this book.One of the aims of this chapter is to point out a contrast between Shannon’saim of achieving reliable communication over a noisy channel and the apparentaim of many in the world of coding theory. Many coding theorists take astheir fundamental problem the task of packing as many spheres as possible,with radius as large as possible, into an N-dimensional space, with no spheresoverlapping. Prizes are awarded to people who find packings that squeeze in anextra few spheres. While this is a fascinating mathematical topic, we shall seethat the aim of maximizing the distance between codewords in a code has onlya tenuous relationship to Shannon’s aim of reliable communication.205
Page 1 and 2:
Copyright Cambridge University Pres
Page 3 and 4:
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167: Copyright Cambridge University Pres
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 316 and 317:
Page 318 and 319:
Page 320 and 321:
Page 322 and 323:
Page 324 and 325:
Page 326 and 327:
Page 328 and 329:
Page 330 and 331:
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
Page 338 and 339:
Page 340 and 341:
Page 342 and 343:
Page 344 and 345:
Page 346 and 347:
Page 348 and 349:
Page 350 and 351:
Page 352 and 353:
Page 354 and 355:
Page 356 and 357:
Page 358 and 359:
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
Page 390 and 391:
Page 392 and 393:
Page 394 and 395:
Page 396 and 397:
Page 398 and 399:
Page 400 and 401:
Page 402 and 403:
Page 404 and 405:
Page 406 and 407:
Page 408 and 409:
Page 410 and 411:
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
Page 444 and 445:
Page 446 and 447:
Page 448 and 449:
Page 450 and 451:
Page 452 and 453:
Page 454 and 455:
Page 456 and 457:
Page 458 and 459:
Page 460 and 461:
Page 462 and 463:
Page 464 and 465:
Page 466 and 467:
Page 468 and 469:
Page 470 and 471:
Page 472 and 473:
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
¡¤¢¢¤¨©¢£¡¢£¨¢£ ©Co
Page 516 and 517:
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
Page 524 and 525:
Page 526 and 527:
Page 528 and 529:
Page 530 and 531:
Page 532 and 533:
Page 534 and 535:
Page 536 and 537:
Page 538 and 539:
Page 540 and 541:
Page 542 and 543:
Page 544 and 545:
Page 546 and 547:
Page 548 and 549:
Page 550 and 551:
Page 552 and 553:
Page 554 and 555:
ttttCopyright Cambridge University
Page 556 and 557:
Page 558 and 559:
Page 560 and 561:
Page 562 and 563:
Page 564 and 565:
Page 566 and 567:
Page 568 and 569:
Page 570 and 571:
Page 572 and 573:
Page 574 and 575:
Page 576 and 577:
Page 578 and 579:
Page 580 and 581:
Page 582 and 583:
Page 584 and 585:
Page 586 and 587:
Page 588 and 589:
Page 590 and 591:
Page 592 and 593:
Page 594 and 595:
Page 596 and 597:
Page 598 and 599:
Page 600 and 601:
Page 602 and 603:
Page 604 and 605:
Page 606 and 607:
Page 608 and 609:
Page 610 and 611:
Page 612 and 613:
Page 614 and 615:
Page 616 and 617:
Page 618 and 619:
Page 620 and 621:
Page 622 and 623:
Page 624 and 625:
Page 626 and 627:
Page 628 and 629:
Page 630 and 631:
Page 632 and 633:
Page 634 and 635:
Page 636 and 637:
Page 638 and 639:
Page 640:
show all

Information Theory, Inference, and Learning ... - Inference Group

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?