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.460 37 — Bayesian Inference and Sampling TheoryCamp two runs a finger across the χ 2 table found at the back of any goodsampling theory book and finds χ 2 .10 = 2.71. Interpolating between χ2 .10 andχ 2 .05, camp two reports ‘the p-value is p = 0.07’.Notice that this answer does not say how much more effective A is than B,it simply says that A is ‘significantly’ different from B. And here, ‘significant’means only ‘statistically significant’, not practically significant.The man in the street, reading the statement that ‘the treatment was significantlydifferent from the control (p = 0.07)’, might come to the conclusionthat ‘there is a 93% chance that the treatments differ in effectiveness’. Butwhat ‘p = 0.07’ actually means is ‘if you did this experiment many times, andthe two treatments had equal effectiveness, then 7% of the time you wouldfind a value of χ 2 more extreme than the one that happened here’. This hasalmost nothing to do with what we want to know, which is how likely it isthat treatment A is better than B.Let me through, I’m a BayesianOK, now let’s infer what we really want to know. We scrap the hypothesisthat the two treatments have exactly equal effectivenesses, since we do notbelieve it. There are two unknown parameters, p A+ and p B+ , which are theprobabilities that people given treatments A and B, respectively, contract thedisease.Given the data, we can infer these two probabilities, and we can answerquestions of interest by examining the posterior distribution.The posterior distribution isP (p A+ , p B+ | {F i }) = P ({F i} | p A+ , p B+ )P (p A+ , p B+ ). (37.11)P ({F i })The likelihood function isP ({F i } | p A+ , p B+ ) ==( ) ( )NAp F A+A+F pF A− NBA−p F B+B+ A+ F pF B−B−(37.12)B+( ) ( )30 10p 11A+p 29A− p 33B+p 7 B−. (37.13)What prior distribution should we use? The prior distribution gives us theopportunity to include knowledge from other experiments, or a prior beliefthat the two parameters p A+ and p B+ , while different from each other, areexpected to have similar values.Here we will use the simplest vanilla prior distribution, a uniform distributionover each parameter.P (p A+ , p B+ ) = 1. (37.14)We can now plot the posterior distribution. Given the assumption of a separableprior on p A+ and p B+ , the posterior distribution is also separable:P (p A+ , p B+ | {F i }) = P (p A+ | F A+ , F A− )P (p B+ | F B+ , F B− ). (37.15)The two posterior distributions are shown in figure 37.1 (except the graphsare not normalized) and the joint posterior probability is shown in figure 37.2.If we want to know the answer to the question ‘how probable is it that p A+is smaller than p B+ ?’, we can answer exactly that question by computing theposterior probabilityP (p A+ < p B+ | Data), (37.16)
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.37.1: A medical example 461Figure 37.1. Posteriorprobabilities of the twoeffectivenesses. Treatment A –solid line; B – dotted line.0 0.2 0.4 0.6 0.8 1p B+0 0.2 0.4 0.6 0.8 110.80.60.40.20p A+0 0.2 0.4 0.6 0.8 1Figure 37.2. Joint posteriorprobability of the twoeffectivenesses – contour plot andsurface plot.0 0.20.40.60.81which is the integral of the joint posterior probability P (p A+ , p B+ | Data)shown in figure 37.2 over the region in which p A+ < p B+ , i.e., the shadedtriangle in figure 37.3. The value of this integral (obtained by a straightforwardnumerical integration of the likelihood function (37.13) over the relevantregion) is P (p A+
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:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
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: Copyright Cambridge University Pres
Page 514 and 515: ¡¤¢¢¤¨©¢£¡¢£¨¢£ ©Co
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?