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0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 35 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 32 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 30 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 27 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 22 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 21 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 18 0 R] +endobj + +16 0 obj +<< + /Type /StructElem + /S /Document + /P 10 0 R + /K [412 0 R 411 0 R 404 0 R 402 0 R 401 0 R 397 0 R 395 0 R 394 0 R 388 0 R 387 0 R 386 0 R 385 0 R 384 0 R 382 0 R 381 0 R 379 0 R 355 0 R 351 0 R 345 0 R 344 0 R 343 0 R 339 0 R 336 0 R 335 0 R 334 0 R 327 0 R 324 0 R 281 0 R 278 0 R 276 0 R 275 0 R 271 0 R 270 0 R 260 0 R 258 0 R 247 0 R 196 0 R 195 0 R 190 0 R 186 0 R 185 0 R 184 0 R 183 0 R 182 0 R 178 0 R 177 0 R 176 0 R 175 0 R 174 0 R 172 0 R 168 0 R 167 0 R 164 0 R 162 0 R 160 0 R 159 0 R 157 0 R 136 0 R 134 0 R 131 0 R 129 0 R 128 0 R 127 0 R 106 0 R 105 0 R 103 0 R 102 0 R 101 0 R 99 0 R 97 0 R 96 0 R 93 0 R 91 0 R 90 0 R 86 0 R 81 0 R 80 0 R 75 0 R 54 0 R 51 0 R 46 0 R 25 0 R 22 0 R 20 0 R 19 0 R 17 0 R] +>> +endobj + +17 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [18 0 R] +>> +endobj + +18 0 obj +<< + /Type /StructElem + /S /Strong + /P 17 0 R + /K [794] + /Pg 462 0 R +>> +endobj + +19 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793] + /Pg 462 0 R +>> +endobj + +20 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [21 0 R] +>> +endobj + +21 0 obj +<< + /Type /StructElem + /S /Strong + /P 20 0 R + /K [740] + /Pg 462 0 R +>> +endobj + +22 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [24 0 R 706 23 0 R] + /Pg 462 0 R +>> +endobj + +23 0 obj +<< + /Type /StructElem + /S /Formula + /P 22 0 R + /K [707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739] + /Pg 462 0 R +>> +endobj + +24 0 obj +<< + /Type /StructElem + /S /Formula + /P 22 0 R + /K [667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705] + /Pg 462 0 R +>> +endobj + +25 0 obj +<< + /Type /StructElem + /S /L + /P 16 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [41 0 R 36 0 R 31 0 R 26 0 R] +>> +endobj + +26 0 obj +<< + /Type /StructElem + /S /LI + /P 25 0 R + /K [30 0 R 27 0 R] +>> +endobj + +27 0 obj +<< + /Type /StructElem + /S /LBody + /P 26 0 R + /K [29 0 R 644 28 0 R] + /Pg 462 0 R +>> +endobj + +28 0 obj +<< + /Type /StructElem + /S /Formula + /P 27 0 R + /K [645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666] + /Pg 462 0 R +>> +endobj + +29 0 obj +<< + /Type /StructElem + /S /Formula + /P 27 0 R + /K [637 638 639 640 641 642 643] + /Pg 462 0 R +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /Lbl + /P 26 0 R + /K [636] + /Pg 462 0 R +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /LI + /P 25 0 R + /K [35 0 R 32 0 R] +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /LBody + /P 31 0 R + /K [34 0 R 572 33 0 R] + /Pg 462 0 R +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /Formula + /P 32 0 R + /K [573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635] + /Pg 462 0 R +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /Formula + /P 32 0 R + /K [565 566 567 568 569 570 571] + /Pg 462 0 R +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /Lbl + /P 31 0 R + /K [564] + /Pg 462 0 R +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /LI + /P 25 0 R + /K [40 0 R 37 0 R] +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /LBody + /P 36 0 R + /K [39 0 R 541 38 0 R] + /Pg 462 0 R +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Formula + /P 37 0 R + /K [542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563] + /Pg 462 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /Formula + /P 37 0 R + /K [534 535 536 537 538 539 540] + /Pg 462 0 R +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /Lbl + /P 36 0 R + /K [533] + /Pg 462 0 R +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /LI + /P 25 0 R + /K [45 0 R 42 0 R] +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /LBody + /P 41 0 R + /K [44 0 R 469 43 0 R] + /Pg 462 0 R +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /Formula + /P 42 0 R + /K [470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532] + /Pg 462 0 R +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /Formula + /P 42 0 R + /K [462 463 464 465 466 467 468] + /Pg 462 0 R +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /Lbl + /P 41 0 R + /K [461] + /Pg 462 0 R +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [50 0 R 432 49 0 R 440 48 0 R 455 47 0 R 460] + /Pg 462 0 R +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /Formula + /P 46 0 R + /K [456 457 458 459] + /Pg 462 0 R +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /Formula + /P 46 0 R + /K [441 442 443 444 445 446 447 448 449 450 451 452 453 454] + /Pg 462 0 R +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /Formula + /P 46 0 R + /K [433 434 435 436 437 438 439] + /Pg 462 0 R +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /Strong + /P 46 0 R + /K [431] + /Pg 462 0 R +>> +endobj + +51 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [53 0 R 397 52 0 R] + /Pg 462 0 R +>> +endobj + +52 0 obj +<< + /Type /StructElem + /S /Formula + /P 51 0 R + /K [398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430] + /Pg 462 0 R +>> +endobj + +53 0 obj +<< + /Type /StructElem + /S /Formula + /P 51 0 R + /K [358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396] + /Pg 462 0 R +>> +endobj + +54 0 obj +<< + /Type /StructElem + /S /L + /P 16 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [70 0 R 65 0 R 60 0 R 55 0 R] +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /LI + /P 54 0 R + /K [59 0 R 56 0 R] +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /LBody + /P 55 0 R + /K [58 0 R 335 57 0 R] + /Pg 462 0 R +>> +endobj + +57 0 obj +<< + /Type /StructElem + /S /Formula + /P 56 0 R + /K [336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357] + /Pg 462 0 R +>> +endobj + +58 0 obj +<< + /Type /StructElem + /S /Formula + /P 56 0 R + /K [328 329 330 331 332 333 334] + /Pg 462 0 R +>> +endobj + +59 0 obj +<< + /Type /StructElem + /S /Lbl + /P 55 0 R + /K [327] + /Pg 462 0 R +>> +endobj + +60 0 obj +<< + /Type /StructElem + /S /LI + /P 54 0 R + /K [64 0 R 61 0 R] +>> +endobj + +61 0 obj +<< + /Type /StructElem + /S /LBody + /P 60 0 R + /K [63 0 R 263 62 0 R] + /Pg 462 0 R +>> +endobj + +62 0 obj +<< + /Type /StructElem + /S /Formula + /P 61 0 R + /K [264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326] + /Pg 462 0 R +>> +endobj + +63 0 obj +<< + /Type /StructElem + /S /Formula + /P 61 0 R + /K [256 257 258 259 260 261 262] + /Pg 462 0 R +>> +endobj + +64 0 obj +<< + /Type /StructElem + /S /Lbl + /P 60 0 R + /K [255] + /Pg 462 0 R +>> +endobj + +65 0 obj +<< + /Type /StructElem + /S /LI + /P 54 0 R + /K [69 0 R 66 0 R] +>> +endobj + +66 0 obj +<< + /Type /StructElem + /S /LBody + /P 65 0 R + /K [68 0 R 232 67 0 R] + /Pg 462 0 R +>> +endobj + +67 0 obj +<< + /Type /StructElem + /S /Formula + /P 66 0 R + /K [233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254] + /Pg 462 0 R +>> +endobj + +68 0 obj +<< + /Type /StructElem + /S /Formula + /P 66 0 R + /K [225 226 227 228 229 230 231] + /Pg 462 0 R +>> +endobj + +69 0 obj +<< + /Type /StructElem + /S /Lbl + /P 65 0 R + /K [224] + /Pg 462 0 R +>> +endobj + +70 0 obj +<< + /Type /StructElem + /S /LI + /P 54 0 R + /K [74 0 R 71 0 R] +>> +endobj + +71 0 obj +<< + /Type /StructElem + /S /LBody + /P 70 0 R + /K [73 0 R 160 72 0 R] + /Pg 462 0 R +>> +endobj + +72 0 obj +<< + /Type /StructElem + /S /Formula + /P 71 0 R + /K [161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223] + /Pg 462 0 R +>> +endobj + +73 0 obj +<< + /Type /StructElem + /S /Formula + /P 71 0 R + /K [153 154 155 156 157 158 159] + /Pg 462 0 R +>> +endobj + +74 0 obj +<< + /Type /StructElem + /S /Lbl + /P 70 0 R + /K [152] + /Pg 462 0 R +>> +endobj + +75 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [79 0 R 123 78 0 R 131 77 0 R 146 76 0 R 151] + /Pg 462 0 R +>> +endobj + +76 0 obj +<< + /Type /StructElem + /S /Formula + /P 75 0 R + /K [147 148 149 150] + /Pg 462 0 R +>> +endobj + +77 0 obj +<< + /Type /StructElem + /S /Formula + /P 75 0 R + /K [132 133 134 135 136 137 138 139 140 141 142 143 144 145] + /Pg 462 0 R +>> +endobj + +78 0 obj +<< + /Type /StructElem + /S /Formula + /P 75 0 R + /K [124 125 126 127 128 129 130] + /Pg 462 0 R +>> +endobj + +79 0 obj +<< + /Type /StructElem + /S /Strong + /P 75 0 R + /K [122] + /Pg 462 0 R +>> +endobj + +80 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121] + /Pg 462 0 R +>> +endobj + +81 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [0 85 0 R 2 84 0 R 17 83 0 R 32 33 82 0 R 77 78] + /Pg 462 0 R +>> +endobj + +82 0 obj +<< + /Type /StructElem + /S /Formula + /P 81 0 R + /K [34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76] + /Pg 462 0 R +>> +endobj + +83 0 obj +<< + /Type /StructElem + /S /Formula + /P 81 0 R + /K [18 19 20 21 22 23 24 25 26 27 28 29 30 31] + /Pg 462 0 R +>> +endobj + +84 0 obj +<< + /Type /StructElem + /S /Formula + /P 81 0 R + /K [3 4 5 6 7 8 9 10 11 12 13 14 15 16] + /Pg 462 0 R +>> +endobj + +85 0 obj +<< + /Type /StructElem + /S /Formula + /P 81 0 R + /K [1] + /Pg 462 0 R +>> +endobj + +86 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [89 0 R 87 0 R] +>> +endobj + +87 0 obj +<< + /Type /StructElem + /S /Formula + /P 86 0 R + /K [88 0 R] +>> +endobj + +88 0 obj +<< + /Type /StructElem + /S /Strong + /P 87 0 R + /K [391 392 393 394 395 396 397 398 399 400 401 402 403 404] + /Pg 460 0 R +>> +endobj + +89 0 obj +<< + /Type /StructElem + /S /Strong + /P 86 0 R + /K [390] + /Pg 460 0 R +>> +endobj + +90 0 obj +<< + /Type /StructElem + /S /Span + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [385 386 387 388 389] + /Pg 460 0 R +>> +endobj + +91 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [92 0 R] +>> +endobj + +92 0 obj +<< + /Type /StructElem + /S /Strong + /P 91 0 R + /K [384] + /Pg 460 0 R +>> +endobj + +93 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [95 0 R 353 354 355 356 357 358 359 360 361 362 363 94 0 R 383] + /Pg 460 0 R +>> +endobj + +94 0 obj +<< + /Type /StructElem + /S /Formula + /P 93 0 R + /K [364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382] + /Pg 460 0 R +>> +endobj + +95 0 obj +<< + /Type /StructElem + /S /Strong + /P 93 0 R + /K [352] + /Pg 460 0 R +>> +endobj + +96 0 obj +<< + /Type /StructElem + /S /H1 + /P 16 0 R + /T + /K [350 351] + /Pg 460 0 R +>> +endobj + +97 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [98 0 R] +>> +endobj + +98 0 obj +<< + /Type /StructElem + /S /Strong + /P 97 0 R + /K [349] + /Pg 460 0 R +>> +endobj + +99 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [314 100 0 R 348] + /Pg 460 0 R +>> +endobj + +100 0 obj +<< + /Type /StructElem + /S /Formula + /P 99 0 R + /K [315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347] + /Pg 460 0 R +>> +endobj + +101 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313] + /Pg 460 0 R +>> +endobj + +102 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286] + /Pg 460 0 R +>> +endobj + +103 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [245 104 0 R 256 257 258] + /Pg 460 0 R +>> +endobj + +104 0 obj +<< + /Type /StructElem + /S /Formula + /P 103 0 R + /K [246 247 248 249 250 251 252 253 254 255] + /Pg 460 0 R +>> +endobj + +105 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244] + /Pg 460 0 R +>> +endobj + +106 0 obj +<< + /Type /StructElem + /S /L + /P 16 0 R + /A [<< + /O /List + /ListNumbering /Decimal + >>] + /K [122 0 R 117 0 R 112 0 R 107 0 R] +>> +endobj + +107 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [111 0 R 108 0 R] +>> +endobj + +108 0 obj +<< + /Type /StructElem + /S /LBody + /P 107 0 R + /K [110 0 R 191 109 0 R] + /Pg 460 0 R +>> +endobj + +109 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208] + /Pg 460 0 R +>> +endobj + +110 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [184 185 186 187 188 189 190] + /Pg 460 0 R +>> +endobj + +111 0 obj +<< + /Type /StructElem + /S /Lbl + /P 107 0 R + /K [183] + /Pg 460 0 R +>> +endobj + +112 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [116 0 R 113 0 R] +>> +endobj + +113 0 obj +<< + /Type /StructElem + /S /LBody + /P 112 0 R + /K [115 0 R 165 114 0 R] + /Pg 460 0 R +>> +endobj + +114 0 obj +<< + /Type /StructElem + /S /Formula + /P 113 0 R + /K [166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182] + /Pg 460 0 R +>> +endobj + +115 0 obj +<< + /Type /StructElem + /S /Formula + /P 113 0 R + /K [158 159 160 161 162 163 164] + /Pg 460 0 R +>> +endobj + +116 0 obj +<< + /Type /StructElem + /S /Lbl + /P 112 0 R + /K [157] + /Pg 460 0 R +>> +endobj + +117 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [121 0 R 118 0 R] +>> +endobj + +118 0 obj +<< + /Type /StructElem + /S /LBody + /P 117 0 R + /K [120 0 R 139 119 0 R] + /Pg 460 0 R +>> +endobj + +119 0 obj +<< + /Type /StructElem + /S /Formula + /P 118 0 R + /K [140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156] + /Pg 460 0 R +>> +endobj + +120 0 obj +<< + /Type /StructElem + /S /Formula + /P 118 0 R + /K [132 133 134 135 136 137 138] + /Pg 460 0 R +>> +endobj + +121 0 obj +<< + /Type /StructElem + /S /Lbl + /P 117 0 R + /K [131] + /Pg 460 0 R +>> +endobj + +122 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [126 0 R 123 0 R] +>> +endobj + +123 0 obj +<< + /Type /StructElem + /S /LBody + /P 122 0 R + /K [125 0 R 113 124 0 R] + /Pg 460 0 R +>> +endobj + +124 0 obj +<< + /Type /StructElem + /S /Formula + /P 123 0 R + /K [114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130] + /Pg 460 0 R +>> +endobj + +125 0 obj +<< + /Type /StructElem + /S /Formula + /P 123 0 R + /K [106 107 108 109 110 111 112] + /Pg 460 0 R +>> +endobj + +126 0 obj +<< + /Type /StructElem + /S /Lbl + /P 122 0 R + /K [105] + /Pg 460 0 R +>> +endobj + +127 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [100 101 102 103 104] + /Pg 460 0 R +>> +endobj + +128 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O 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190 191] + /Pg 456 0 R +>> +endobj + +286 0 obj +<< + /Type /StructElem + /S /L + /P 283 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [305 0 R 300 0 R 293 0 R 287 0 R] +>> +endobj + +287 0 obj +<< + /Type /StructElem + /S /LI + /P 286 0 R + /K [292 0 R 288 0 R] +>> +endobj + +288 0 obj +<< + /Type /StructElem + /S /LBody + /P 287 0 R + /K [157 291 0 R 159 290 0 R 161 162 163 164 289 0 R 166] + /Pg 456 0 R +>> +endobj + +289 0 obj +<< + /Type /StructElem + /S /Formula + /P 288 0 R + /K [165] + /Pg 456 0 R +>> +endobj + +290 0 obj +<< + /Type /StructElem + /S /Formula + /P 288 0 R + /K [160] + /Pg 456 0 R +>> +endobj + +291 0 obj +<< + /Type /StructElem + /S /Formula + /P 288 0 R + /K [158] + /Pg 456 0 R +>> +endobj + +292 0 obj +<< + /Type /StructElem + /S /Lbl + /P 287 0 R + /K [156] + /Pg 456 0 R +>> +endobj + +293 0 obj +<< + /Type /StructElem + /S /LI + /P 286 0 R + /K [299 0 R 294 0 R] +>> +endobj + +294 0 obj +<< + /Type /StructElem + /S /LBody + /P 293 0 R + /K 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/Pg 454 0 R +>> +endobj + +345 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [294 350 0 R 307 349 0 R 312 313 314 348 0 R 316 317 318 347 0 R 320 346 0 R 322] + /Pg 454 0 R +>> +endobj + +346 0 obj +<< + /Type /StructElem + /S /Formula + /P 345 0 R + /K [321] + /Pg 454 0 R +>> +endobj + +347 0 obj +<< + /Type /StructElem + /S /Formula + /P 345 0 R + /K [319] + /Pg 454 0 R +>> +endobj + +348 0 obj +<< + /Type /StructElem + /S /Formula + /P 345 0 R + /K [315] + /Pg 454 0 R +>> +endobj + +349 0 obj +<< + /Type /StructElem + /S /Formula + /P 345 0 R + /K [308 309 310 311] + /Pg 454 0 R +>> +endobj + +350 0 obj +<< + /Type /StructElem + /S /Formula + /P 345 0 R + /K [295 296 297 298 299 300 301 302 303 304 305 306] + /Pg 454 0 R +>> +endobj + +351 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [354 0 R 352 0 R] +>> +endobj + +352 0 obj +<< + /Type /StructElem + /S /Formula + /P 351 0 R + /K [353 0 R] +>> +endobj + +353 0 obj +<< + /Type /StructElem + /S /Strong + /P 352 0 R 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0 R + /A [<< + /O /Table + /Headers [] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [277] + /Pg 454 0 R +>> +endobj + +367 0 obj +<< + /Type /StructElem + /S /TD + /P 363 0 R + /A [<< + /O /Table + /Headers [] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [276] + /Pg 454 0 R +>> +endobj + +368 0 obj +<< + /Type /StructElem + /S /TD + /P 363 0 R + /A [<< + /O /Table + /Headers [] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [275] + /Pg 454 0 R +>> +endobj + +369 0 obj +<< + /Type /StructElem + /S /TR + /P 355 0 R + /K [378 0 R 377 0 R 375 0 R 373 0 R 370 0 R] +>> +endobj + +370 0 obj +<< + /Type /StructElem + /S /TD + /P 369 0 R + /A [<< + /O /Table + /Headers [] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [372 0 R 272 371 0 R 274] + /Pg 454 0 R +>> +endobj + +371 0 obj +<< + /Type /StructElem + /S /Formula + /P 370 0 R + /K [273] + /Pg 454 0 R +>> +endobj + +372 0 obj +<< + /Type /StructElem + /S /Formula + /P 370 0 R + /K [271] + /Pg 454 0 R +>> 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0 R 263 264] + /Pg 454 0 R +>> +endobj + +380 0 obj +<< + /Type /StructElem + /S /Strong + /P 379 0 R + /K [262] + /Pg 454 0 R +>> +endobj + +381 0 obj +<< + /Type /StructElem + /S /H1 + /P 16 0 R + /T + /K [260 261] + /Pg 454 0 R +>> +endobj + +382 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [383 0 R] +>> +endobj + +383 0 obj +<< + /Type /StructElem + /S /Strong + /P 382 0 R + /K [259] + /Pg 454 0 R +>> +endobj + +384 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258] + /Pg 454 0 R +>> +endobj + +385 0 obj +<< + /Type /StructElem + /S /P + /P 16 0 R + /K [222] + /Pg 454 0 R +>> +endobj + +386 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 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0 R + /ID [(56cDPmhV9s3zbCLrJhRD3g==) (56cDPmhV9s3zbCLrJhRD3g==)] +>> +startxref +213490 +%%EOF \ No newline at end of file diff --git a/bayesian/main.typ b/bayesian/main.typ new file mode 100644 index 0000000..e9f5b1d --- /dev/null +++ b/bayesian/main.typ @@ -0,0 +1,289 @@ +#import "@preview/cetz:0.3.3" + +#set text(font: ("Noto Sans CJK SC", "Noto Serif CJK SC"), lang: "zh") +#set page(paper: "a4", margin: (x: 2cm, y: 2cm)) +#set heading(numbering: "1.") + +#let answer(body) = block( + fill: luma(240), + stroke: (left: 2pt + blue), + inset: 10pt, + radius: 2pt, + width: 100%, + body +) + +#align(center)[ + #text(size: 18pt, weight: "bold")[第13-14章 概率与贝叶斯网络 习题解答] + + #v(1em) + // *生成时间:* #datetime.today().display() +] + += 题目 1 +*题意:* 已知 $P(a) = 0.3$, $P(b | a) = 0.2$, $P(c | a) = 0.5$。比较 $P(a and b)$ 与 $P(a and c)$。 + +#answer[ + 根据乘法公式 $P(x and y) = P(x)P(y|x)$: + $ P(a and b) &= P(a) dot P(b | a) = 0.3 times 0.2 = 0.06 \ + P(a and c) &= P(a) dot P(c | a) = 0.3 times 0.5 = 0.15 $ + 因为 $0.06 < 0.15$,所以 *$P(a and b) < P(a and c)$*。 + + *答案:B* +] + += 题目 2 +*题意:* $P(a or b)=0.7$, $P(a)=0.4$, $P(b)=0.5$,求 $P(a and b)$。 + +#answer[ + 根据概率加法公式(容斥原理): + $ P(a or b) = P(a) + P(b) - P(a and b) $ + 代入已知数值: + $ 0.7 = 0.4 + 0.5 - P(a and b) \ + P(a and b) = 0.9 - 0.7 = 0.2 $ + + *答案:0.2* +] + += 题目 3 +*数据:* 全联合概率分布表如下(Cavity, Toothache, Catch): + +#table( + columns: (auto, auto, auto, auto, auto), + inset: 8pt, + align: center, + [], [T, Catch], [T, $not$ Catch], [$not$ T, Catch], [$not$ T, $not$ Catch], + [Cavity], [0.108], [0.012], [0.072], [0.008], + [$not$ Cavity], [0.016], [0.064], [0.144], [0.576] +) + +#answer[ + *1. 计算 $P("Toothache" or not "Cavity")$* + + 利用补集思想:$P(A) = 1 - P(not A)$。 + 事件的补集是 $not "Toothache" and "Cavity"$。 + 对应表格中的项为:(Cavity, $not$ T, Catch) 和 (Cavity, $not$ T, $not$ Catch)。 + $ P(not T and "Cav") = 0.072 + 0.008 = 0.080 $ + $ P(T or not "Cav") = 1 - 0.080 = 0.92 $ + + *2. 计算 $P(not "toothache" or "catch")$* + + 补集是 $"toothache" and not "catch"$。 + 对应表格列为 (T, $not$ Catch),即第 2 列。 + $ P(T and not "Catch") = 0.012 ("Cav") + 0.064 (not "Cav") = 0.076 $ + $ P(not T or "Catch") = 1 - 0.076 = 0.924 $ + + *3. 计算 $P("Toothache" or not "Cavity" | not "catch")$ 和 $P(not "catch" | "Toothache" or not "Cavity")$* + + 令 $A = "Toothache" or not "Cavity"$, $B = not "catch"$。 + + - *计算 $P(B)$*: Sum of all $not$ Catch columns (Col 2, Col 4). + $P(B) = (0.012 + 0.064) + (0.008 + 0.576) = 0.076 + 0.584 = 0.66$ + - *计算 $P(A and B)$*: 在 $not$ Catch 列中满足 $T or not "Cav"$ 的项。 + - (Cav, T, $not$ C): 0.012 (满足 T) + - ($not$ Cav, T, $not$ C): 0.064 (满足 T) + - ($not$ Cav, $not$ T, $not$ C): 0.576 (满足 $not$ Cav) + - (Cav, $not$ T, $not$ C): 0.008 (不满足,既无T也无$not$Cav) + Sum $= 0.012 + 0.064 + 0.576 = 0.652$ + + 结果 1: $P(A | B) = 0.652 / 0.66 approx 0.9879$ + 结果 2: $P(B | A) = P(A and B) / P(A) = 0.652 / 0.92 approx 0.7087$ (其中 P(A) 来自第1小问) + + *4. 判断 Cavity 与 Toothache 是否独立* + + $ P("Cav") &= 0.108+0.012+0.072+0.008 = 0.2 \ + P("Toothache") &= 0.108+0.012+0.016+0.064 = 0.2 \ + P("Cav" and "T") &= 0.108 + 0.012 = 0.12 $ + 检验:$P("Cav") times P("T") = 0.2 times 0.2 = 0.04$。 + 因为 $0.12 != 0.04$,所以两者 *不独立*。 +] + += 题目 4 +*题意:* 设 $X, Y$ 条件独立于 $Z$(即 $X perp Y | Z$),$Y, W$ 条件独立于 $Z$(即 $Y perp W | Z$)。问 $X, W$ 是否条件独立于 $Z$? + +#answer[ + *结论:不能。* 条件独立性不具备传递性。 + + *反例证明:* + 假设 $Z$ 是一个公平的硬币投掷(0 或 1)。 + 令 $X = Z$($X$ 完全依赖于 $Z$)。 + 令 $W = Z$($W$ 完全依赖于 $Z$)。 + 令 $Y$ 是一个与 $X, W, Z$ 都完全独立的随机变量(例如掷骰子)。 + + 1. *检查 $X, Y$ 关于 $Z$ 的独立性*:给定 $Z$, $X$ 变为常数。常数与任何变量独立,故成立。 + 2. *检查 $Y, W$ 关于 $Z$ 的独立性*:同理,给定 $Z$,$W$ 变为常数,故成立。 + 3. *检查 $X, W$ 关于 $Z$ 的独立性*: + 给定 $Z$,$X$ 和 $W$ 的值完全确定且相同(例如若 $Z=1$,则 $X=1, W=1$)。虽然在 $Z$ 固定的情况下它们的方差为0(技术上可视作独立),但如果我们考虑一种因果结构: + 令 $X$ 和 $W$ 为同一变量的两个副本。显然它们是强相关的。 + + 更直观的例子:$X$ 和 $W$ 互为因果或由共同隐变量控制,而它们都与 $Y$ 独立。仅仅知道它们分别与 $Y$ 独立,无法切断 $X$ 和 $W$ 之间的联系。 +] + += 题目 5 +*题意:* 罕见病检测。$P(D)=0.001$。 +检测A:$P(+|D)=0.95, P(+|not D)=0.05$。 +检测B:$P(+|D)=0.90, P(+|not D)=0.10$。 + +#answer[ + *a. 单次检测 A 为阳性,求 $P(D|A+)$* + + 使用贝叶斯公式: + $ P(D|A+) = (P(A+|D)P(D)) / P(A+) $ + $ P(A+) &= P(A+|D)P(D) + P(A+|not D)P(not D) \ + &= 0.95 times 0.001 + 0.05 times 0.999 \ + &= 0.00095 + 0.04995 = 0.0509 $ + $ P(D|A+) &= 0.00095 / 0.0509 approx 0.0187 (1.87%) $ + + *b. 两次检测 A 和 B 均为阳性,求 $P(D|A+, B+)$* + + 由于检测独立(Naive Bayes 假设): + $ P(A+, B+ | D) = 0.95 times 0.90 = 0.855 $ + $ P(A+, B+ | not D) = 0.05 times 0.10 = 0.005 $ + + $ P(D | A+, B+) &= (P(A+, B+ | D)P(D)) / (P(A+, B+ | D)P(D) + P(A+, B+ | not D)P(not D)) \ + &= (0.855 times 0.001) / (0.855 times 0.001 + 0.005 times 0.999) \ + &= 0.000855 / (0.000855 + 0.004995) \ + &= 0.000855 / 0.00585 approx 0.1462 (14.62%) $ + + *c. 解释* + + 联合检测通过引入第二次独立测试,极大地降低了*假阳性率*(从 0.05 降至 $0.05 times 0.10 = 0.005$)。虽然真阳性率也略有下降,但分母中占主导地位的假阳性项(由 $P(not D)$ 权重放大)被大幅削减,从而显著提升了后验概率。 +] + += 题目 6 +*场景:* 骑车(B) 取决于 天气(W) 和 熬夜(S)。 +$P(W="Sun")=0.7, P(S="Yes")=0.4$。 +CPT 已知。 + +#answer[ + *a. 贝叶斯网络与 CPT* + + 结构:$W -> B <- S$ (V-structure) + + #cetz.canvas({ + import cetz.draw: * + + let r = 0.8 + // Nodes + circle((0, 0), radius: r, name: "B") + content("B", "B (骑车)") + + circle((-2, 3), radius: r, name: "W") + content("W", "W (天气)") + + circle((2, 3), radius: r, name: "S") + content("S", "S (熬夜)") + + // Edges + line("W", "B", mark: (end: ">")) + line("S", "B", mark: (end: ">")) + }) + + *条件概率表 (CPT) for B:* + #table( + columns: 3, + align: center, + [W (天气)], [S (熬夜)], [P(B=是 | W, S)], + [晴 (Sun)], [否 (No)], [0.9], + [晴 (Sun)], [是 (Yes)], [0.6], + [雨 (Rain)], [否 (No)], [0.2], + [雨 (Rain)], [是 (Yes)], [0.1] + ) + + *b. 推理:已知骑车(B=Yes),求天气概率* + + 我们需要计算 $P(W | B="Yes")$。根据贝叶斯法则:$P(W|B) = P(B|W)P(W) / P(B)$。 + + 首先计算边缘概率 $P(B="Yes")$。 + $ P(B=y) = sum_(w, s) P(B=y|w,s)P(w)P(s) $ + 因 W 和 S 独立,联合概率直接相乘。 + + 1. $W="Sun", S="No"$: $0.9 times 0.7 times 0.6 = 0.378$ + 2. $W="Sun", S="Yes"$: $0.6 times 0.7 times 0.4 = 0.168$ + 3. $W="Rain", S="No"$: $0.2 times 0.3 times 0.6 = 0.036$ + 4. $W="Rain", S="Yes"$: $0.1 times 0.3 times 0.4 = 0.012$ + + $ P(B="Yes") = 0.378 + 0.168 + 0.036 + 0.012 = 0.594 $ + + 计算 $P(W="Sun" | B="Yes")$: + 分子为 W=Sun 的所有情况之和(项1 + 项2): + $ P(B=y, W="Sun") = 0.378 + 0.168 = 0.546 $ + $ P(W="Sun" | B="Yes") = 0.546 / 0.594 approx 0.919 $ + + 同理,$P(W="Rain" | B="Yes") = (0.036 + 0.012) / 0.594 approx 0.081$。 + + *结论:这天最可能是晴天。* +] + += 题目 7 +*智能健康助手:* 变量 S(吸烟), G(基因), L(肺癌), C(咳嗽), X(胸片)。 +关系:$S->L, G->L, S->C, L->C, L->X$。 + +#answer[ + *问题 1:贝叶斯网络图* + + #align(center)[ + #cetz.canvas({ + import cetz.draw: * + // Manually positioning nodes for clarity + let r = 0.5 + circle((0, 0), radius: r, name: "L") + content("L", "L") + + circle((-2, 2), radius: r, name: "S") + content("S", "S") + + circle((2, 2), radius: r, name: "G") + content("G", "G") + + circle((-1, -2), radius: r, name: "C") + content("C", "C") + + circle((1, -2), radius: r, name: "X") + content("X", "X") + + line("S", "L", mark: (end: ">")) + line("G", "L", mark: (end: ">")) + line("L", "X", mark: (end: ">")) + line("L", "C", mark: (end: ">")) + + // S -> C 直接边 + line("S", "C", mark: (end: ">")) + }) + ] + + *问题 2:概率推理 $P(L="Yes" | C="No", X="Yes")$* + + 令 $alpha$ 为归一化常数。我们需要计算 $P(L=y, C=n, X=y)$ 和 $P(L=n, C=n, X=y)$。 + 公式分解:$P(S,G,L,C,X) = P(S)P(G)P(L|S,G)P(C|L,S)P(X|L)$。 + 求和消除 S, G: + $ P(L, C, X) = P(X|L) sum_S sum_G P(C|L,S) P(L|S,G) P(S) P(G) $ + + *Case 1: L = Yes* (且 $C=n, X=y$) + 因子 $P(X=y|L=y) = 0.9$。 + 内部求和 $Sigma_(L=y)$: + - $S=y, G=h$: $P(C=n|L=y,S=y)P(L=y|S=y,G=h)P(S=y)P(G=h) = 0.2 times 0.6 times 0.3 times 0.2 = 0.0072$ + - $S=y, G=l$: $0.2 times 0.3 times 0.3 times 0.8 = 0.0144$ + - $S=n, G=h$: $P(C=n|L=y,S=n)P(L=y|S=n,G=h)P(S=n)P(G=h) = 0.4 times 0.4 times 0.7 times 0.2 = 0.0224$ + - $S=n, G=l$: $0.4 times 0.1 times 0.7 times 0.8 = 0.0224$ + + $Sigma_(L=y) = 0.0072 + 0.0144 + 0.0224 + 0.0224 = 0.0664$ + $P(L=y, C=n, X=y) = 0.9 times 0.0664 = 0.05976$ + + *Case 2: L = No* (且 $C=n, X=y$) + 因子 $P(X=y|L=n) = 0.1$。 + 内部求和 $Sigma_(L=n)$: + - $S=y, G=h$: $P(C=n|L=n,S=y)P(L=n|S=y,G=h)P(S=y)P(G=h) = 0.7 times 0.4 times 0.3 times 0.2 = 0.0168$ + - $S=y, G=l$: $0.7 times 0.7 times 0.3 times 0.8 = 0.1176$ + - $S=n, G=h$: $P(C=n|L=n,S=n)P(L=n|S=n,G=h)P(S=n)P(G=h) = 0.9 times 0.6 times 0.7 times 0.2 = 0.0756$ + - $S=n, G=l$: $0.9 times 0.9 times 0.7 times 0.8 = 0.4536$ + + $Sigma_(L=n) = 0.0168 + 0.1176 + 0.0756 + 0.4536 = 0.6636$ + $P(L=n, C=n, X=y) = 0.1 times 0.6636 = 0.06636$ + + *最终归一化:* + $ P(L="Yes" | ...) &= 0.05976 / (0.05976 + 0.06636) \ + &= 0.05976 / 0.12612 approx 0.4738 $ + + *答案:47.38%* +]