File size: 83,291 Bytes
6fa4bc9 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 |
{
"paper_id": "N06-1044",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T14:45:42.512391Z"
},
"title": "Estimation of Consistent Probabilistic Context-free Grammars",
"authors": [
{
"first": "Mark-Jan",
"middle": [],
"last": "Nederhof",
"suffix": "",
"affiliation": {},
"email": "markjan.nederhof@mpi.nl"
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": "",
"affiliation": {},
"email": "satta@dei.unipd.it"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "We consider several empirical estimators for probabilistic context-free grammars, and show that the estimated grammars have the so-called consistency property, under the most general conditions. Our estimators include the widely applied expectation maximization method, used to estimate probabilistic context-free grammars on the basis of unannotated corpora. This solves a problem left open in the literature, since for this method the consistency property has been shown only under restrictive assumptions on the rules of the source grammar.",
"pdf_parse": {
"paper_id": "N06-1044",
"_pdf_hash": "",
"abstract": [
{
"text": "We consider several empirical estimators for probabilistic context-free grammars, and show that the estimated grammars have the so-called consistency property, under the most general conditions. Our estimators include the widely applied expectation maximization method, used to estimate probabilistic context-free grammars on the basis of unannotated corpora. This solves a problem left open in the literature, since for this method the consistency property has been shown only under restrictive assumptions on the rules of the source grammar.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "Probabilistic context-free grammars are one of the most widely used formalisms in current work in statistical natural language parsing and stochastic language modeling. An important property for a probabilistic context-free grammar is that it be consistent, that is, the grammar should assign probability of one to the set of all finite strings or parse trees that it generates. In other words, the grammar should not lose probability mass with strings or trees of infinite length.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Several methods for the empirical estimation of probabilistic context-free grammars have been proposed in the literature, based on the optimization of some function on the probabilities of the observed data, such as the maximization of the likelihood of a tree bank or a corpus of unannotated sentences. It has been conjectured in (Wetherell, 1980 ) that these methods always provide probabilistic context-free grammars with the consistency property. A first result in this direction was presented in (Chaudhuri et al., 1983) , by showing that a probabilistic contextfree grammar estimated by maximizing the likelihood of a sample of parse trees is always consistent.",
"cite_spans": [
{
"start": 331,
"end": 347,
"text": "(Wetherell, 1980",
"ref_id": "BIBREF14"
},
{
"start": 501,
"end": 525,
"text": "(Chaudhuri et al., 1983)",
"ref_id": "BIBREF4"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In later work by (S\u00e1nchez and Bened\u00ed, 1997) and (Chi and Geman, 1998) , the result was independently extended to expectation maximization, which is an unsupervised method exploited to estimate probabilistic context-free grammars by finding local maxima of the likelihood of a sample of unannotated sentences. The proof in (S\u00e1nchez and Bened\u00ed, 1997) makes use of spectral analysis of expectation matrices, while the proof in (Chi and Geman, 1998 ) is based on a simpler counting argument. Both these proofs assume restrictions on the underlying context-free grammars. More specifically, in (Chi and Geman, 1998) empty rules and unary rules are not allowed, thus excluding infinite ambiguity, that is, the possibility that some string in the input sample has an infinite number of derivations in the grammar. The treatment of general form contextfree grammars has been an open problem so far.",
"cite_spans": [
{
"start": 17,
"end": 43,
"text": "(S\u00e1nchez and Bened\u00ed, 1997)",
"ref_id": "BIBREF11"
},
{
"start": 48,
"end": 69,
"text": "(Chi and Geman, 1998)",
"ref_id": "BIBREF5"
},
{
"start": 322,
"end": 348,
"text": "(S\u00e1nchez and Bened\u00ed, 1997)",
"ref_id": "BIBREF11"
},
{
"start": 424,
"end": 444,
"text": "(Chi and Geman, 1998",
"ref_id": "BIBREF5"
},
{
"start": 589,
"end": 610,
"text": "(Chi and Geman, 1998)",
"ref_id": "BIBREF5"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In this paper we consider several estimation methods for probabilistic context-free grammars, and we show that the resulting grammars have the consistency property. Our proofs are applicable under the most general conditions, and our results also include the expectation maximization method, thus solving the open problem discussed above. We use an alternative proof technique with respect to pre-vious work, based on an already known renormalization construction for probabilistic context-free grammars, which has been used in the context of language modeling.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "The structure of this paper is as follows. We provide some preliminary definitions in Section 2, followed in Section 3 by a brief overview of the estimation methods we investigate in this paper. In Section 4 we prove some properties of a renormalization technique for probabilistic context-free grammars, and use this property to show our main results in Section 5. Section 6 closes with some concluding remarks.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In this paper we use mostly standard notation, as for instance in (Hopcroft and Ullman, 1979) and (Booth and Thompson, 1973) , which we summarize below.",
"cite_spans": [
{
"start": 66,
"end": 93,
"text": "(Hopcroft and Ullman, 1979)",
"ref_id": "BIBREF9"
},
{
"start": 98,
"end": 124,
"text": "(Booth and Thompson, 1973)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "A context-free grammar (CFG) is a 4-tuple G = (N, \u03a3, S, R) where N and \u03a3 are finite disjoint sets of nonterminal and terminal symbols, respectively, S \u2208 N is the start symbol and R is a finite set of rules. Each rule has the form A \u2192 \u03b1, where A \u2208 N and \u03b1 \u2208 (\u03a3 \u222a N ) * . We write V for set \u03a3 \u222a N .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "Each CFG G is associated with a left-most derive relation \u21d2, defined on triples consisting of two strings \u03b3, \u03b4 \u2208 V * and a rule \u03c0 \u2208 R. We write \u03b3 \u03c0 \u21d2 \u03b4 if and only if \u03b3 = uA\u03b3 and \u03b4 = u\u03b1\u03b3 , for some u \u2208 \u03a3 * , \u03b3 \u2208 V * , and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u03c0 = (A \u2192 \u03b1). A left- most derivation for G is a string d = \u03c0 1 \u2022 \u2022 \u2022 \u03c0 m , m \u2265 0, such that \u03b3 0 \u03c0 1 \u21d2 \u03b3 1 \u03c0 2 \u21d2 \u2022 \u2022 \u2022 \u03c0m \u21d2 \u03b3 m , for some \u03b3 0 , . . . , \u03b3 m \u2208 V * ; d = \u03b5 (",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "where \u03b5 denotes the empty string) is also a left-most derivation. In the remainder of this paper, we will let the term derivation always refer to left-most derivation. If",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u03b3 0 \u03c0 1 \u21d2 \u2022 \u2022 \u2022 \u03c0m \u21d2 \u03b3 m for some \u03b3 0 , . . . , \u03b3 m \u2208 V * , then we say that d = \u03c0 1 \u2022 \u2022 \u2022 \u03c0 m derives \u03b3 m from \u03b3 0 and we write \u03b3 0 d \u21d2 \u03b3 m ; d = \u03b5 derives any \u03b3 0 \u2208 V * from itself. A (left-most) derivation d such that S d \u21d2 w, w \u2208 \u03a3 * ,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "is called a complete derivation. If d is a complete derivation, we write y(d) to denote the (unique) string w \u2208 \u03a3 * such that S d \u21d2 w. We define D(G) to be the set of all complete derivations for G. The language generated by G is the set of all strings derived by complete derivations, i.e.,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "L(G) = {y(d) | d \u2208 D(G)}. It is well-known that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "there is a one-to-one correspondence between complete derivations and parse trees for strings in L(G).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "For X \u2208 V and \u03b1 \u2208 V * , we write f (X, \u03b1) to denote the number of occurrences of X in \u03b1. For (A \u2192 \u03b1) \u2208 R and a derivation d, f (A \u2192 \u03b1, d) denotes the number of occurrences of A \u2192 \u03b1 in d.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "We let f (A, d) = \u03b1 f (A \u2192 \u03b1, d).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "A probabilistic CFG (PCFG) is a pair G = (G, p G ), where G is a CFG and p G is a function from R to real numbers in the interval [0, 1] . We say that G is proper if, for every A \u2208 N , we have",
"cite_spans": [
{
"start": 130,
"end": 133,
"text": "[0,",
"ref_id": null
},
{
"start": 134,
"end": 136,
"text": "1]",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "A\u2192\u03b1 p G (A \u2192 \u03b1) = 1.",
"eq_num": "(1)"
}
],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "Function p G can be used to assign probabilities to derivations of the underlying CFG G, in the following way.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "For d = \u03c0 1 \u2022 \u2022 \u2022 \u03c0 m \u2208 R * , m \u2265 0, we define p G (d) = m i=1 p G (\u03c0 i ).",
"eq_num": "(2)"
}
],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "Note that p G (\u03b5) = 1. The probability of a string w \u2208 \u03a3 * is defined as",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p G (w) = y(d)=w p G (d).",
"eq_num": "(3)"
}
],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "A PCFG is consistent if",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "w p G (w) = 1.",
"eq_num": "(4)"
}
],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "Consistency implies that the PCFG defines a probability distribution over both sets D(G) and L(G). If a PCFG is proper, then consistency means that no probability mass is lost in derivations of infinite length. All PCFGs in this paper are implicitly assumed to be proper, unless otherwise stated.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "In this section we give a brief overview of some estimation methods for PCFGs. These methods will be later investigated to show that they always provide consistent PCFGs. In natural language processing applications, estimation of a PCFG is usually carried out on the basis of a tree bank, which in this paper we assume to be a sample, that is, a finite multiset, of complete derivations. Let D be such a sample, and let D be the underlying set of derivations. For d \u2208 D, we let f (d, D) be the multiplicity of d in D, that is, the number of occurrences of d in D. We define",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "f (A \u2192 \u03b1, D) = d\u2208D f (d, D) \u2022 f (A \u2192 \u03b1, d),",
"eq_num": "(5)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "let f (A, D) = \u03b1 f (A \u2192 \u03b1, D).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "Consider a CFG G = (N, \u03a3, R, S) defined by all and only the nonterminals, terminals and rules observed in D. The criterion of maximum likelihood estimation (MLE) prescribes the construction of a PCFG",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "G = (G, p G ) such that p G maximizes the likelihood of D, defined as p G (D) = d\u2208D p G (d) f (d,D) ,",
"eq_num": "(6)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "subject to the properness conditions \u03b1 p G (A \u2192 \u03b1) = 1 for each A \u2208 N . The maximization problem above has a unique solution, provided by the estimator (see for instance (Chi and Geman, 1998) )",
"cite_spans": [
{
"start": 170,
"end": 191,
"text": "(Chi and Geman, 1998)",
"ref_id": "BIBREF5"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p G (A \u2192 \u03b1) = f (A \u2192 \u03b1, D) f (A, D) .",
"eq_num": "(7)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "We refer to this as the supervised MLE method.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "In applications in which a tree bank is not available, one might still use the MLE criterion to train a PCFG in an unsupervised way, on the basis of a sample of unannotated sentences, also called a corpus. Let us call C such a sample and C the underlying set of sentences. For w \u2208 C, we let f (w, C) be the multiplicity of w in C.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "Assume a CFG G = (N, \u03a3, R, S) that is able to generate all of the sentences in C, and possibly more. The MLE criterion prescribes the construction of a PCFG G = (G, p G ) such that p G maximizes the likelihood of C, defined as",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p G (C) = w\u2208C p G (w) f (w,C) ,",
"eq_num": "(8)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "subject to the properness conditions as in the supervised case above. The above maximization problem provides a system of |R| nonlinear equations (see (Chi and Geman, 1998) )",
"cite_spans": [
{
"start": 151,
"end": 172,
"text": "(Chi and Geman, 1998)",
"ref_id": "BIBREF5"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p G (A \u2192 \u03b1) = w\u2208C f (w, C) \u2022 E p G (d | w) f (A \u2192 \u03b1, d) w\u2208C f (w, C) \u2022 E p G (d | w) f (A, d) ,",
"eq_num": "(9)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "where E p denotes an expectation computed under distribution p, and p G (d | w) is the probability of derivation d conditioned by sentence w (so that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "p G (d | w) > 0 only if y(d) = w)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": ". The solution to the above system is not unique, because of the nonlinearity. Furthermore, each solution of (9) identifies a point where the curve in (8) has partial derivatives of zero, but this does not necessarily correspond to a local maximum, let alone an absolute maximum. (A point with partial derivatives of zero that is not a local maximum could be a local minimum or even a so-called saddle point.) In practice, this system is typically solved by means of an iterative algorithm called inside/outside (Charniak, 1993) , which implements the expectation maximization (EM) method (Dempster et al., 1977) . Starting with an initial function p G that probabilistically extends G, a so-called growth transformation is computed, defined as",
"cite_spans": [
{
"start": 512,
"end": 528,
"text": "(Charniak, 1993)",
"ref_id": "BIBREF3"
},
{
"start": 589,
"end": 612,
"text": "(Dempster et al., 1977)",
"ref_id": "BIBREF8"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "p G (A \u2192 \u03b1) = w\u2208C f (w, C)\u2022 y(d)=w p G (d) p G (w) \u2022f (A \u2192 \u03b1, d) w\u2208C f (w, C)\u2022 y(d)=w p G (d) p G (w) \u2022f (A, d)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": ". 10Following (Baum, 1972) , one can show that p G (C) \u2265 p G (C). Thus, by iterating the growth transformation above, we are guaranteed to reach a local maximum for (8), or possibly a saddle point. We refer to this as the unsupervised MLE method. We now discuss a third estimation method for PCFGs, which was proposed in (Corazza and Satta, 2006) . This method can be viewed as a generalization of the supervised MLE method to probability distributions defined over infinite sets of complete derivations. Let D be an infinite set of complete derivations using nonterminal symbols in N , start symbol S \u2208 N and terminal symbols in \u03a3. We assume that the set of rules that are observed in D is drawn from some finite set R. Let p D be a probability distribution defined over D, that is, a function from set D to interval",
"cite_spans": [
{
"start": 14,
"end": 26,
"text": "(Baum, 1972)",
"ref_id": "BIBREF1"
},
{
"start": 321,
"end": 346,
"text": "(Corazza and Satta, 2006)",
"ref_id": "BIBREF7"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "[0, 1] such that d\u2208D p D (d) = 1.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "Consider the CFG G = (N, \u03a3, R, S). Note that D \u2286 D(G). We wish to extend G to some PCFG G = (G, p G ) in such a way that p D is approximated by p G (viewed as a distribution over complete derivations) as well as possible according to some criterion. One possible criterion is minimization of the cross-entropy between p D and p G , defined as the expectation, under distribution p D , of the information of the derivations in D computed under distribution p G , that is",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "H(p D || p G ) = E p D log 1 p G (d) = \u2212 d\u2208D p D (d) \u2022 log p G (d). (11)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "We thus want to assign to the parameters p G (A \u2192 \u03b1), A \u2192 \u03b1 \u2208 R, the values that minimize (11), subject to the conditions \u03b1 p G (A \u2192 \u03b1) = 1 for each A \u2208 N . Note that minimization of the cross-entropy above is equivalent to minimization of the Kullback-Leibler distance between p D and p G . Also note that the likelihood of an infinite set of derivations would always be zero and therefore cannot be considered here.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "The solution to the above minimization problem provides the estimator",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p G (A \u2192 \u03b1) = E p D f (A \u2192 \u03b1, d) E p D f (A, d) .",
"eq_num": "(12)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "A proof of this result appears in (Corazza and Satta, 2006) , and is briefly summarized in Appendix A, in order to make this paper self-contained. We call the above estimator the cross-entropy minimization method. The cross-entropy minimization method can be viewed as a generalization of the supervised MLE method in (7), as shown in what follows. Let D and D be defined as for the supervised MLE method. We define a distribution over D as",
"cite_spans": [
{
"start": 34,
"end": 59,
"text": "(Corazza and Satta, 2006)",
"ref_id": "BIBREF7"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p D (d) = f (d, D) |D| .",
"eq_num": "(13)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "Distribution p D is usually called the empirical distribution associated with D. Applying the estimator in (12) to p D , we obtain",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p G (A \u2192 \u03b1) = = d\u2208D p D (d) \u2022 f (A \u2192 \u03b1, d) d\u2208D p D (d) \u2022 f (A, d) = d\u2208D f (d,D) |D| \u2022 f (A \u2192 \u03b1, d) d\u2208D f (d,D) |D| \u2022 f (A, d) = d\u2208D f (d, D) \u2022 f (A \u2192 \u03b1, d) d\u2208D f (d, D) \u2022 f (A, d) .",
"eq_num": "(14)"
}
],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "This is the supervised MLE estimator in (7). This reminds us of the well-known fact that maximizing the likelihood of a (finite) sample through a PCFG distribution amounts to minimizing the cross-entropy between the empirical distribution of the sample and the PCFG distribution itself.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Estimation of PCFGs",
"sec_num": "3"
},
{
"text": "In this section we recall a renormalization technique for PCFGs that was used before in (Abney et al., 1999) , (Chi, 1999) and (Nederhof and Satta, 2003) for different purposes, and is exploited in the next section to prove our main results. In the remainder of this section, we assume a fixed, not necessarily proper PCFG G = (G, p G ), with G = (N, \u03a3, S, R).",
"cite_spans": [
{
"start": 88,
"end": 108,
"text": "(Abney et al., 1999)",
"ref_id": "BIBREF0"
},
{
"start": 111,
"end": 122,
"text": "(Chi, 1999)",
"ref_id": "BIBREF6"
},
{
"start": 127,
"end": 153,
"text": "(Nederhof and Satta, 2003)",
"ref_id": "BIBREF10"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Renormalization",
"sec_num": "4"
},
{
"text": "We define the renormalization of G as the PCFG",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Renormalization",
"sec_num": "4"
},
{
"text": "R(G) = (G, p R ) with p R specified by p R (A \u2192 \u03b1) = p G (A \u2192 \u03b1) \u2022 d,w p G (\u03b1 d \u21d2 w) d,w p G (A d \u21d2 w)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Renormalization",
"sec_num": "4"
},
{
"text": ". 15It is not difficult to see that R(G) is a proper PCFG. We now show an important property of R(G), discussed before in (Nederhof and Satta, 2003) in the context of so-called weighted context-free grammars.",
"cite_spans": [
{
"start": 122,
"end": 148,
"text": "(Nederhof and Satta, 2003)",
"ref_id": "BIBREF10"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Renormalization",
"sec_num": "4"
},
{
"text": "A d \u21d2 w, A \u2208 N and w \u2208 \u03a3 * , we have p R (A d \u21d2 w) = p G (A d \u21d2 w) d ,w p G (A d \u21d2 w )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": ". 16Proof. The proof is by induction on the length of d, written |d|. If |d| = 1 we must have d = (A \u2192 w), and thus p R (d) = p R (A \u2192 w). In this case, the statement of the lemma directly follows from (15). Assume now |d| > 1 and let \u03c0 = (A \u2192 \u03b1) be the first rule used in d. Note that there must be at least one nonterminal symbol in \u03b1. We can then write \u03b1 as",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "u 0 A 1 u 1 A 2 \u2022 \u2022 \u2022 u q\u22121 A q u q , for q \u2265 1, A i \u2208 N , 1 \u2264 i \u2264 q, and u j \u2208 \u03a3 * , 0 \u2264 j \u2264 q.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "In words, A 1 , . . . , A q are all of the occurrences of nonterminals in \u03b1, as they appear from left to right. Consequently, we can write d in the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "d = \u03c0 \u2022 d 1 \u2022 \u2022 \u2022 d q for some derivations d i , 1 \u2264 i \u2264 q, with A i d i \u21d2 w i , |d i | \u2265 1 and with w = u 0 w 1 u 1 w 2 \u2022 \u2022 \u2022 u q\u22121 w q u q . Below we use the fact that p R (u j \u03b5 \u21d2 u j ) = p G (u j \u03b5",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "\u21d2 u j ) = 1 for each j with 0 \u2264 j \u2264 q, and further using the definition of p R and the inductive hypothesis, we can write",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p R (A d \u21d2 w) = = p R (A \u2192 \u03b1) \u2022 q i=1 p R (A i d i \u21d2 w i ) = p G (A \u2192 \u03b1) \u2022 d ,w p G (\u03b1 d \u21d2 w ) d ,w p G (A d \u21d2 w ) \u2022 \u2022 q i=1 p R (A i d i \u21d2 w i ) = p G (A \u2192 \u03b1) \u2022 d ,w p G (\u03b1 d \u21d2 w ) d ,w p G (A d \u21d2 w ) \u2022 \u2022 q i=1 p G (A i d i \u21d2 w i ) d ,w p G (A i d \u21d2 w ) = p G (A \u2192 \u03b1) \u2022 d ,w p G (\u03b1 d \u21d2 w ) d ,w p G (A d \u21d2 w ) \u2022 \u2022 q i=1 p G (A i d i \u21d2 w i ) q i=1 d ,w p G (A i d \u21d2 w ) = p G (A \u2192 \u03b1) \u2022 d ,w p G (\u03b1 d \u21d2 w ) d ,w p G (A d \u21d2 w ) \u2022 \u2022 q i=1 p G (A i d i \u21d2 w i ) d ,w p G (\u03b1 d \u21d2 w ) = p G (A \u2192 \u03b1) \u2022 q i=1 p G (A i d i \u21d2 w i ) d ,w p G (A d \u21d2 w ) \u2022 = p G (A d \u21d2 w) d ,w p G (A d \u21d2 w ) .",
"eq_num": "(17)"
}
],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "As an easy corollary of Lemma 1, we have that R(G) is a consistent PCFG, as we can write",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "d,w p R (S d \u21d2 w) = = d,w p G (S d \u21d2 w) d ,w p G (S d \u21d2 w ) = d,w p G (S d \u21d2 w) d ,w p G (S d \u21d2 w ) = 1.",
"eq_num": "(18)"
}
],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "5 Consistency",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "In this section we prove the main results of this paper, namely that all of the estimation methods discussed in Section 3 always provide consistent PCFGs. We start with a technical lemma, central to our results, showing that a PCFG that minimizes the cross-entropy with a distribution over any set of derivations must be consistent.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "Lemma 2 Let G = (G, p G ) be a proper PCFG and let p D be a probability distribution defined over some set",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "D \u2286 D(G). If G minimizes function H(p D || p G ), then G is consistent.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "Proof. Let G = (N, \u03a3, S, R), and assume that G is not consistent. We establish a contradiction. Since G is not consistent, we must",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "have d,w p G (S d \u21d2 w) < 1. Let then R(G) = (G, p R ) be the renormalization of G, defined as in (15). For any derivation S d \u21d2 w, w \u2208 \u03a3 * , with d in D,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "we can use Lemma 1 and write",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p R (S d \u21d2 w) = = 1 d ,w p G (S d \u21d2 w ) \u2022 p G (S d \u21d2 w) > p G (S d \u21d2 w).",
"eq_num": "(19)"
}
],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "In words, every complete derivation d in D has a probability in R(G) that is strictly greater than in G. But this means",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "H(p D || p R ) < H(p D || p G )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": ", against our hypothesis. Therefore, G is consistent and p G is a probability distribution over set D(G).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "Thus function H(p D || p G ) can be interpreted as the cross-entropy. (Observe that in the statement of the lemma we have avoided the term 'cross-entropy', since cross-entropies are only defined for probability distributions.) Lemma 2 directly implies that the cross-entropy minimization method in (12) always provides a consistent PCFG, since it minimizes cross-entropy for a distribution defined over a subset of D(G). We have already seen in Section 3 that the supervised MLE method is a special case of the cross-entropy minimization method. Thus we can also conclude that a PCFG trained with the supervised MLE method is always consistent. This provides an alternative proof of a property that was first shown in (Chaudhuri et al., 1983) , as discussed in Section 1.",
"cite_spans": [
{
"start": 718,
"end": 742,
"text": "(Chaudhuri et al., 1983)",
"ref_id": "BIBREF4"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "We now prove the same result for the unsupervised MLE method, without any restrictive assumption on the rules of our CFGs. This solves a problem that was left open in the literature (Chi and Geman, 1998) ; see again Section 1 for discussion. Let C and C be defined as in Section 3. We define the empirical distribution of C as",
"cite_spans": [
{
"start": 182,
"end": 203,
"text": "(Chi and Geman, 1998)",
"ref_id": "BIBREF5"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "p C (w) = f (w, C) |C| .",
"eq_num": "(20)"
}
],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "Let G = (N, \u03a3, S, R) be a CFG such that C \u2286 L(G). Let D(C) be the set of all complete derivations for G that generate sentences in C, that is,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "D(C) = {d | d \u2208 D(G), y(d) \u2208 C}.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "Further, assume some probabilistic extension",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "G = (G, p G ) of G, such that p G (d) > 0 for every d \u2208 D(C). We define a distribution over D(C) by p D(C) (d) = p C (y(d)) \u2022 p G (d) p G (y(d))",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": ". 21It is not difficult to verify that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "d\u2208D(C) p D(C) (d) = 1.",
"eq_num": "(22)"
}
],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "We now apply to G the estimator in (12), in order to obtain a new PCFG\u011c = (G,p G ) that minimizes the cross-entropy between p D(C) andp G . According to Lemma 2, we have that\u011c is a consistent PCFG. Distributionp G is specified b\u0177",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "p G (A \u2192 \u03b1) = = d\u2208D(C) p D(C) (d)\u2022f (A \u2192 \u03b1, d) d\u2208D(C) p D(C) (d)\u2022f (A, d) = d\u2208D(C) f (y(d),C) |C| \u2022 p G (d) p G (y(d)) \u2022f (A \u2192 \u03b1, d) d\u2208D(C) f (y(d),C) |C| \u2022 p G (d) p G (y(d)) \u2022f (A, d) = w\u2208C f (w, C)\u2022 y(d)=w p G (d) p G (w) \u2022f (A \u2192 \u03b1, d) w\u2208C f (w, C)\u2022 y(d)=w p G (d) p G (w) \u2022f (A, d) = w\u2208C f (w, C)\u2022E p G (d | w) f (A \u2192 \u03b1, d) w\u2208C f (w, C)\u2022E p G (d | w) f (A, d)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": ".",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "Since distribution p G was arbitrarily chosen, subject to the only restriction that p G (d) > 0 for every d \u2208 D(C), we have that (23) is the growth estimator (10) already discussed in Section 3. In fact, for each w \u2208 L(G) and d \u2208 D(G), we have w) . We conclude with the desired result, namely that a general form PCFG obtained at any iteration of the EM method for the unsupervised MLE is always consistent.",
"cite_spans": [
{
"start": 244,
"end": 246,
"text": "w)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "p G (d | w) = p G (d) p G (",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1 For each derivation d with",
"sec_num": null
},
{
"text": "In this paper we have investigated a number of methods for the empirical estimation of probabilistic context-free grammars, and have shown that the resulting grammars have the so-called consistency property. This property guarantees that all the probability mass of the grammar is used for the finite strings it derives. Thus if the grammar is used in combination with other probabilistic models, as for instance in a speech processing system, consistency allows us to combine or compare scores from different modules in a sound way.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusions",
"sec_num": "6"
},
{
"text": "To obtain our results, we have used a novel proof technique that exploits an already known construction for the renormalization of probabilistic contextfree grammars. Our proof technique seems more intuitive than arguments previously used in the literature to prove the consistency property, based on counting arguments or on spectral analysis. It is not difficult to see that our proof technique can also be used with probabilistic rewriting formalisms whose underlying derivations can be characterized by means of context-free rewriting. This is for instance the case with probabilistic tree-adjoining grammars (Schabes, 1992; Sarkar, 1998) , for which consistency results have not yet been shown in the literature.",
"cite_spans": [
{
"start": 613,
"end": 628,
"text": "(Schabes, 1992;",
"ref_id": "BIBREF13"
},
{
"start": 629,
"end": 642,
"text": "Sarkar, 1998)",
"ref_id": "BIBREF12"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusions",
"sec_num": "6"
},
{
"text": "In order to make this paper self-contained, we sketch a proof of the claim in Section 3 that the estimator in (12) minimizes the cross entropy in (11). A full proof appears in (Corazza and Satta, 2006) .",
"cite_spans": [
{
"start": 176,
"end": 201,
"text": "(Corazza and Satta, 2006)",
"ref_id": "BIBREF7"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "Let D, p D and G = (N, \u03a3, R, S) be defined as in Section 3. We want to find a proper PCFG G = (G, p G ) such that the cross-entropy H(p D || p G ) is minimal. We use Lagrange multipliers \u03bb A for each A \u2208 N and define the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u2207 = A\u2208N \u03bb A \u2022 ( \u03b1 p G (A \u2192 \u03b1) \u2212 1) + \u2212 d\u2208D p D (d) \u2022 log p G (d).",
"eq_num": "(24)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "We now consider all the partial derivatives of \u2207. For each A \u2208 N we have",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u2202\u2207 \u2202\u03bb A = \u03b1 p G (A \u2192 \u03b1) \u2212 1.",
"eq_num": "(25)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "For each (A \u2192 \u03b1) \u2208 R we have",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u2202\u2207 \u2202p G (A \u2192 \u03b1) = = \u03bb A \u2212 \u2202 \u2202p G (A \u2192 \u03b1) d\u2208D p D (d) \u2022 log p G (d) = \u03bb A \u2212 d\u2208D p D (d) \u2022 \u2202 \u2202p G (A \u2192 \u03b1) log p G (d) = \u03bb A \u2212 d\u2208D p D (d) \u2022 \u2202 \u2202p G (A \u2192 \u03b1) log (B\u2192\u03b2)\u2208R p G (B \u2192 \u03b2) f (B\u2192\u03b2,d) = \u03bb A \u2212 d\u2208D p D (d) \u2022 \u2202 \u2202p G (A \u2192 \u03b1) (B\u2192\u03b2)\u2208R f (B \u2192 \u03b2, d) \u2022 log p G (B \u2192 \u03b2) = \u03bb A \u2212 d\u2208D p D (d) \u2022 (B\u2192\u03b2)\u2208R f (B \u2192 \u03b2, d) \u2022 \u2202 \u2202p G (A \u2192 \u03b1) log p G (B \u2192 \u03b2) = \u03bb A \u2212 d\u2208D p D (d) \u2022 f (A \u2192 \u03b1, d) \u2022 \u2022 1 ln(2) \u2022 1 p G (A \u2192 \u03b1) = \u03bb A \u2212 1 ln(2) \u2022 1 p G (A \u2192 \u03b1) \u2022 \u2022 d\u2208D p D (d) \u2022 f (A \u2192 \u03b1, d) = \u03bb A \u2212 1 ln(2) \u2022 1 p G (A \u2192 \u03b1) \u2022 \u2022 E p D f (A \u2192 \u03b1, d).",
"eq_num": "(26)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "By setting to zero all of the above partial derivatives, we obtain a system of |N |+|R| equations, which we must solve. From \u2202\u2207 \u2202p G (A\u2192\u03b1) = 0 we obtain",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u03bb A \u2022 ln(2) \u2022 p G (A \u2192 \u03b1) = E p D f (A \u2192 \u03b1, d).",
"eq_num": "(27)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "We sum over all strings \u03b1 such that (A \u2192 \u03b1) \u2208 R, deriving",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u03bb A \u2022 ln(2) \u2022 \u03b1 p G (A \u2192 \u03b1) = = \u03b1 E p D f (A \u2192 \u03b1, d) = \u03b1 d\u2208D p D (d) \u2022 f (A \u2192 \u03b1, d) = d\u2208D p D (d) \u2022 \u03b1 f (A \u2192 \u03b1, d) = d\u2208D p D (d) \u2022 f (A, d) = E p D f (A, d).",
"eq_num": "(28)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "From each equation \u2202\u2207 \u2202\u03bb A = 0 we obtain \u03b1 p G (A \u2192 \u03b1) = 1 for each A \u2208 N (our original constraints). Combining this with (28) we obtain",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u03bb A \u2022 ln(2) = E p D f (A, d).",
"eq_num": "(29)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "Replacing 29into (27) we obtain, for every rule",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "(A \u2192 \u03b1) \u2208 R, p G (A \u2192 \u03b1) = E p D f (A \u2192 \u03b1, d) E p D f (A, d) .",
"eq_num": "(30)"
}
],
"section": "A Cross-entropy minimization",
"sec_num": null
},
{
"text": "This is the estimator introduced in Section 3.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Cross-entropy minimization",
"sec_num": null
}
],
"back_matter": [],
"bib_entries": {
"BIBREF0": {
"ref_id": "b0",
"title": "Relating probabilistic grammars and automata",
"authors": [
{
"first": "D",
"middle": [],
"last": "Abney",
"suffix": ""
},
{
"first": "F",
"middle": [],
"last": "Mcallester",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Pereira",
"suffix": ""
}
],
"year": 1999,
"venue": "37th Annual Meeting of the Association for Computational Linguistics, Proceedings of the Conference",
"volume": "",
"issue": "",
"pages": "542--549",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Abney, D. McAllester, and F. Pereira. 1999. Relating probabilistic grammars and automata. In 37th Annual Meeting of the Association for Computational Linguis- tics, Proceedings of the Conference, pages 542-549, Maryland, USA, June.",
"links": null
},
"BIBREF1": {
"ref_id": "b1",
"title": "An inequality and associated maximization technique in statistical estimations of probabilistic functions of Markov processes",
"authors": [
{
"first": "L",
"middle": [
"E"
],
"last": "Baum",
"suffix": ""
}
],
"year": 1972,
"venue": "Inequalities",
"volume": "3",
"issue": "",
"pages": "1--8",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "L. E. Baum. 1972. An inequality and associated max- imization technique in statistical estimations of prob- abilistic functions of Markov processes. Inequalities, 3:1-8.",
"links": null
},
"BIBREF2": {
"ref_id": "b2",
"title": "Applying probabilistic measures to abstract languages",
"authors": [
{
"first": "T",
"middle": [
"L"
],
"last": "Booth",
"suffix": ""
},
{
"first": "R",
"middle": [
"A"
],
"last": "Thompson",
"suffix": ""
}
],
"year": 1973,
"venue": "IEEE Transactions on Computers, C",
"volume": "22",
"issue": "5",
"pages": "442--450",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "T.L. Booth and R.A. Thompson. 1973. Applying prob- abilistic measures to abstract languages. IEEE Trans- actions on Computers, C-22(5):442-450, May.",
"links": null
},
"BIBREF3": {
"ref_id": "b3",
"title": "Statistical Language Learning",
"authors": [
{
"first": "E",
"middle": [],
"last": "Charniak",
"suffix": ""
}
],
"year": 1993,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "E. Charniak. 1993. Statistical Language Learning. MIT Press.",
"links": null
},
"BIBREF4": {
"ref_id": "b4",
"title": "Solution of an open problem on probabilistic grammars",
"authors": [
{
"first": "R",
"middle": [],
"last": "Chaudhuri",
"suffix": ""
},
{
"first": "S",
"middle": [],
"last": "Pham",
"suffix": ""
},
{
"first": "O",
"middle": [
"N"
],
"last": "Garcia",
"suffix": ""
}
],
"year": 1983,
"venue": "IEEE Transactions on Computers",
"volume": "32",
"issue": "8",
"pages": "748--750",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "R. Chaudhuri, S. Pham, and O. N. Garcia. 1983. Solution of an open problem on probabilistic grammars. IEEE Transactions on Computers, 32(8):748-750.",
"links": null
},
"BIBREF5": {
"ref_id": "b5",
"title": "Estimation of probabilistic context-free grammars",
"authors": [
{
"first": "Z",
"middle": [],
"last": "Chi",
"suffix": ""
},
{
"first": "S",
"middle": [],
"last": "Geman",
"suffix": ""
}
],
"year": 1998,
"venue": "Computational Linguistics",
"volume": "24",
"issue": "2",
"pages": "299--305",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Z. Chi and S. Geman. 1998. Estimation of probabilis- tic context-free grammars. Computational Linguistics, 24(2):299-305.",
"links": null
},
"BIBREF6": {
"ref_id": "b6",
"title": "Statistical properties of probabilistic context-free grammars",
"authors": [
{
"first": "Z",
"middle": [],
"last": "Chi",
"suffix": ""
}
],
"year": 1999,
"venue": "Computational Linguistics",
"volume": "25",
"issue": "1",
"pages": "131--160",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Z. Chi. 1999. Statistical properties of probabilistic context-free grammars. Computational Linguistics, 25(1):131-160.",
"links": null
},
"BIBREF7": {
"ref_id": "b7",
"title": "Cross-entropy and estimation of probabilistic context-free grammars",
"authors": [
{
"first": "A",
"middle": [],
"last": "Corazza",
"suffix": ""
},
{
"first": "G",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 2006,
"venue": "Proc. of HLT/NAACL 2006 Conference (this volume)",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "A. Corazza and G. Satta. 2006. Cross-entropy and es- timation of probabilistic context-free grammars. In Proc. of HLT/NAACL 2006 Conference (this volume), New York.",
"links": null
},
"BIBREF8": {
"ref_id": "b8",
"title": "Maximum likelihood from incomplete data via the EM algorithm",
"authors": [
{
"first": "A",
"middle": [
"P"
],
"last": "Dempster",
"suffix": ""
},
{
"first": "N",
"middle": [
"M"
],
"last": "Laird",
"suffix": ""
},
{
"first": "D",
"middle": [
"B"
],
"last": "Rubin",
"suffix": ""
}
],
"year": 1977,
"venue": "Journal of the Royal Statistical Society, B",
"volume": "39",
"issue": "",
"pages": "1--38",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "A. P. Dempster, N. M. Laird, and D. B. Rubin. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B, 39:1-38.",
"links": null
},
"BIBREF9": {
"ref_id": "b9",
"title": "Introduction to Automata Theory, Languages, and Computation",
"authors": [
{
"first": "J",
"middle": [
"E"
],
"last": "Hopcroft",
"suffix": ""
},
{
"first": "J",
"middle": [
"D"
],
"last": "Ullman",
"suffix": ""
}
],
"year": 1979,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "J.E. Hopcroft and J.D. Ullman. 1979. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.",
"links": null
},
"BIBREF10": {
"ref_id": "b10",
"title": "Probabilistic parsing as intersection",
"authors": [
{
"first": "M.-J",
"middle": [],
"last": "Nederhof",
"suffix": ""
},
{
"first": "G",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 2003,
"venue": "8th International Workshop on Parsing Technologies",
"volume": "",
"issue": "",
"pages": "137--148",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "M.-J. Nederhof and G. Satta. 2003. Probabilistic pars- ing as intersection. In 8th International Workshop on Parsing Technologies, pages 137-148, LORIA, Nancy, France, April.",
"links": null
},
"BIBREF11": {
"ref_id": "b11",
"title": "Consistency of stochastic context-free grammars from probabilistic estimation based on growth transformations",
"authors": [
{
"first": "J.-A",
"middle": [],
"last": "S\u00e1nchez",
"suffix": ""
},
{
"first": "J.-M",
"middle": [],
"last": "Bened\u00ed",
"suffix": ""
}
],
"year": 1997,
"venue": "IEEE Transactions on Pattern Analysis and Machine Intelligence",
"volume": "19",
"issue": "9",
"pages": "1052--1055",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "J.-A. S\u00e1nchez and J.-M. Bened\u00ed. 1997. Consistency of stochastic context-free grammars from probabilis- tic estimation based on growth transformations. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 19(9):1052-1055, September.",
"links": null
},
"BIBREF12": {
"ref_id": "b12",
"title": "Conditions on consistency of probabilistic tree adjoining grammars",
"authors": [
{
"first": "A",
"middle": [],
"last": "Sarkar",
"suffix": ""
}
],
"year": 1998,
"venue": "Proc. of the 36 th ACL",
"volume": "",
"issue": "",
"pages": "1164--1170",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "A. Sarkar. 1998. Conditions on consistency of proba- bilistic tree adjoining grammars. In Proc. of the 36 th ACL, pages 1164-1170, Montreal, Canada.",
"links": null
},
"BIBREF13": {
"ref_id": "b13",
"title": "Stochastic lexicalized tree-adjoining grammars",
"authors": [
{
"first": "Y",
"middle": [],
"last": "Schabes",
"suffix": ""
}
],
"year": 1992,
"venue": "Proc. of the 14 th COLING",
"volume": "",
"issue": "",
"pages": "426--432",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Y. Schabes. 1992. Stochastic lexicalized tree-adjoining grammars. In Proc. of the 14 th COLING, pages 426- 432, Nantes, France.",
"links": null
},
"BIBREF14": {
"ref_id": "b14",
"title": "Probabilistic languages: A review and some open questions",
"authors": [
{
"first": "C",
"middle": [
"S"
],
"last": "Wetherell",
"suffix": ""
}
],
"year": 1980,
"venue": "Computing Surveys",
"volume": "12",
"issue": "4",
"pages": "361--379",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "C. S. Wetherell. 1980. Probabilistic languages: A re- view and some open questions. Computing Surveys, 12(4):361-379.",
"links": null
}
},
"ref_entries": {}
}
} |