File size: 110,362 Bytes
a67be9a |
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 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171 2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343 2344 2345 2346 2347 2348 2349 2350 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379 2380 2381 2382 2383 2384 2385 2386 2387 2388 2389 2390 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470 2471 2472 2473 2474 2475 2476 2477 2478 2479 2480 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496 2497 2498 2499 2500 2501 2502 2503 2504 2505 2506 2507 2508 2509 2510 2511 2512 2513 2514 2515 2516 2517 2518 2519 2520 2521 2522 2523 2524 2525 2526 2527 2528 2529 2530 2531 2532 2533 2534 2535 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 2568 2569 2570 2571 2572 2573 2574 2575 2576 2577 2578 2579 2580 2581 2582 2583 2584 2585 2586 2587 2588 2589 2590 2591 2592 2593 2594 2595 2596 2597 2598 2599 2600 2601 2602 2603 2604 2605 2606 2607 2608 2609 2610 2611 2612 2613 2614 2615 2616 2617 2618 2619 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665 2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2721 2722 2723 2724 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 2790 2791 2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808 2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834 2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847 2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887 2888 2889 2890 2891 2892 2893 2894 2895 2896 2897 2898 2899 2900 2901 2902 2903 2904 2905 2906 2907 2908 2909 2910 2911 2912 2913 2914 2915 2916 2917 2918 2919 2920 2921 2922 2923 2924 2925 2926 2927 2928 2929 2930 2931 2932 2933 2934 2935 2936 2937 2938 2939 2940 2941 2942 2943 2944 2945 2946 2947 2948 2949 2950 2951 2952 2953 2954 2955 2956 2957 2958 2959 2960 2961 2962 2963 2964 2965 2966 2967 2968 2969 2970 2971 2972 2973 2974 2975 2976 2977 2978 2979 2980 2981 2982 2983 2984 2985 2986 2987 2988 2989 2990 2991 2992 2993 2994 2995 2996 2997 2998 2999 3000 3001 3002 3003 3004 3005 3006 3007 3008 3009 3010 3011 3012 3013 3014 3015 3016 3017 3018 3019 3020 3021 3022 3023 3024 3025 3026 3027 3028 3029 3030 3031 3032 3033 3034 3035 3036 3037 3038 3039 3040 3041 3042 3043 3044 3045 3046 3047 3048 3049 3050 3051 3052 3053 3054 3055 3056 3057 3058 3059 3060 3061 3062 3063 3064 3065 3066 3067 3068 3069 3070 3071 3072 3073 3074 3075 3076 3077 3078 3079 3080 3081 3082 3083 3084 3085 3086 3087 3088 3089 3090 3091 3092 3093 3094 3095 3096 3097 3098 3099 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120 3121 3122 3123 3124 3125 3126 3127 3128 3129 3130 3131 3132 3133 3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146 3147 3148 3149 3150 3151 3152 3153 3154 3155 3156 3157 3158 3159 3160 3161 3162 3163 3164 3165 3166 3167 3168 3169 3170 3171 3172 3173 3174 3175 3176 3177 3178 3179 3180 3181 3182 3183 3184 3185 3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 3256 3257 3258 3259 3260 3261 3262 3263 3264 3265 3266 3267 3268 3269 3270 3271 3272 3273 3274 3275 3276 3277 3278 3279 3280 3281 3282 3283 3284 3285 3286 3287 3288 3289 3290 3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301 3302 3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320 3321 3322 3323 3324 3325 3326 3327 3328 3329 3330 3331 3332 3333 3334 3335 3336 3337 3338 3339 3340 3341 3342 3343 3344 3345 3346 3347 3348 3349 3350 3351 3352 3353 3354 3355 3356 3357 3358 3359 3360 3361 3362 3363 3364 3365 3366 3367 3368 3369 3370 3371 3372 3373 3374 3375 3376 3377 3378 3379 3380 3381 3382 3383 3384 3385 3386 3387 3388 3389 3390 3391 3392 3393 3394 3395 3396 3397 3398 3399 3400 3401 3402 3403 3404 3405 3406 3407 3408 3409 3410 3411 3412 3413 3414 3415 3416 3417 3418 3419 3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432 3433 3434 3435 3436 3437 3438 3439 3440 3441 3442 3443 3444 3445 3446 3447 3448 3449 3450 3451 3452 3453 3454 3455 3456 3457 3458 3459 3460 3461 3462 3463 3464 3465 3466 3467 3468 3469 3470 3471 3472 3473 3474 3475 3476 3477 3478 3479 3480 3481 3482 3483 3484 3485 3486 3487 3488 3489 3490 3491 3492 3493 3494 3495 3496 3497 3498 3499 3500 3501 3502 3503 3504 3505 3506 3507 3508 3509 3510 3511 3512 3513 3514 3515 3516 3517 3518 3519 3520 3521 3522 3523 3524 3525 3526 3527 3528 3529 3530 3531 3532 3533 3534 3535 3536 3537 3538 3539 3540 3541 3542 3543 3544 3545 3546 3547 3548 3549 3550 3551 3552 3553 3554 3555 3556 3557 3558 3559 3560 3561 3562 3563 3564 3565 3566 3567 3568 3569 3570 3571 3572 3573 3574 3575 3576 3577 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588 3589 3590 3591 3592 3593 3594 3595 3596 3597 3598 3599 3600 3601 3602 3603 3604 3605 3606 3607 3608 3609 3610 3611 3612 3613 3614 3615 3616 3617 3618 3619 3620 3621 3622 3623 3624 3625 3626 3627 3628 3629 3630 3631 3632 3633 3634 3635 3636 3637 3638 3639 3640 3641 3642 3643 3644 3645 3646 3647 3648 3649 3650 3651 3652 3653 3654 3655 3656 3657 3658 3659 3660 3661 3662 3663 3664 3665 3666 3667 3668 3669 3670 3671 3672 3673 3674 3675 3676 3677 3678 3679 3680 3681 3682 3683 3684 3685 3686 3687 3688 3689 3690 3691 3692 3693 3694 3695 3696 3697 3698 3699 3700 3701 3702 3703 3704 3705 3706 3707 3708 3709 3710 3711 3712 3713 3714 3715 3716 3717 3718 3719 3720 3721 3722 3723 3724 3725 3726 3727 3728 3729 3730 3731 3732 3733 3734 3735 3736 3737 3738 3739 3740 3741 3742 3743 3744 3745 3746 3747 3748 3749 3750 3751 3752 3753 3754 3755 3756 3757 3758 3759 3760 3761 3762 3763 3764 3765 3766 3767 3768 3769 3770 3771 3772 3773 3774 3775 3776 3777 3778 3779 3780 3781 3782 3783 3784 3785 3786 3787 3788 3789 3790 3791 3792 3793 3794 3795 3796 3797 3798 3799 3800 3801 3802 3803 3804 3805 3806 3807 3808 3809 3810 3811 3812 3813 3814 3815 3816 3817 3818 3819 3820 3821 3822 3823 3824 3825 3826 3827 3828 3829 3830 3831 3832 3833 3834 3835 3836 3837 3838 3839 3840 3841 3842 3843 3844 3845 3846 3847 3848 3849 3850 3851 3852 3853 3854 3855 3856 3857 3858 3859 3860 3861 3862 3863 3864 3865 3866 3867 3868 3869 3870 3871 3872 3873 3874 3875 3876 3877 3878 3879 3880 3881 3882 3883 3884 3885 3886 3887 3888 3889 3890 3891 3892 3893 3894 3895 3896 3897 3898 3899 3900 3901 3902 3903 3904 3905 3906 3907 3908 3909 3910 3911 3912 3913 3914 3915 3916 3917 3918 3919 3920 3921 3922 3923 3924 3925 3926 3927 3928 3929 3930 3931 3932 3933 3934 3935 3936 3937 3938 3939 3940 3941 3942 3943 3944 3945 3946 3947 3948 3949 3950 3951 3952 3953 3954 3955 3956 3957 3958 3959 3960 3961 3962 3963 3964 3965 3966 3967 3968 3969 3970 3971 3972 3973 3974 3975 3976 3977 3978 3979 3980 3981 3982 3983 3984 3985 3986 3987 3988 3989 3990 3991 3992 3993 3994 3995 3996 3997 3998 3999 4000 4001 4002 4003 4004 4005 4006 4007 4008 4009 4010 4011 4012 4013 4014 4015 4016 4017 4018 4019 4020 4021 4022 4023 4024 4025 4026 4027 4028 4029 4030 4031 4032 4033 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046 4047 4048 4049 4050 4051 4052 4053 4054 4055 4056 4057 4058 4059 4060 4061 4062 4063 4064 4065 4066 4067 4068 4069 4070 4071 4072 4073 4074 4075 4076 4077 4078 4079 4080 4081 4082 4083 4084 4085 4086 4087 4088 4089 4090 4091 4092 4093 4094 4095 4096 4097 4098 4099 4100 4101 4102 4103 4104 4105 4106 4107 4108 4109 4110 4111 4112 4113 4114 4115 4116 4117 4118 4119 4120 4121 4122 4123 4124 4125 4126 4127 4128 4129 4130 4131 4132 4133 4134 4135 4136 4137 4138 4139 4140 4141 4142 4143 4144 4145 4146 4147 4148 4149 4150 4151 4152 4153 4154 4155 4156 4157 4158 4159 4160 4161 4162 4163 4164 4165 4166 4167 4168 4169 4170 4171 4172 4173 4174 4175 4176 4177 4178 4179 4180 4181 4182 4183 4184 4185 4186 4187 4188 4189 4190 4191 4192 4193 4194 4195 4196 4197 4198 4199 4200 4201 4202 4203 4204 4205 4206 4207 4208 4209 4210 4211 4212 4213 4214 4215 4216 4217 4218 4219 4220 4221 4222 4223 4224 4225 4226 4227 4228 4229 4230 4231 4232 4233 4234 4235 4236 4237 4238 4239 4240 4241 4242 4243 4244 4245 4246 4247 4248 4249 4250 4251 4252 4253 4254 4255 4256 4257 4258 4259 4260 4261 4262 4263 4264 4265 4266 4267 4268 4269 4270 4271 4272 4273 4274 4275 4276 4277 4278 4279 4280 4281 4282 4283 4284 4285 4286 4287 4288 4289 4290 4291 4292 4293 4294 4295 4296 4297 4298 4299 4300 4301 4302 4303 4304 4305 4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340 4341 4342 4343 4344 4345 4346 4347 4348 4349 4350 4351 4352 4353 4354 4355 4356 4357 4358 4359 4360 4361 4362 4363 4364 4365 4366 4367 4368 4369 4370 4371 4372 4373 4374 4375 4376 4377 4378 4379 4380 4381 4382 4383 4384 4385 4386 4387 4388 4389 4390 4391 4392 4393 4394 4395 4396 4397 4398 4399 4400 4401 4402 4403 4404 4405 4406 4407 4408 4409 4410 4411 4412 4413 4414 4415 4416 4417 4418 4419 4420 4421 4422 4423 4424 4425 4426 4427 4428 4429 4430 4431 4432 4433 4434 4435 4436 4437 4438 4439 4440 4441 4442 4443 4444 4445 4446 4447 4448 4449 4450 4451 4452 4453 4454 4455 4456 4457 4458 4459 4460 4461 4462 4463 4464 4465 4466 4467 4468 4469 4470 4471 4472 4473 4474 4475 4476 4477 4478 4479 4480 4481 4482 4483 4484 4485 4486 4487 4488 4489 4490 4491 4492 4493 4494 4495 4496 4497 4498 4499 4500 4501 4502 4503 4504 4505 4506 4507 4508 4509 4510 4511 4512 4513 4514 4515 4516 4517 4518 4519 4520 4521 4522 4523 4524 4525 4526 4527 4528 4529 4530 4531 4532 4533 4534 4535 4536 4537 4538 4539 4540 4541 4542 4543 4544 4545 4546 4547 4548 4549 4550 4551 4552 4553 4554 4555 4556 4557 4558 4559 4560 4561 4562 4563 4564 4565 4566 4567 4568 4569 4570 4571 4572 4573 4574 4575 4576 4577 4578 4579 4580 4581 4582 4583 4584 4585 4586 4587 4588 4589 4590 4591 4592 4593 4594 4595 4596 4597 4598 4599 4600 4601 4602 4603 4604 4605 4606 4607 4608 4609 4610 4611 4612 4613 4614 4615 4616 4617 4618 4619 4620 4621 4622 4623 4624 4625 4626 4627 4628 4629 4630 4631 4632 4633 4634 4635 4636 4637 4638 4639 4640 4641 4642 4643 4644 4645 4646 4647 4648 4649 4650 4651 4652 4653 4654 4655 4656 4657 4658 4659 4660 4661 4662 4663 4664 4665 4666 4667 4668 4669 4670 4671 4672 4673 4674 4675 4676 4677 4678 4679 4680 4681 4682 4683 4684 4685 4686 4687 4688 4689 4690 4691 4692 4693 4694 4695 4696 4697 4698 4699 4700 4701 4702 4703 4704 4705 4706 4707 4708 4709 4710 4711 4712 4713 4714 4715 4716 4717 4718 4719 4720 4721 4722 4723 4724 4725 4726 4727 4728 4729 4730 4731 4732 4733 4734 4735 4736 4737 4738 4739 4740 4741 4742 4743 4744 4745 4746 4747 4748 4749 4750 4751 4752 4753 4754 4755 4756 4757 4758 4759 4760 4761 4762 4763 4764 4765 4766 4767 4768 4769 4770 4771 4772 4773 4774 4775 4776 4777 4778 4779 4780 4781 4782 4783 4784 4785 4786 4787 4788 4789 4790 4791 4792 4793 4794 4795 4796 4797 4798 4799 4800 4801 4802 4803 4804 4805 4806 4807 4808 4809 4810 4811 4812 4813 4814 4815 4816 4817 4818 4819 4820 4821 4822 4823 4824 4825 4826 4827 4828 4829 4830 4831 4832 4833 4834 4835 4836 4837 4838 4839 4840 4841 4842 4843 4844 4845 4846 4847 4848 4849 4850 4851 4852 4853 4854 4855 4856 4857 4858 4859 4860 4861 4862 4863 4864 4865 4866 4867 4868 4869 4870 4871 4872 4873 4874 4875 4876 4877 4878 4879 4880 4881 4882 4883 4884 4885 4886 4887 4888 4889 4890 4891 4892 4893 4894 4895 4896 4897 4898 4899 4900 4901 4902 4903 4904 4905 4906 4907 4908 4909 4910 4911 4912 4913 4914 4915 4916 4917 4918 4919 4920 4921 4922 4923 4924 4925 4926 4927 4928 4929 4930 4931 4932 4933 4934 4935 4936 4937 4938 4939 4940 4941 4942 4943 4944 4945 4946 4947 4948 4949 4950 4951 4952 4953 4954 4955 4956 4957 4958 4959 4960 4961 4962 4963 4964 4965 4966 4967 4968 4969 4970 4971 4972 4973 4974 4975 4976 4977 4978 4979 4980 4981 4982 4983 4984 4985 4986 4987 4988 4989 4990 4991 4992 4993 4994 4995 4996 4997 4998 4999 5000 5001 5002 5003 5004 5005 5006 5007 5008 5009 5010 5011 5012 5013 5014 5015 5016 5017 5018 5019 5020 5021 5022 5023 5024 5025 5026 5027 5028 5029 5030 5031 5032 5033 5034 5035 5036 5037 5038 5039 5040 5041 5042 5043 5044 5045 5046 5047 5048 5049 5050 5051 5052 5053 5054 5055 5056 5057 5058 5059 5060 5061 5062 5063 5064 5065 5066 5067 5068 5069 5070 5071 5072 5073 5074 5075 5076 5077 5078 5079 5080 5081 5082 5083 5084 5085 5086 5087 5088 5089 5090 5091 5092 5093 5094 5095 5096 5097 5098 5099 5100 5101 5102 5103 5104 5105 5106 5107 5108 5109 5110 5111 5112 5113 5114 5115 5116 5117 5118 5119 5120 5121 5122 5123 5124 5125 5126 5127 5128 5129 5130 5131 5132 5133 5134 5135 5136 5137 5138 5139 5140 5141 5142 5143 5144 5145 5146 5147 5148 5149 5150 5151 5152 5153 5154 5155 5156 5157 5158 5159 5160 5161 5162 5163 5164 5165 5166 5167 5168 5169 5170 5171 5172 5173 5174 5175 5176 5177 5178 5179 5180 5181 5182 5183 5184 5185 5186 5187 5188 5189 5190 5191 5192 5193 5194 5195 5196 5197 5198 5199 5200 5201 5202 5203 5204 5205 5206 5207 5208 5209 5210 5211 5212 5213 5214 5215 5216 5217 5218 5219 5220 5221 5222 5223 5224 5225 5226 5227 5228 5229 5230 5231 5232 5233 5234 5235 5236 5237 5238 5239 5240 5241 5242 5243 5244 5245 5246 5247 5248 5249 5250 5251 5252 5253 5254 5255 5256 5257 5258 5259 5260 5261 5262 5263 5264 5265 5266 5267 5268 5269 5270 5271 5272 5273 5274 5275 5276 5277 5278 5279 5280 5281 5282 5283 5284 5285 5286 5287 5288 5289 5290 5291 5292 5293 5294 5295 5296 5297 5298 5299 5300 5301 5302 5303 5304 5305 5306 5307 5308 5309 5310 5311 5312 5313 5314 5315 5316 5317 5318 5319 5320 5321 5322 5323 5324 5325 5326 5327 5328 5329 5330 5331 5332 5333 5334 5335 5336 5337 5338 5339 5340 5341 5342 5343 |
WEBVTT Kind: captions; Language: en-US
NOTE
Created on 2024-02-07T20:52:10.2470009Z by ClassTranscribe
00:01:22.340 --> 00:01:22.750
Good morning.
00:01:24.260 --> 00:01:27.280
Alright, so I'm going to just first
00:01:27.280 --> 00:01:29.738
finish up what I was, what I was going
00:01:29.738 --> 00:01:31.660
to cover at the end of the last lecture
00:01:31.660 --> 00:01:32.980
about Cannon.
00:01:33.640 --> 00:01:36.550
And then I'll talk about probabilities
00:01:36.550 --> 00:01:37.540
and Naive Bayes.
00:01:38.260 --> 00:01:39.940
And so I wanted to give an example of
00:01:39.940 --> 00:01:41.930
how K&N is used in practice.
00:01:42.530 --> 00:01:44.880
Here's one example of using it for face
00:01:44.880 --> 00:01:45.920
recognition.
00:01:46.750 --> 00:01:48.480
A lot of times when it's used in
00:01:48.480 --> 00:01:50.030
practice, there's a lot of feature
00:01:50.030 --> 00:01:51.780
learning that goes on ahead of the
00:01:51.780 --> 00:01:52.588
nearest neighbor.
00:01:52.588 --> 00:01:54.510
So nearest neighbor itself is really
00:01:54.510 --> 00:01:55.125
simple.
00:01:55.125 --> 00:01:58.530
It's efficacy depends on learning good
00:01:58.530 --> 00:02:00.039
representation so that.
00:02:00.800 --> 00:02:02.640
Data points that are near each other
00:02:02.640 --> 00:02:04.410
actually have similar labels.
00:02:05.450 --> 00:02:07.385
Here's one example.
00:02:07.385 --> 00:02:10.550
They want to try to be able to
00:02:10.550 --> 00:02:12.330
recognize whether two faces are the
00:02:12.330 --> 00:02:13.070
same person.
00:02:13.820 --> 00:02:16.460
And so the method is that you Detect
00:02:16.460 --> 00:02:18.940
facial features and then use those
00:02:18.940 --> 00:02:21.630
feature detections to align the image
00:02:21.630 --> 00:02:23.300
so that the face looks more frontal.
00:02:24.060 --> 00:02:26.480
Then they use a CNN convolutional
00:02:26.480 --> 00:02:29.240
neural network to train Features that
00:02:29.240 --> 00:02:32.600
will be good for recognizing faces.
00:02:32.600 --> 00:02:34.360
And the way they did that is that they
00:02:34.360 --> 00:02:37.950
first collected hundreds of Faces from
00:02:37.950 --> 00:02:40.300
a few thousand different people.
00:02:40.300 --> 00:02:41.680
I think it was their employees of
00:02:41.680 --> 00:02:42.250
Facebook.
00:02:43.030 --> 00:02:46.420
And they trained a classifier to say
00:02:46.420 --> 00:02:48.970
which, given a face, which of these
00:02:48.970 --> 00:02:50.960
people does the face belong to.
00:02:52.030 --> 00:02:54.340
And from that, they learn a
00:02:54.340 --> 00:02:55.210
REPRESENTATION.
00:02:55.210 --> 00:02:57.030
Those classifiers aren't very useful,
00:02:57.030 --> 00:02:59.300
because nobody's interested in seeing
00:02:59.300 --> 00:03:00.230
given a face.
00:03:00.230 --> 00:03:01.843
Which of the Facebook employees is that
00:03:01.843 --> 00:03:02.914
they want to know?
00:03:02.914 --> 00:03:04.932
Like, is it you want to know?
00:03:04.932 --> 00:03:07.460
Like, organize your photo album or see
00:03:07.460 --> 00:03:08.800
whether you've been tagged in another
00:03:08.800 --> 00:03:09.960
photo or something like that?
00:03:10.630 --> 00:03:12.050
And so then they throw out the
00:03:12.050 --> 00:03:13.980
Classifier and they just use the
00:03:13.980 --> 00:03:16.280
feature representation that was learned
00:03:16.280 --> 00:03:21.070
and use nearest neighbor to identify a
00:03:21.070 --> 00:03:22.510
person that's been detected in a
00:03:22.510 --> 00:03:23.090
photograph.
00:03:24.830 --> 00:03:26.540
So in their paper, they showed that
00:03:26.540 --> 00:03:28.565
this performs similarly to humans in
00:03:28.565 --> 00:03:30.470
this data set called label faces in the
00:03:30.470 --> 00:03:31.970
wild where you're trying to recognize
00:03:31.970 --> 00:03:32.560
celebrities.
00:03:34.140 --> 00:03:35.770
But it can be used for many things.
00:03:35.770 --> 00:03:37.516
So you can organize photo albums, you
00:03:37.516 --> 00:03:40.360
can detect faces and then you try to
00:03:40.360 --> 00:03:41.970
match Faces across the photos.
00:03:41.970 --> 00:03:44.175
So then you can organize like which
00:03:44.175 --> 00:03:46.320
photos have a particular person.
00:03:47.070 --> 00:03:49.950
Again, you can't identify celebrities
00:03:49.950 --> 00:03:51.860
or famous people by building up a
00:03:51.860 --> 00:03:54.919
database of faces of famous people.
00:03:55.870 --> 00:03:58.110
And you can also alert, alert somebody
00:03:58.110 --> 00:04:00.100
if somebody else uploads a photo of
00:04:00.100 --> 00:04:00.330
them.
00:04:00.330 --> 00:04:02.922
So you can see if somebody uploads a
00:04:02.922 --> 00:04:05.364
photo, then you can detect faces, you
00:04:05.364 --> 00:04:07.830
can see what their friends network is,
00:04:07.830 --> 00:04:10.056
see what other which of their faces
00:04:10.056 --> 00:04:12.220
have been uploaded and then Detect the
00:04:12.220 --> 00:04:14.330
other users whose faces have been
00:04:14.330 --> 00:04:16.580
uploaded and ask them for permission to
00:04:16.580 --> 00:04:17.930
like make this photo public.
00:04:19.750 --> 00:04:22.020
So this algorithm is actually used by
00:04:22.020 --> 00:04:22.560
Facebook.
00:04:22.560 --> 00:04:24.340
It has been for several years.
00:04:24.340 --> 00:04:28.640
They're limiting some of its use more
00:04:28.640 --> 00:04:30.544
recently, but they've been.
00:04:30.544 --> 00:04:32.010
But it's been used really heavily.
00:04:32.680 --> 00:04:34.410
And of course they have expanded
00:04:34.410 --> 00:04:36.365
training data because whenever anybody
00:04:36.365 --> 00:04:37.940
uploads photos then they can
00:04:37.940 --> 00:04:40.353
automatically detect them and add them
00:04:40.353 --> 00:04:42.360
to the database.
00:04:42.360 --> 00:04:45.150
So here the use of KN is important
00:04:45.150 --> 00:04:47.220
because KNN doesn't require any
00:04:47.220 --> 00:04:47.490
training.
00:04:47.490 --> 00:04:49.295
So every time somebody uploads a new
00:04:49.295 --> 00:04:50.930
face you can update the model just by
00:04:50.930 --> 00:04:54.430
adding this four 4096 dimensional
00:04:54.430 --> 00:04:56.646
feature vector that corresponds to the
00:04:56.646 --> 00:05:00.230
face and then use it in like based on
00:05:00.230 --> 00:05:02.550
the friend networks to.
00:05:02.910 --> 00:05:04.840
To recognize faces that are associated
00:05:04.840 --> 00:05:05.410
with somebody.
00:05:07.530 --> 00:05:11.270
I won't take time to discuss it now,
00:05:11.270 --> 00:05:13.473
but it's worth thinking about some of
00:05:13.473 --> 00:05:15.710
the consequences of the way that the
00:05:15.710 --> 00:05:17.888
algorithm was trained and the way that
00:05:17.888 --> 00:05:18.620
it's deployed.
00:05:18.620 --> 00:05:19.600
So for example.
00:05:20.510 --> 00:05:22.680
If you think about that, it was that
00:05:22.680 --> 00:05:24.650
the initial Features were learned on
00:05:24.650 --> 00:05:26.030
Facebook employees.
00:05:26.030 --> 00:05:27.440
That's not a very.
00:05:28.070 --> 00:05:29.630
That's not very representative
00:05:29.630 --> 00:05:32.120
demographic of the US the employees
00:05:32.120 --> 00:05:35.000
tend to be younger and.
00:05:35.490 --> 00:05:38.446
Probably skew towards male might skew
00:05:38.446 --> 00:05:40.210
towards certain ethnicities.
00:05:40.820 --> 00:05:43.210
And so the Algorithm may be much better
00:05:43.210 --> 00:05:45.030
at recognizing some kinds of Faces than
00:05:45.030 --> 00:05:46.016
other faces.
00:05:46.016 --> 00:05:47.628
And then, of course, there's lots and
00:05:47.628 --> 00:05:49.495
lots of ethical issues that surround
00:05:49.495 --> 00:05:51.830
the use of face recognition and its
00:05:51.830 --> 00:05:52.610
applications.
00:05:53.930 --> 00:05:55.550
Of course, like in many ways, this is
00:05:55.550 --> 00:05:58.150
used to help people maintain privacy.
00:05:58.150 --> 00:06:00.080
But even the use of recognition at all
00:06:00.080 --> 00:06:03.120
raises privacy concerns, and that's why
00:06:03.120 --> 00:06:04.860
they've limited the use to some extent.
00:06:06.470 --> 00:06:08.060
So just something to think about.
00:06:09.980 --> 00:06:13.430
So just to recap kann, the key
00:06:13.430 --> 00:06:16.480
assumptions of K&N are that K nearest
00:06:16.480 --> 00:06:18.260
neighbors that Samples with similar
00:06:18.260 --> 00:06:19.730
features will have similar output
00:06:19.730 --> 00:06:20.695
predictions.
00:06:20.695 --> 00:06:23.290
And for most of the Distance measures
00:06:23.290 --> 00:06:25.590
you implicitly assumes that all the
00:06:25.590 --> 00:06:27.200
dimensions are equally important.
00:06:27.200 --> 00:06:29.820
So it requires some kind of scaling or
00:06:29.820 --> 00:06:31.500
learning to be really effective.
00:06:33.540 --> 00:06:35.620
The parameters are just the data
00:06:35.620 --> 00:06:36.080
itself.
00:06:36.080 --> 00:06:37.870
You don't really have to learn any kind
00:06:37.870 --> 00:06:40.526
of statistics of the data.
00:06:40.526 --> 00:06:42.270
The data are the parameters.
00:06:43.820 --> 00:06:46.160
The designs are mainly the choice of K
00:06:46.160 --> 00:06:48.130
if you have higher K then it gets
00:06:48.130 --> 00:06:49.360
smoother Prediction.
00:06:50.340 --> 00:06:51.730
You can decide how you're going to
00:06:51.730 --> 00:06:54.400
combine predictions if K is greater
00:06:54.400 --> 00:06:56.750
than one, usually it's just voting or
00:06:56.750 --> 00:06:57.280
averaging.
00:06:58.610 --> 00:07:00.920
You can try to design the features and
00:07:00.920 --> 00:07:03.450
that's where things can get a lot more
00:07:03.450 --> 00:07:03.930
creative.
00:07:04.680 --> 00:07:06.770
And you can choose a Distance function.
00:07:08.900 --> 00:07:12.370
So this K&N is useful in many cases.
00:07:12.370 --> 00:07:14.520
So if you have very few examples per
00:07:14.520 --> 00:07:16.605
class then it can be applied even if
00:07:16.605 --> 00:07:17.320
you just have one.
00:07:18.080 --> 00:07:20.290
It can also work if you have many
00:07:20.290 --> 00:07:21.560
Examples per class.
00:07:22.200 --> 00:07:24.910
It's best if the features are all
00:07:24.910 --> 00:07:26.960
roughly equally important, because K&N
00:07:26.960 --> 00:07:28.540
itself doesn't really learn which
00:07:28.540 --> 00:07:29.449
features are important.
00:07:31.570 --> 00:07:33.910
It's good if the training data is
00:07:33.910 --> 00:07:34.585
changing frequently.
00:07:34.585 --> 00:07:37.520
In the face recognition Example face,
00:07:37.520 --> 00:07:38.830
there's no way that Facebook will
00:07:38.830 --> 00:07:41.160
collect everybody's Faces up front.
00:07:41.160 --> 00:07:43.030
People keep on joining and leaving the
00:07:43.030 --> 00:07:45.480
social network, and so they and they
00:07:45.480 --> 00:07:47.080
don't want to have to keep retraining
00:07:47.080 --> 00:07:49.850
models every time somebody uploads a
00:07:49.850 --> 00:07:52.005
image with a new face in it or tags a
00:07:52.005 --> 00:07:52.615
new face.
00:07:52.615 --> 00:07:54.990
And so the ability to instantly update
00:07:54.990 --> 00:07:56.330
your model is very important.
00:07:58.160 --> 00:07:59.850
You can apply it to classification or
00:07:59.850 --> 00:08:01.740
regression whether you have discrete or
00:08:01.740 --> 00:08:04.570
continuous values, and its most
00:08:04.570 --> 00:08:06.020
powerful when you do some feature
00:08:06.020 --> 00:08:08.180
learning as an upfront operation.
00:08:10.130 --> 00:08:12.210
So there's cases where it has its
00:08:12.210 --> 00:08:13.330
downsides though.
00:08:13.330 --> 00:08:15.650
One is that if you have a lot of
00:08:15.650 --> 00:08:18.250
examples that are available per class,
00:08:18.250 --> 00:08:20.360
then usually training a Logistic
00:08:20.360 --> 00:08:23.690
regressor other Linear Classifier will
00:08:23.690 --> 00:08:26.200
outperform because it's able to learn
00:08:26.200 --> 00:08:27.990
the importance of different Features.
00:08:28.950 --> 00:08:32.125
Also, K&N requires that you store all
00:08:32.125 --> 00:08:34.692
the training data and that may require
00:08:34.692 --> 00:08:38.153
a lot of storage and it requires a lot
00:08:38.153 --> 00:08:40.145
of computation, and that you have to
00:08:40.145 --> 00:08:42.200
compare each new input to all of the
00:08:42.200 --> 00:08:43.750
inputs in your training data.
00:08:43.750 --> 00:08:45.525
So in the case of Facebook for example,
00:08:45.525 --> 00:08:47.745
they don't need if somebody uploads, if
00:08:47.745 --> 00:08:49.780
they detect a face in somebody's image,
00:08:49.780 --> 00:08:51.520
they don't need to compare it to the
00:08:51.520 --> 00:08:53.410
other, like 2 billion Facebook users.
00:08:53.410 --> 00:08:55.176
They just would compare it to people in
00:08:55.176 --> 00:08:56.570
the person's social network, which will
00:08:56.570 --> 00:08:58.900
be a much smaller number of Faces.
00:08:58.970 --> 00:09:01.240
So they're able to limit the
00:09:01.240 --> 00:09:02.190
computation that way.
00:09:05.940 --> 00:09:08.760
And then finally, to recap what we
00:09:08.760 --> 00:09:12.180
learned on Thursday, there's a basic
00:09:12.180 --> 00:09:14.420
machine learning process, which is that
00:09:14.420 --> 00:09:16.170
you've got training data, validation
00:09:16.170 --> 00:09:17.260
data and TestData.
00:09:18.160 --> 00:09:19.980
Given the training data, which are
00:09:19.980 --> 00:09:22.730
pairs of Features and labels, you fit
00:09:22.730 --> 00:09:25.060
the parameters of your Model.
00:09:25.060 --> 00:09:26.950
Then you use the validation Model to
00:09:26.950 --> 00:09:28.670
check how good the Model is and maybe
00:09:28.670 --> 00:09:29.805
check many models.
00:09:29.805 --> 00:09:31.960
You choose the best one and then you
00:09:31.960 --> 00:09:33.590
get your final estimate of performance
00:09:33.590 --> 00:09:34.410
on the TestData.
00:09:36.790 --> 00:09:39.670
We talked about KNN, which is simple
00:09:39.670 --> 00:09:42.040
but effective Classifier and regressor
00:09:42.040 --> 00:09:44.140
that predicts the label of the most
00:09:44.140 --> 00:09:45.540
similar training Example.
00:09:46.770 --> 00:09:49.110
And then we talked about kind of
00:09:49.110 --> 00:09:51.110
patterns of error and what causes
00:09:51.110 --> 00:09:51.580
errors.
00:09:51.580 --> 00:09:53.780
So it's important to remember that as
00:09:53.780 --> 00:09:56.069
you get more training, more training
00:09:56.070 --> 00:09:57.830
samples, you would expect that fitting
00:09:57.830 --> 00:09:58.962
the training data gets harder.
00:09:58.962 --> 00:10:01.500
So your error will tend to go up while
00:10:01.500 --> 00:10:03.390
your error on the TestData will get
00:10:03.390 --> 00:10:05.535
lower because the training data better
00:10:05.535 --> 00:10:07.010
represents the TestData or better
00:10:07.010 --> 00:10:08.430
represents the full distribution.
00:10:09.770 --> 00:10:11.840
And there's many reasons why at the end
00:10:11.840 --> 00:10:13.250
of training your Algorithm, you're
00:10:13.250 --> 00:10:14.720
still going to have error in most
00:10:14.720 --> 00:10:15.220
cases.
00:10:15.880 --> 00:10:17.400
It could be that the problem is
00:10:17.400 --> 00:10:20.940
intrinsically difficult, or it's
00:10:20.940 --> 00:10:22.590
impossible to have 0 error.
00:10:22.590 --> 00:10:24.232
It could be that you're Model has
00:10:24.232 --> 00:10:24.845
limited power.
00:10:24.845 --> 00:10:27.370
It could be that your Model has plenty
00:10:27.370 --> 00:10:29.015
of power, but you have limited data so
00:10:29.015 --> 00:10:30.710
you can't Estimate the parameters
00:10:30.710 --> 00:10:31.290
exactly.
00:10:32.050 --> 00:10:33.100
And it could be that there's
00:10:33.100 --> 00:10:34.550
differences in the training test
00:10:34.550 --> 00:10:35.280
distribution.
00:10:37.020 --> 00:10:38.980
And then finally it's important to
00:10:38.980 --> 00:10:41.315
remember that this Model fitting, the
00:10:41.315 --> 00:10:42.980
model design and fitting is just one
00:10:42.980 --> 00:10:44.750
part of a larger processing collecting
00:10:44.750 --> 00:10:46.600
data and fitting it into an
00:10:46.600 --> 00:10:47.610
application.
00:10:47.610 --> 00:10:51.230
So both the cases of in Facebook's case
00:10:51.230 --> 00:10:54.160
for example they had pre training stage
00:10:54.160 --> 00:10:56.663
which is like training a classifier and
00:10:56.663 --> 00:10:58.852
then they use that in a different, they
00:10:58.852 --> 00:11:01.370
use it in a different way as a nearest
00:11:01.370 --> 00:11:05.320
neighbor recognizer on their pool of
00:11:05.320 --> 00:11:06.010
user data.
00:11:07.070 --> 00:11:10.384
And so they're kind of building a model
00:11:10.384 --> 00:11:11.212
using it.
00:11:11.212 --> 00:11:13.700
They're building a model one way and
00:11:13.700 --> 00:11:15.150
then using it in a different way.
00:11:15.150 --> 00:11:16.660
So often that's the case that you have
00:11:16.660 --> 00:11:17.590
to kind of be creative.
00:11:18.360 --> 00:11:20.580
About how you collect data and how you
00:11:20.580 --> 00:11:23.800
can get the model that you need to
00:11:23.800 --> 00:11:24.860
solve your application.
00:11:28.010 --> 00:11:30.033
Alright, so now I'm going to move on to
00:11:30.033 --> 00:11:31.640
the main topic of today's lecture,
00:11:31.640 --> 00:11:34.880
which is probabilities and the night
00:11:34.880 --> 00:11:35.935
based Classifier.
00:11:35.935 --> 00:11:39.690
So the knight based Classifier is
00:11:39.690 --> 00:11:41.220
unlike nearest neighbor, it's not.
00:11:41.990 --> 00:11:44.020
Usually like the final approach that
00:11:44.020 --> 00:11:46.080
somebody takes, but it's sometimes a
00:11:46.080 --> 00:11:49.460
piece of a piece of how somebody is
00:11:49.460 --> 00:11:51.210
estimating probabilities as part of
00:11:51.210 --> 00:11:51.870
their approach.
00:11:52.690 --> 00:11:55.610
And it's a good introduction to
00:11:55.610 --> 00:11:56.630
Probabilistic models.
00:11:59.220 --> 00:12:02.525
So with the nearest neighbor
00:12:02.525 --> 00:12:04.670
classifier, that's an instance based
00:12:04.670 --> 00:12:05.960
Classifier, which means that you're
00:12:05.960 --> 00:12:07.800
assigning labels just based on matching
00:12:07.800 --> 00:12:08.515
other instances.
00:12:08.515 --> 00:12:11.160
The instances the data are the Model.
00:12:12.260 --> 00:12:14.590
Now we're going to start talking about
00:12:14.590 --> 00:12:15.910
Probabilistic models.
00:12:15.910 --> 00:12:18.290
In a Probabilistic Model, you choose
00:12:18.290 --> 00:12:21.060
the label that is most likely given the
00:12:21.060 --> 00:12:21.630
Features.
00:12:21.630 --> 00:12:23.390
So that's kind of an intuitive thing to
00:12:23.390 --> 00:12:25.510
do if you want to know.
00:12:26.520 --> 00:12:28.690
Which if you're looking at an image and
00:12:28.690 --> 00:12:30.390
trying to classify it into a Digit, it
00:12:30.390 --> 00:12:32.074
makes sense that you would assign it to
00:12:32.074 --> 00:12:34.000
the Digit that is most likely given the
00:12:34.000 --> 00:12:35.940
Features given the pixel intensities.
00:12:36.610 --> 00:12:38.170
But of course, like the challenge is
00:12:38.170 --> 00:12:40.030
modeling this probability function, how
00:12:40.030 --> 00:12:42.590
do you Model the probability of the
00:12:42.590 --> 00:12:44.000
label given the data?
00:12:45.340 --> 00:12:47.520
So this is just a very compact way of
00:12:47.520 --> 00:12:48.135
writing that.
00:12:48.135 --> 00:12:50.270
So I have Y star is the predicted
00:12:50.270 --> 00:12:53.150
label, and that's equal to the argmax
00:12:53.150 --> 00:12:53.836
over Y.
00:12:53.836 --> 00:12:55.770
So it's the Y that maximizes
00:12:55.770 --> 00:12:56.950
probability of Y given X.
00:12:56.950 --> 00:12:59.250
So you assign the label that's most
00:12:59.250 --> 00:13:00.590
likely given the data.
00:13:03.170 --> 00:13:05.210
So I just want to do a very brief
00:13:05.210 --> 00:13:08.240
review of some probability things.
00:13:08.240 --> 00:13:10.730
Hopefully this looks familiar, but it's
00:13:10.730 --> 00:13:12.920
still useful to refresh on it.
00:13:13.720 --> 00:13:15.290
So first Joint and conditional
00:13:15.290 --> 00:13:16.260
probability.
00:13:16.260 --> 00:13:19.040
If you say probability of X&Y then that
00:13:19.040 --> 00:13:20.900
means the probability that both of
00:13:20.900 --> 00:13:24.180
those values are true at the same time,
00:13:24.180 --> 00:13:25.030
so.
00:13:26.330 --> 00:13:28.400
So if you say like the probability that
00:13:28.400 --> 00:13:29.290
it's sunny.
00:13:29.980 --> 00:13:32.540
And it's rainy, then that's probably a
00:13:32.540 --> 00:13:33.910
very low probability, because those
00:13:33.910 --> 00:13:35.700
usually don't happen at the same time.
00:13:35.700 --> 00:13:37.635
Both X&Y are true.
00:13:37.635 --> 00:13:40.396
That's equal to the probability of X
00:13:40.396 --> 00:13:42.179
given Y times probability of Y.
00:13:42.180 --> 00:13:45.725
So probability of X given Y is the
00:13:45.725 --> 00:13:48.700
probability that X is true given the
00:13:48.700 --> 00:13:50.956
known values of Y times the probability
00:13:50.956 --> 00:13:52.280
that Y is true.
00:13:52.970 --> 00:13:54.789
And that's also equal to probability of
00:13:54.790 --> 00:13:56.769
Y given X times probability of X.
00:13:56.770 --> 00:13:59.450
So you can take a Joint probability and
00:13:59.450 --> 00:14:01.580
turn it into a conditional probability
00:14:01.580 --> 00:14:04.370
times the probability of their meaning
00:14:04.370 --> 00:14:06.190
variables, the condition variables.
00:14:07.010 --> 00:14:08.660
And you can apply that down a chain.
00:14:08.660 --> 00:14:11.341
So probability of ABC is probability of
00:14:11.341 --> 00:14:13.531
a given BC times probability of B given
00:14:13.531 --> 00:14:14.900
C times probability of C.
00:14:17.320 --> 00:14:18.730
And then it's important to remember
00:14:18.730 --> 00:14:21.110
Bayes rule, which is a way of relating
00:14:21.110 --> 00:14:23.160
probability of X given Y and
00:14:23.160 --> 00:14:24.869
probability of Y given X.
00:14:25.520 --> 00:14:27.440
So of X given Y.
00:14:28.100 --> 00:14:30.516
Is equal to probability of Y given X
00:14:30.516 --> 00:14:32.222
times probability of X over probability
00:14:32.222 --> 00:14:35.090
of Y and you can get that by saying
00:14:35.090 --> 00:14:38.595
probability of X given Y is probability
00:14:38.595 --> 00:14:41.599
of X&Y over probability of Y.
00:14:41.600 --> 00:14:43.730
So what was done here is you multiply
00:14:43.730 --> 00:14:45.910
this by probability of Y and then
00:14:45.910 --> 00:14:47.771
divide it by probability of Y and
00:14:47.771 --> 00:14:49.501
probability of X given Y times
00:14:49.501 --> 00:14:51.519
probability of Y is probability of X&Y.
00:14:52.600 --> 00:14:54.390
And then the probability of X&Y is
00:14:54.390 --> 00:14:56.030
broken out into probability of Y given
00:14:56.030 --> 00:14:57.209
X times probability of X.
00:14:59.150 --> 00:15:01.040
So often it's the case that you want to
00:15:01.040 --> 00:15:03.484
kind of switch things you the label and
00:15:03.484 --> 00:15:06.339
you want to know the likelihood of the
00:15:06.339 --> 00:15:08.350
Features, but you have like a
00:15:08.350 --> 00:15:10.544
likelihood for that, but you want a
00:15:10.544 --> 00:15:11.830
likelihood the other way of the
00:15:11.830 --> 00:15:13.654
probability of the label given the
00:15:13.654 --> 00:15:13.868
Features.
00:15:13.868 --> 00:15:15.529
And so you use Bayes rule to kind of
00:15:15.530 --> 00:15:17.550
turn the tables on your likelihood
00:15:17.550 --> 00:15:17.950
function.
00:15:20.620 --> 00:15:25.810
So using using using these rules of
00:15:25.810 --> 00:15:26.530
probability.
00:15:27.210 --> 00:15:29.830
We can show that if I want to find the
00:15:29.830 --> 00:15:33.250
Y that maximizes the likelihood of the
00:15:33.250 --> 00:15:34.690
label given the data.
00:15:35.370 --> 00:15:38.490
That's equivalent to finding the Y that
00:15:38.490 --> 00:15:41.240
maximizes the likelihood of the data
00:15:41.240 --> 00:15:44.520
given the label times the probability
00:15:44.520 --> 00:15:45.210
of the label.
00:15:45.920 --> 00:15:47.690
So in other words, if you wanted to
00:15:47.690 --> 00:15:50.030
say, well, what is the probability that
00:15:50.030 --> 00:15:53.550
my face is Derek given my facial
00:15:53.550 --> 00:15:54.220
features?
00:15:54.950 --> 00:15:56.100
That's the top.
00:15:56.100 --> 00:15:58.323
That's equivalent to saying what's the
00:15:58.323 --> 00:16:00.400
probability that it's me without
00:16:00.400 --> 00:16:02.635
looking at the Features times the
00:16:02.635 --> 00:16:04.270
probability of my Features given that
00:16:04.270 --> 00:16:04.870
it's me?
00:16:04.870 --> 00:16:05.980
Those are the same.
00:16:06.330 --> 00:16:09.770
Those the why that maximizes that is
00:16:09.770 --> 00:16:11.150
going to be the same so.
00:16:12.990 --> 00:16:15.230
And the reason for that is derived down
00:16:15.230 --> 00:16:15.720
here.
00:16:15.720 --> 00:16:17.473
So I can take Y given X.
00:16:17.473 --> 00:16:20.686
So argmax of Y given X is the as argmax
00:16:20.686 --> 00:16:23.029
of Y given X times probability of X.
00:16:23.780 --> 00:16:26.000
And the reason for that is just that
00:16:26.000 --> 00:16:27.880
probability of X doesn't depend on Y.
00:16:27.880 --> 00:16:31.140
So I can multiply multiply this thing
00:16:31.140 --> 00:16:33.092
in the argmax by anything that doesn't
00:16:33.092 --> 00:16:35.410
depend on Y and it's going to be
00:16:35.410 --> 00:16:37.890
unchanged because it's just going to.
00:16:38.870 --> 00:16:41.460
The way that maximizes it will be the
00:16:41.460 --> 00:16:41.780
same.
00:16:43.410 --> 00:16:44.940
So then I turn that.
00:16:45.530 --> 00:16:47.810
I turned that into the Joint Y&X and
00:16:47.810 --> 00:16:48.940
then I broke it out again.
00:16:49.900 --> 00:16:51.300
Right, so the reason why this is
00:16:51.300 --> 00:16:54.430
important is that I can choose to
00:16:54.430 --> 00:16:57.562
either Model directly the probability
00:16:57.562 --> 00:17:00.659
of the label given the data, or I can
00:17:00.659 --> 00:17:02.231
choose the Model the probability of the
00:17:02.231 --> 00:17:03.129
data given the label.
00:17:03.910 --> 00:17:06.172
In a Naive Bayes, we're going to Model
00:17:06.172 --> 00:17:07.950
probability the data given the label,
00:17:07.950 --> 00:17:09.510
and then in the next class we'll talk
00:17:09.510 --> 00:17:11.425
about logistic regression where we try
00:17:11.425 --> 00:17:12.930
to directly Model the probability of
00:17:12.930 --> 00:17:14.000
the label given the data.
00:17:22.090 --> 00:17:24.760
All right, so let's just.
00:17:26.170 --> 00:17:29.400
Do a simple probability exercise just
00:17:29.400 --> 00:17:31.430
to kind of make sure that.
00:17:33.430 --> 00:17:34.730
That we get.
00:17:37.010 --> 00:17:38.230
So let's say.
00:17:39.620 --> 00:17:41.060
Here I have a feature.
00:17:41.060 --> 00:17:41.970
Doesn't really matter what the
00:17:41.970 --> 00:17:43.440
Features, but let's say that it's
00:17:43.440 --> 00:17:45.233
whether something is larger than £10
00:17:45.233 --> 00:17:48.210
and I collected a bunch of different
00:17:48.210 --> 00:17:50.530
animals, cats and dogs and measured
00:17:50.530 --> 00:17:50.770
them.
00:17:51.450 --> 00:17:53.130
And I want to train something that will
00:17:53.130 --> 00:17:54.510
tell me whether or not something is a
00:17:54.510 --> 00:17:54.810
cat.
00:17:55.730 --> 00:17:57.370
And so.
00:17:58.190 --> 00:18:00.985
Or a dog, and so I have like 40
00:18:00.985 --> 00:18:03.280
different cats and 45 different dogs,
00:18:03.280 --> 00:18:04.860
and I measured whether or not they're
00:18:04.860 --> 00:18:06.693
bigger than £10.
00:18:06.693 --> 00:18:10.270
So first, given this empirical
00:18:10.270 --> 00:18:12.505
distribution, given these samples that
00:18:12.505 --> 00:18:15.120
I have, what's the probability that Y
00:18:15.120 --> 00:18:15.810
is a cat?
00:18:22.430 --> 00:18:25.970
So it's actually 40 / 85 because it's
00:18:25.970 --> 00:18:26.960
going to be.
00:18:27.640 --> 00:18:29.030
Let me see if I can write on this.
00:18:36.840 --> 00:18:37.330
OK.
00:18:39.520 --> 00:18:40.460
That's not what I wanted.
00:18:43.970 --> 00:18:45.500
If I can get the pen to work.
00:18:48.610 --> 00:18:50.360
OK, it doesn't work that well.
00:18:55.010 --> 00:18:56.250
OK, forget that.
00:18:56.250 --> 00:18:57.420
Alright, I'll write it on the board.
00:18:57.420 --> 00:18:59.639
So it's 40 / 85.
00:19:01.780 --> 00:19:05.010
So it's 40 / 40 + 45.
00:19:05.920 --> 00:19:08.595
And that's because there's 40 cats and
00:19:08.595 --> 00:19:09.888
there's 45 dogs.
00:19:09.888 --> 00:19:13.040
So I take the count of all the cats and
00:19:13.040 --> 00:19:14.970
divide it by the count of all the data
00:19:14.970 --> 00:19:16.635
in total, all the cats and dogs.
00:19:16.635 --> 00:19:17.860
So that's 40 / 85.
00:19:18.580 --> 00:19:20.470
And what's the probability that Y is a
00:19:20.470 --> 00:19:22.810
cat given that X is false?
00:19:29.380 --> 00:19:31.510
So it's right?
00:19:31.510 --> 00:19:34.240
So it's 15 / 20 or 3 / 4.
00:19:34.240 --> 00:19:35.890
And that's because given that X is
00:19:35.890 --> 00:19:37.620
false, I'm just in this one column
00:19:37.620 --> 00:19:40.799
here, so it's 15 / 15 / 20.
00:19:42.090 --> 00:19:45.110
And what's the probability that X is
00:19:45.110 --> 00:19:46.650
false given that Y is a cat?
00:19:49.320 --> 00:19:51.570
Right, 15 / 480 because if I know that
00:19:51.570 --> 00:19:53.500
Y is a Cat, then I'm in the top row, so
00:19:53.500 --> 00:19:55.590
it's just 15 divided by all the cats,
00:19:55.590 --> 00:19:56.650
so 15 / 40.
00:19:58.320 --> 00:20:00.737
OK, and it's important to remember that
00:20:00.737 --> 00:20:03.119
Y given X is different than X given Y.
00:20:05.110 --> 00:20:08.276
Right, so some other simple rules of
00:20:08.276 --> 00:20:08.572
probability.
00:20:08.572 --> 00:20:11.150
One is the law of total probability.
00:20:11.150 --> 00:20:13.060
That is, if you sum over all the values
00:20:13.060 --> 00:20:16.020
of a variable, then the sum of those
00:20:16.020 --> 00:20:17.630
probabilities is equal to 1.
00:20:18.240 --> 00:20:20.450
And if this were a continuous variable,
00:20:20.450 --> 00:20:21.840
it would just be an integral over the
00:20:21.840 --> 00:20:23.716
domain of X over all the values of X
00:20:23.716 --> 00:20:26.180
and then the integral over P of X is
00:20:26.180 --> 00:20:26.690
equal to 1.
00:20:27.980 --> 00:20:29.470
Then I've got Marginalization.
00:20:29.470 --> 00:20:31.990
So if I have a joint probability of two
00:20:31.990 --> 00:20:34.150
variables and I want to get rid of one
00:20:34.150 --> 00:20:34.520
of them.
00:20:35.280 --> 00:20:37.630
Then I take this sum over all the
00:20:37.630 --> 00:20:39.290
values of 1 and the variables.
00:20:39.290 --> 00:20:41.052
In this case it's the sum over all the
00:20:41.052 --> 00:20:41.900
values of X.
00:20:42.570 --> 00:20:46.268
Of X&Y and that's going to be equal to
00:20:46.268 --> 00:20:46.910
P of Y.
00:20:53.440 --> 00:20:55.380
And then finally independence.
00:20:55.380 --> 00:20:59.691
So A is independent of B if and only if
00:20:59.691 --> 00:21:02.414
the probability of A&B is equal to the
00:21:02.414 --> 00:21:04.115
probability of a times the probability
00:21:04.115 --> 00:21:04.660
of B.
00:21:05.430 --> 00:21:07.974
Or another way to write it is that
00:21:07.974 --> 00:21:10.142
probability that what this implies is
00:21:10.142 --> 00:21:12.500
that probability of a given B is equal
00:21:12.500 --> 00:21:13.890
to probability of a.
00:21:13.890 --> 00:21:15.680
So if I just divide both sides by
00:21:15.680 --> 00:21:17.250
probability of B then I get that.
00:21:18.160 --> 00:21:20.855
Or probability of B given A equals
00:21:20.855 --> 00:21:22.010
probability of B.
00:21:22.010 --> 00:21:24.150
So these things are the top one.
00:21:24.150 --> 00:21:25.700
Might not be something that pops into
00:21:25.700 --> 00:21:26.420
your head right away.
00:21:26.420 --> 00:21:28.450
It's not necessarily as intuitive, but
00:21:28.450 --> 00:21:30.001
these are pretty intuitive that
00:21:30.001 --> 00:21:32.376
probability of a given B equals
00:21:32.376 --> 00:21:33.564
probability of a.
00:21:33.564 --> 00:21:36.050
So in other words, whether or not a is
00:21:36.050 --> 00:21:37.470
true doesn't depend on B at all.
00:21:38.720 --> 00:21:40.430
And whether or not B is true doesn't
00:21:40.430 --> 00:21:42.360
depend on A at all, and then you can
00:21:42.360 --> 00:21:44.810
easily get to the one up there just by
00:21:44.810 --> 00:21:47.410
multiplying here both sides by
00:21:47.410 --> 00:21:48.100
probability of a.
00:21:56.140 --> 00:21:59.180
Alright, so in some of the slides
00:21:59.180 --> 00:22:00.650
there's going to be a bunch of like
00:22:00.650 --> 00:22:02.760
indices, so I just wanted to try to be
00:22:02.760 --> 00:22:04.370
consistent in the way that I use them.
00:22:05.030 --> 00:22:07.674
And also like usually verbally say what
00:22:07.674 --> 00:22:10.543
the what the variables mean, but when I
00:22:10.543 --> 00:22:14.300
say XI mean the ith feature so I is a
00:22:14.300 --> 00:22:15.085
feature index.
00:22:15.085 --> 00:22:18.619
When I say XNI mean the nth sample, so
00:22:18.620 --> 00:22:20.520
north is the sample index and Lynn
00:22:20.520 --> 00:22:21.590
would be the nth label.
00:22:22.370 --> 00:22:24.993
So if I say X and I, then that's the
00:22:24.993 --> 00:22:26.760
ith feature of the nth label.
00:22:26.760 --> 00:22:29.763
So for digits for example, would be the
00:22:29.763 --> 00:22:33.720
ith pixel of the nth Digit Example.
00:22:35.070 --> 00:22:37.580
I used this delta here to indicate with
00:22:37.580 --> 00:22:39.900
some expression inside to indicate that
00:22:39.900 --> 00:22:42.780
it returns true or returns one if the
00:22:42.780 --> 00:22:44.850
expression inside it is true and 0
00:22:44.850 --> 00:22:45.410
otherwise.
00:22:46.200 --> 00:22:48.110
And I'll Use V for a feature value.
00:22:55.320 --> 00:22:57.900
So if I want to Estimate the
00:22:57.900 --> 00:22:59.830
probabilities of some function, I can
00:22:59.830 --> 00:23:00.578
just do it by counting.
00:23:00.578 --> 00:23:02.760
So if I want to say what is the
00:23:02.760 --> 00:23:04.950
probability that X equals some value
00:23:04.950 --> 00:23:07.600
and I have capital N Samples, then I
00:23:07.600 --> 00:23:09.346
can just take a sum over all the
00:23:09.346 --> 00:23:11.350
samples and count for how many of them
00:23:11.350 --> 00:23:14.030
XN equals V so that's kind of intuitive
00:23:14.030 --> 00:23:14.480
if I have.
00:23:15.870 --> 00:23:17.750
If I have a month full of days and I
00:23:17.750 --> 00:23:19.280
want to say what's the probability that
00:23:19.280 --> 00:23:21.610
one of those days is sunny, then I can
00:23:21.610 --> 00:23:23.809
just take a sum over all the I can
00:23:23.810 --> 00:23:25.370
count how many sunny days there were
00:23:25.370 --> 00:23:26.908
divided by the total number of days and
00:23:26.908 --> 00:23:27.930
that gives me an Estimate.
00:23:31.930 --> 00:23:35.340
But what if I have 100 variables?
00:23:35.340 --> 00:23:36.380
So if I have.
00:23:37.310 --> 00:23:39.220
For example, in the digits case I have
00:23:39.220 --> 00:23:42.840
784 different and pixel intensities.
00:23:43.710 --> 00:23:46.350
And there's no way I can count over all
00:23:46.350 --> 00:23:48.222
possible combinations of pixel
00:23:48.222 --> 00:23:49.000
intensities, right?
00:23:49.000 --> 00:23:51.470
Even if I were to turn them into binary
00:23:51.470 --> 00:23:56.070
values, there would be 2 to the 784
00:23:56.070 --> 00:23:58.107
different combinations of pixel
00:23:58.107 --> 00:23:58.670
intensities.
00:23:58.670 --> 00:24:01.635
So you would need like data samples
00:24:01.635 --> 00:24:03.520
that are equal to like number of atoms
00:24:03.520 --> 00:24:05.300
in the universe or something like that
00:24:05.300 --> 00:24:07.415
in order to even begin to Estimate it.
00:24:07.415 --> 00:24:08.900
And that would that would only be
00:24:08.900 --> 00:24:10.460
giving you very few samples per
00:24:10.460 --> 00:24:11.050
combination.
00:24:12.860 --> 00:24:15.407
So obviously, like jointly modeling a
00:24:15.407 --> 00:24:17.799
whole bunch of different, the
00:24:17.800 --> 00:24:19.431
probability of a whole bunch of
00:24:19.431 --> 00:24:20.740
different variables is usually
00:24:20.740 --> 00:24:23.490
impossible, and even approximating it,
00:24:23.490 --> 00:24:24.880
it's very challenging.
00:24:24.880 --> 00:24:26.260
You have to try to solve for the
00:24:26.260 --> 00:24:28.036
dependency structures and then solve
00:24:28.036 --> 00:24:30.236
for different combinations of variables
00:24:30.236 --> 00:24:30.699
and.
00:24:31.550 --> 00:24:33.740
And then worry about the dependencies
00:24:33.740 --> 00:24:35.040
that aren't fully accounted for.
00:24:35.880 --> 00:24:37.670
And so it's just really difficult to
00:24:37.670 --> 00:24:40.160
estimate the probability of all your
00:24:40.160 --> 00:24:41.810
Features given the label.
00:24:42.900 --> 00:24:43.610
Jointly.
00:24:44.440 --> 00:24:47.540
And so that's the Naive Bayes Model
00:24:47.540 --> 00:24:48.240
comes in.
00:24:48.240 --> 00:24:50.430
It makes us greatly simplifying
00:24:50.430 --> 00:24:51.060
assumption.
00:24:51.730 --> 00:24:54.132
Which is that all of the features are
00:24:54.132 --> 00:24:56.010
independent given the label, so it
00:24:56.010 --> 00:24:57.480
doesn't mean the Features are
00:24:57.480 --> 00:24:57.840
independent.
00:24:57.940 --> 00:25:00.200
Unconditionally, but they're
00:25:00.200 --> 00:25:02.370
independent given the label, so.
00:25:03.550 --> 00:25:05.716
So because of because they're
00:25:05.716 --> 00:25:06.149
independent.
00:25:06.150 --> 00:25:08.400
Remember that probability of A&B equals
00:25:08.400 --> 00:25:11.173
probability of a * b times probability
00:25:11.173 --> 00:25:12.603
B if they're independent.
00:25:12.603 --> 00:25:15.160
So probability of X that's like a Joint
00:25:15.160 --> 00:25:17.920
X, all the Features given Y is equal to
00:25:17.920 --> 00:25:20.501
the product over all the features of
00:25:20.501 --> 00:25:22.919
probability of each feature given Y.
00:25:24.880 --> 00:25:28.866
And so then I can make my Classifier as
00:25:28.866 --> 00:25:30.450
the Y star.
00:25:30.450 --> 00:25:32.880
The most likely label is the one that
00:25:32.880 --> 00:25:35.415
maximizes this joint probability of
00:25:35.415 --> 00:25:37.930
probability of X given Y times
00:25:37.930 --> 00:25:38.779
probability of Y.
00:25:39.810 --> 00:25:42.715
And this thing, the joint probability
00:25:42.715 --> 00:25:44.985
of X given Y would be really hard to
00:25:44.985 --> 00:25:45.240
Estimate.
00:25:45.240 --> 00:25:47.490
You need tons of data, but this is not
00:25:47.490 --> 00:25:49.120
so hard to Estimate because you're just
00:25:49.120 --> 00:25:50.590
estimating the probability of 1
00:25:50.590 --> 00:25:51.590
variable at a time.
00:25:57.200 --> 00:25:59.190
So for example if I.
00:25:59.810 --> 00:26:01.900
In the Digit Example, this would be
00:26:01.900 --> 00:26:03.860
saying that the I'm going to choose the
00:26:03.860 --> 00:26:07.310
label that maximizes the product of
00:26:07.310 --> 00:26:09.220
likelihoods of each of the pixel
00:26:09.220 --> 00:26:09.980
intensities.
00:26:10.690 --> 00:26:12.555
So I'm just going to consider each
00:26:12.555 --> 00:26:13.170
pixel.
00:26:13.170 --> 00:26:15.170
How likely is each pixel to have its
00:26:15.170 --> 00:26:16.959
intensity given the label?
00:26:16.960 --> 00:26:18.230
And then I choose the label that
00:26:18.230 --> 00:26:20.132
maximizes that, taking the product of
00:26:20.132 --> 00:26:21.760
all the all those likelihoods over the
00:26:21.760 --> 00:26:22.140
pixels.
00:26:23.210 --> 00:26:23.690
So.
00:26:24.650 --> 00:26:26.880
Obviously it's not a perfect Model,
00:26:26.880 --> 00:26:28.210
even if I know that.
00:26:28.210 --> 00:26:30.610
If I'm given that it's a three, knowing
00:26:30.610 --> 00:26:32.759
that one pixel has an intensity of 1
00:26:32.760 --> 00:26:33.920
makes it more likely that the
00:26:33.920 --> 00:26:35.815
neighboring pixel has a likelihood of
00:26:35.815 --> 00:26:36.240
1.
00:26:36.240 --> 00:26:37.630
On the other hand, it's not a terrible
00:26:37.630 --> 00:26:38.710
Model either.
00:26:38.710 --> 00:26:41.028
If I know that it's a 3, then I have a
00:26:41.028 --> 00:26:43.210
pretty good idea of the expected
00:26:43.210 --> 00:26:45.177
intensity of each pixel, so I have a
00:26:45.177 --> 00:26:46.503
pretty good idea of how likely each
00:26:46.503 --> 00:26:47.920
pixel is to be a one or a zero.
00:26:50.490 --> 00:26:51.780
In the case of the temperature
00:26:51.780 --> 00:26:53.760
Regression will make a slightly
00:26:53.760 --> 00:26:55.040
different assumption.
00:26:55.040 --> 00:26:57.736
So here we have continuous Features and
00:26:57.736 --> 00:26:59.320
a continuous Prediction.
00:27:00.030 --> 00:27:02.840
So we're going to assume that each
00:27:02.840 --> 00:27:05.490
feature predicts the temperature that
00:27:05.490 --> 00:27:07.690
we're trying to predict the tomorrow's
00:27:07.690 --> 00:27:10.160
Cleveland temperature with some offset
00:27:10.160 --> 00:27:10.673
and variance.
00:27:10.673 --> 00:27:13.100
So for example, if I know yesterday's
00:27:13.100 --> 00:27:14.670
Cleveland temperature, then tomorrow's
00:27:14.670 --> 00:27:16.633
Cleveland temperature is probably about
00:27:16.633 --> 00:27:19.300
the same, but with some variance around
00:27:19.300 --> 00:27:19.577
it.
00:27:19.577 --> 00:27:21.239
If I know the Cleveland temperature
00:27:21.240 --> 00:27:23.520
from three days ago, then tomorrow's is
00:27:23.520 --> 00:27:25.732
also expected to be about the same but
00:27:25.732 --> 00:27:26.525
with higher variance.
00:27:26.525 --> 00:27:28.596
If I know the temperature of Austin,
00:27:28.596 --> 00:27:30.590
TX, then probably Cleveland is a bit
00:27:30.590 --> 00:27:31.819
colder with some variance.
00:27:33.550 --> 00:27:34.940
And so I'm going to use just that
00:27:34.940 --> 00:27:37.100
combination of individual predictions
00:27:37.100 --> 00:27:38.480
to make my final prediction.
00:27:44.170 --> 00:27:48.680
So here is the Naive Bayes Algorithm.
00:27:49.540 --> 00:27:53.250
For training, I Estimate the parameters
00:27:53.250 --> 00:27:55.370
for each of my likelihood functions,
00:27:55.370 --> 00:27:57.290
the probability of each feature given
00:27:57.290 --> 00:27:57.910
the label.
00:27:58.940 --> 00:28:01.878
And I Estimate the parameters for my
00:28:01.878 --> 00:28:02.232
prior.
00:28:02.232 --> 00:28:06.640
The prior is like the my Estimate, my
00:28:06.640 --> 00:28:08.370
likelihood of the label when I don't
00:28:08.370 --> 00:28:10.180
know anything else, just before I look
00:28:10.180 --> 00:28:11.200
at anything.
00:28:11.200 --> 00:28:13.475
So the probability of the label.
00:28:13.475 --> 00:28:14.770
And that's usually really easy to
00:28:14.770 --> 00:28:15.140
Estimate.
00:28:17.020 --> 00:28:19.280
And then at Prediction time, I'm going
00:28:19.280 --> 00:28:22.970
to solve for the label that maximizes
00:28:22.970 --> 00:28:26.330
the probability of X&Y or the and which
00:28:26.330 --> 00:28:28.620
the Naive Bayes assumption is the
00:28:28.620 --> 00:28:31.110
product over I of probability of XI
00:28:31.110 --> 00:28:32.649
given Y times probability of Y.
00:28:36.470 --> 00:28:40.455
The Naive Naive Bayes is that it's just
00:28:40.455 --> 00:28:42.050
the independence assumption.
00:28:42.050 --> 00:28:45.150
It's not an insult to Thomas Bayes that
00:28:45.150 --> 00:28:46.890
he's an idiot or something.
00:28:46.890 --> 00:28:49.970
It's just that we're going to make this
00:28:49.970 --> 00:28:52.140
very simplifying assumption.
00:28:58.170 --> 00:29:00.550
So all right, so the first thing we
00:29:00.550 --> 00:29:02.710
have to deal with is how do we Estimate
00:29:02.710 --> 00:29:03.590
this probability?
00:29:03.590 --> 00:29:06.500
We want to get some probability of each
00:29:06.500 --> 00:29:08.050
feature given the data.
00:29:08.960 --> 00:29:10.990
And the basic principles are that you
00:29:10.990 --> 00:29:12.909
want to choose parameters.
00:29:12.910 --> 00:29:14.550
First you have to have a model for your
00:29:14.550 --> 00:29:16.610
likelihood, and then you have to
00:29:16.610 --> 00:29:19.394
maximize the parameters of that model
00:29:19.394 --> 00:29:21.908
that you have to, sorry, Choose the
00:29:21.908 --> 00:29:22.885
parameters of that Model.
00:29:22.885 --> 00:29:25.180
That makes your training data most
00:29:25.180 --> 00:29:25.600
likely.
00:29:25.600 --> 00:29:27.210
That's the main principle.
00:29:27.210 --> 00:29:29.780
So if I say somebody says maximum
00:29:29.780 --> 00:29:32.390
likelihood estimation or Emily, that's
00:29:32.390 --> 00:29:34.190
like straight up maximizes the
00:29:34.190 --> 00:29:37.865
probability of the data given your
00:29:37.865 --> 00:29:38.800
parameters in your model.
00:29:40.320 --> 00:29:42.480
Sometimes that can result in
00:29:42.480 --> 00:29:44.120
overconfident estimates.
00:29:44.120 --> 00:29:46.210
So for example if I just have like.
00:29:46.970 --> 00:29:47.800
If I.
00:29:48.430 --> 00:29:51.810
If I have like 2 measurements, let's
00:29:51.810 --> 00:29:53.470
say I want to know what's the average
00:29:53.470 --> 00:29:56.044
weight of a bird and I just have two
00:29:56.044 --> 00:29:58.480
birds, and I say it's probably like a
00:29:58.480 --> 00:29:59.585
Gaussian distribution.
00:29:59.585 --> 00:30:02.012
I can Estimate a mean and a variance
00:30:02.012 --> 00:30:05.970
from those two birds, but that Estimate
00:30:05.970 --> 00:30:07.105
could be like way off.
00:30:07.105 --> 00:30:09.100
So often it's a good idea to have some
00:30:09.100 --> 00:30:11.530
kind of Prior or to prevent the
00:30:11.530 --> 00:30:12.780
variance from going too low.
00:30:12.780 --> 00:30:14.740
So if I looked at two birds and I said
00:30:14.740 --> 00:30:16.860
and they both happen to be like 47
00:30:16.860 --> 00:30:17.510
grams.
00:30:17.870 --> 00:30:19.965
I probably wouldn't want to say that
00:30:19.965 --> 00:30:22.966
the mean is 47 and the variance is 0,
00:30:22.966 --> 00:30:25.170
because then I would be saying like if
00:30:25.170 --> 00:30:27.090
there's another bird that has 48 grams,
00:30:27.090 --> 00:30:28.550
that's like infinitely unlikely.
00:30:28.550 --> 00:30:29.880
It's a 0 probability.
00:30:29.880 --> 00:30:31.600
So often you want to have some kind of
00:30:31.600 --> 00:30:34.270
Prior over your variables as well in
00:30:34.270 --> 00:30:37.025
order to prevent likelihoods going to 0
00:30:37.025 --> 00:30:38.430
because you just didn't have enough
00:30:38.430 --> 00:30:40.120
data to correctly Estimate them.
00:30:40.930 --> 00:30:42.650
So it's like Warren Buffett says with
00:30:42.650 --> 00:30:43.230
investing.
00:30:43.850 --> 00:30:45.550
It's not just about maximizing the
00:30:45.550 --> 00:30:47.690
expectation, it's also about making
00:30:47.690 --> 00:30:48.890
sure there are no zeros.
00:30:48.890 --> 00:30:50.190
Because if you have a zero and your
00:30:50.190 --> 00:30:51.670
product of likelihoods, the whole thing
00:30:51.670 --> 00:30:52.090
is 0.
00:30:53.690 --> 00:30:55.995
And if you have a zero, return your
00:30:55.995 --> 00:30:57.900
whole investment at any point, your
00:30:57.900 --> 00:30:59.330
whole bank account is 0.
00:31:03.120 --> 00:31:06.550
All right, so we have so.
00:31:06.920 --> 00:31:08.840
How do we Estimate P of X given Y given
00:31:08.840 --> 00:31:09.340
the data?
00:31:09.340 --> 00:31:10.980
It's always based on maximizing the
00:31:10.980 --> 00:31:11.930
likelihood of the data.
00:31:12.690 --> 00:31:14.360
Over your parameters, but you have
00:31:14.360 --> 00:31:15.940
different solutions depending on your
00:31:15.940 --> 00:31:18.200
Model and.
00:31:18.370 --> 00:31:19.860
I guess it just depends on your Model.
00:31:20.520 --> 00:31:24.180
So for binomial, a binomial is just if
00:31:24.180 --> 00:31:25.790
you have a binary variable, then
00:31:25.790 --> 00:31:27.314
there's some probability that the
00:31:27.314 --> 00:31:29.450
variable is 1 and 1 minus that
00:31:29.450 --> 00:31:31.790
probability that the variable is 0.
00:31:31.790 --> 00:31:36.126
So Theta Ki is the probability that X I
00:31:36.126 --> 00:31:38.510
= 1 given y = K.
00:31:39.510 --> 00:31:40.590
And you can write it.
00:31:40.590 --> 00:31:42.349
It's kind of a weird way.
00:31:42.350 --> 00:31:43.700
I mean it looks like a weird way to
00:31:43.700 --> 00:31:44.390
write it.
00:31:44.390 --> 00:31:46.190
But if you think about it, if XI equals
00:31:46.190 --> 00:31:48.760
one, then the probability is Theta Ki.
00:31:49.390 --> 00:31:51.630
And if XI equals zero, then the
00:31:51.630 --> 00:31:54.160
probability is 1 minus Theta Ki so.
00:31:54.800 --> 00:31:55.440
Makes sense?
00:31:56.390 --> 00:31:58.390
And if I want to Estimate this, all I
00:31:58.390 --> 00:32:00.530
have to do is count over all my data
00:32:00.530 --> 00:32:01.180
Samples.
00:32:01.180 --> 00:32:06.410
How many times does xni equal 1 and y =
00:32:06.410 --> 00:32:06.880
K?
00:32:07.530 --> 00:32:09.310
Divided by the total number of times
00:32:09.310 --> 00:32:10.490
that Y and equals K.
00:32:11.610 --> 00:32:13.290
And then here it is in Python.
00:32:13.290 --> 00:32:15.620
So it's just a sum over all my data.
00:32:15.620 --> 00:32:18.170
I'm looking at the ith feature here,
00:32:18.170 --> 00:32:20.377
checking how many times these equal 1
00:32:20.377 --> 00:32:23.585
and the label is equal to K divided by
00:32:23.585 --> 00:32:25.170
the number of times the label is equal
00:32:25.170 --> 00:32:25.580
to K.
00:32:27.240 --> 00:32:28.780
And if I have a multinomial, it's
00:32:28.780 --> 00:32:31.100
basically the same thing except that I
00:32:31.100 --> 00:32:35.342
sum over the number of times that X and
00:32:35.342 --> 00:32:37.990
I = V, where V could be say, zero to 10
00:32:37.990 --> 00:32:38.840
or something like that.
00:32:39.740 --> 00:32:42.490
And otherwise it's the same.
00:32:42.490 --> 00:32:46.040
So I can Estimate if I have 10
00:32:46.040 --> 00:32:49.576
different variables and I Estimate
00:32:49.576 --> 00:32:52.590
Theta KIV for all 10 variables, then
00:32:52.590 --> 00:32:54.410
the sum of those Theta kives should be
00:32:54.410 --> 00:32:54.624
one.
00:32:54.624 --> 00:32:56.540
So one of those is a constrained
00:32:56.540 --> 00:32:56.910
variable.
00:32:58.820 --> 00:33:00.420
And it will workout that way if you
00:33:00.420 --> 00:33:01.270
Estimate it this way.
00:33:05.970 --> 00:33:08.733
So if we have a continuous variable by
00:33:08.733 --> 00:33:11.730
the way, like, these can be fairly
00:33:11.730 --> 00:33:15.360
easily derived just by writing out the
00:33:15.360 --> 00:33:18.720
likelihood terms and taking a partial
00:33:18.720 --> 00:33:21.068
derivative with respect to the variable
00:33:21.068 --> 00:33:22.930
and setting that equal to 0.
00:33:22.930 --> 00:33:24.810
But it does take like a page of
00:33:24.810 --> 00:33:26.940
equations, so I decided not to subject
00:33:26.940 --> 00:33:27.379
you to it.
00:33:28.260 --> 00:33:30.190
Since since, solving for these is not
00:33:30.190 --> 00:33:30.920
the point right now.
00:33:32.920 --> 00:33:34.730
And so.
00:33:34.800 --> 00:33:36.000
Are.
00:33:36.000 --> 00:33:38.620
Let's say X is a continuous variable.
00:33:38.620 --> 00:33:40.740
Maybe I want to assume that XI is a
00:33:40.740 --> 00:33:44.052
Gaussian given some label, where the
00:33:44.052 --> 00:33:45.770
label is a discrete variable.
00:33:47.220 --> 00:33:51.023
So Gaussians, if you took hopefully you
00:33:51.023 --> 00:33:52.625
took probably your statistics and you
00:33:52.625 --> 00:33:53.940
probably ran into Gaussians all the
00:33:53.940 --> 00:33:54.230
time.
00:33:54.230 --> 00:33:55.820
Gaussians come up a lot for many
00:33:55.820 --> 00:33:56.550
reasons.
00:33:56.550 --> 00:33:58.749
One of them is that if you add a lot of
00:33:58.750 --> 00:34:01.125
random variables together, then if you
00:34:01.125 --> 00:34:02.839
add enough of them, then it will end up
00:34:02.840 --> 00:34:03.000
there.
00:34:03.000 --> 00:34:04.280
Some of them will end up being a
00:34:04.280 --> 00:34:05.320
Gaussian distribution.
00:34:07.080 --> 00:34:09.415
So there's lots of things end up being
00:34:09.415 --> 00:34:09.700
Gaussians.
00:34:09.700 --> 00:34:11.500
Gaussians is a really common noise
00:34:11.500 --> 00:34:13.536
model, and it also is like really easy
00:34:13.536 --> 00:34:14.320
to work with.
00:34:14.320 --> 00:34:16.060
Even though it looks complicated.
00:34:16.060 --> 00:34:17.820
When you take the log of it ends up
00:34:17.820 --> 00:34:19.342
just being a quadratic, which is easy
00:34:19.342 --> 00:34:20.010
to minimize.
00:34:22.250 --> 00:34:24.460
So there's the Gaussian expression on
00:34:24.460 --> 00:34:24.950
the top.
00:34:26.550 --> 00:34:28.420
And I.
00:34:29.290 --> 00:34:30.610
So let me get my.
00:34:33.940 --> 00:34:34.490
There it goes.
00:34:34.490 --> 00:34:37.060
OK, so here's the Gaussian expression
00:34:37.060 --> 00:34:39.260
one over square of 2π Sigma Ki.
00:34:39.260 --> 00:34:42.075
So the parameters here are M UI which
00:34:42.075 --> 00:34:43.830
is mu Ki which is the mean.
00:34:44.980 --> 00:34:47.700
For the KTH label and the ith feature
00:34:47.700 --> 00:34:49.946
in Sigma, Ki is the stair deviation for
00:34:49.946 --> 00:34:52.080
the Keith label and the Ith feature.
00:34:52.900 --> 00:34:54.700
And so the higher the standard
00:34:54.700 --> 00:34:57.090
deviation is, the bigger the Gaussian
00:34:57.090 --> 00:34:57.425
is.
00:34:57.425 --> 00:34:59.920
So if you look at these plots here, the
00:34:59.920 --> 00:35:02.150
it's kind of blurry the.
00:35:02.770 --> 00:35:05.540
The red curve or the actually the
00:35:05.540 --> 00:35:07.130
yellow curve has like the biggest
00:35:07.130 --> 00:35:08.880
distribution, the broadest distribution
00:35:08.880 --> 00:35:10.510
and it has the highest variance or
00:35:10.510 --> 00:35:12.010
highest standard deviation.
00:35:14.070 --> 00:35:15.780
So this is the MLE, the maximum
00:35:15.780 --> 00:35:17.240
likelihood estimate of the mean.
00:35:17.240 --> 00:35:19.809
It's just the sum of all the X's
00:35:19.810 --> 00:35:21.850
divided by the number of X's.
00:35:21.850 --> 00:35:25.109
Or, sorry, it's a sum over all the X's.
00:35:26.970 --> 00:35:30.190
For which Y n = K divided by the total
00:35:30.190 --> 00:35:31.900
number of times that Y n = K.
00:35:32.790 --> 00:35:34.845
Because I'm estimating the conditional
00:35:34.845 --> 00:35:36.120
conditional mean.
00:35:36.760 --> 00:35:41.570
So it's the sum over all the X's time.
00:35:41.570 --> 00:35:44.060
This will be where Y and equals K
00:35:44.060 --> 00:35:45.670
divided by the count of y = K.
00:35:46.320 --> 00:35:48.050
And they're staring deviation squared.
00:35:48.050 --> 00:35:50.650
Or the variance is the sum over all the
00:35:50.650 --> 00:35:53.340
differences of the X and the mean
00:35:53.340 --> 00:35:56.890
squared where Y and equals K divided by
00:35:56.890 --> 00:35:58.890
the number of times that y = K.
00:35:59.640 --> 00:36:01.180
And you have to estimate the mean
00:36:01.180 --> 00:36:02.480
before you Estimate the steering
00:36:02.480 --> 00:36:02.950
deviation.
00:36:02.950 --> 00:36:05.100
And if you take a statistics class,
00:36:05.100 --> 00:36:07.980
you'll probably like prove that this is
00:36:07.980 --> 00:36:09.945
an OK thing to do, that you're relying
00:36:09.945 --> 00:36:11.720
on one Estimate in order to get the
00:36:11.720 --> 00:36:12.720
other Estimate.
00:36:12.720 --> 00:36:14.420
But it does turn out it's OK.
00:36:16.670 --> 00:36:20.220
Alright, so in our homework for the
00:36:20.220 --> 00:36:22.890
temperature Regression, we're going to
00:36:22.890 --> 00:36:26.095
assume that Y minus XI is a Gaussian,
00:36:26.095 --> 00:36:27.930
so we have two continuous variables.
00:36:28.900 --> 00:36:29.710
So.
00:36:30.940 --> 00:36:34.847
The idea is that the temperature of
00:36:34.847 --> 00:36:38.565
some city on someday predicts the
00:36:38.565 --> 00:36:41.530
temperature of Cleveland on some other
00:36:41.530 --> 00:36:41.850
day.
00:36:42.600 --> 00:36:44.600
With some offset and some variance.
00:36:45.830 --> 00:36:48.190
And that is pretty easy to Model.
00:36:48.190 --> 00:36:51.020
So here's Sigma I is then the stair
00:36:51.020 --> 00:36:53.770
deviation of that offset Prediction and
00:36:53.770 --> 00:36:54.910
MU I is the offset.
00:36:55.560 --> 00:36:58.230
And I just have Y minus XI minus MU I
00:36:58.230 --> 00:37:00.166
squared here instead of Justice XI
00:37:00.166 --> 00:37:02.590
minus MU I squared, which would be if I
00:37:02.590 --> 00:37:03.960
just said XI is a Gaussian.
00:37:05.170 --> 00:37:08.820
And the mean is just why the sum of Y
00:37:08.820 --> 00:37:11.603
minus XI divided by north, where north
00:37:11.603 --> 00:37:12.870
is the total number of Samples.
00:37:13.990 --> 00:37:14.820
Because why?
00:37:14.820 --> 00:37:16.618
Is not discrete, so I'm not counting
00:37:16.618 --> 00:37:20.100
over certain over only values X where Y
00:37:20.100 --> 00:37:21.625
is equal to some value, I'm counting
00:37:21.625 --> 00:37:22.550
over all the values.
00:37:23.410 --> 00:37:25.280
And the Syrian deviation or their
00:37:25.280 --> 00:37:28.590
variance is Y minus XI minus MU I
00:37:28.590 --> 00:37:29.630
squared divided by north.
00:37:30.480 --> 00:37:32.300
And here's the Python.
00:37:33.630 --> 00:37:35.840
Here I just use the mean and steering
00:37:35.840 --> 00:37:37.630
deviation functions to get it, but it's
00:37:37.630 --> 00:37:40.470
also not a very long formula if I were
00:37:40.470 --> 00:37:41.340
to write it all out.
00:37:44.020 --> 00:37:46.830
And then X&Y were jointly Gaussian.
00:37:46.830 --> 00:37:49.660
So if I say that I need to jointly
00:37:49.660 --> 00:37:52.850
Model them, then one way to do it is
00:37:52.850 --> 00:37:53.600
by.
00:37:54.460 --> 00:37:56.510
By saying that probability of XI given
00:37:56.510 --> 00:38:00.660
Y is the joint probability of XI and Y.
00:38:00.660 --> 00:38:03.070
So now I have a 2 variable Gaussian
00:38:03.070 --> 00:38:06.780
with A2 variable mean and a two by two
00:38:06.780 --> 00:38:07.900
covariance matrix.
00:38:08.920 --> 00:38:11.210
Divided by the probability of Y, which
00:38:11.210 --> 00:38:12.700
is a 1D Gaussian.
00:38:12.700 --> 00:38:14.636
Just the Gaussian over probability of
00:38:14.636 --> 00:38:14.999
Y.
00:38:15.000 --> 00:38:16.340
And if you were to write out all the
00:38:16.340 --> 00:38:18.500
math for it would simplify into some
00:38:18.500 --> 00:38:21.890
other Gaussian equation, but it's
00:38:21.890 --> 00:38:23.360
easier to think about it this way.
00:38:27.660 --> 00:38:28.140
Alright.
00:38:28.140 --> 00:38:31.660
And then what if XI is continuous but
00:38:31.660 --> 00:38:32.770
it's not Gaussian?
00:38:33.920 --> 00:38:35.750
And why is discrete?
00:38:35.750 --> 00:38:37.763
There's one simple thing I can do is I
00:38:37.763 --> 00:38:40.770
can just first turn X into a discrete.
00:38:40.860 --> 00:38:41.490
00:38:42.280 --> 00:38:45.060
Into a discrete function, so.
00:38:46.810 --> 00:38:48.640
For example if.
00:38:49.590 --> 00:38:52.260
Let me venture with my pen again, but.
00:39:08.410 --> 00:39:08.810
Can't do it.
00:39:08.810 --> 00:39:09.170
I want.
00:39:15.140 --> 00:39:15.490
OK.
00:39:16.820 --> 00:39:20.930
So for example, X has a range from.
00:39:21.120 --> 00:39:22.130
From zero to 1.
00:39:22.810 --> 00:39:26.332
That's the case for our intensities of
00:39:26.332 --> 00:39:28.340
the pixel, intensities of amnesty.
00:39:29.180 --> 00:39:31.830
I can just set a threshold for example
00:39:31.830 --> 00:39:38.230
of 0.5 and if X is greater than 05 then
00:39:38.230 --> 00:39:40.369
I'm going to say that it's equal to 1.
00:39:41.030 --> 00:39:43.860
NFX is less than five, then I'm going
00:39:43.860 --> 00:39:45.050
to say it's equal to 0.
00:39:45.050 --> 00:39:46.440
So now I turn my continuous
00:39:46.440 --> 00:39:49.350
distribution into a binary distribution
00:39:49.350 --> 00:39:51.040
and now I can just Estimate it using
00:39:51.040 --> 00:39:52.440
the Bernoulli equation.
00:39:53.100 --> 00:39:54.910
Or I could turn X into 10 different
00:39:54.910 --> 00:39:57.280
values by just multiplying X by 10 and
00:39:57.280 --> 00:39:58.050
taking the floor.
00:39:58.050 --> 00:39:59.560
So now the values are zero to 9.
00:40:01.490 --> 00:40:04.150
So that's one that's actually the one
00:40:04.150 --> 00:40:06.110
of the easiest way to deal with the
00:40:06.110 --> 00:40:08.190
continuous variable that's not
00:40:08.190 --> 00:40:08.850
Gaussian.
00:40:12.900 --> 00:40:15.950
Sometimes X will be like text, so for
00:40:15.950 --> 00:40:18.800
example it could be like blue, orange
00:40:18.800 --> 00:40:19.430
or green.
00:40:20.080 --> 00:40:22.070
And then you just need to Map those
00:40:22.070 --> 00:40:25.390
different text tokens into integers.
00:40:25.390 --> 00:40:26.441
So I might say blue.
00:40:26.441 --> 00:40:28.654
I'm going to say I'm going to Map blue
00:40:28.654 --> 00:40:30.620
into zero, orange into one, green into
00:40:30.620 --> 00:40:32.580
two, and then I can just Solve by
00:40:32.580 --> 00:40:33.060
counting.
00:40:36.610 --> 00:40:38.830
And then finally I need to also
00:40:38.830 --> 00:40:40.380
Estimate the probability of Y.
00:40:41.060 --> 00:40:42.990
One common thing to do is just to say
00:40:42.990 --> 00:40:45.880
that Y is equally likely to be all the
00:40:45.880 --> 00:40:46.860
possible labels.
00:40:47.550 --> 00:40:49.440
And that can be a good thing to do,
00:40:49.440 --> 00:40:51.169
because maybe our training distribution
00:40:51.170 --> 00:40:52.870
isn't even, but you don't think you're
00:40:52.870 --> 00:40:54.310
training distribution will be the same
00:40:54.310 --> 00:40:55.790
as the test distribution.
00:40:55.790 --> 00:40:58.340
So then you say that probability of Y
00:40:58.340 --> 00:41:00.470
is uniform even though it's not uniform
00:41:00.470 --> 00:41:00.920
in training.
00:41:01.630 --> 00:41:03.530
If it's uniform, you can just ignore it
00:41:03.530 --> 00:41:05.910
because it won't have any effect on
00:41:05.910 --> 00:41:07.060
which Y is most likely.
00:41:07.980 --> 00:41:09.860
FY is discrete and non uniform.
00:41:09.860 --> 00:41:11.810
You can just solve it by counting how
00:41:11.810 --> 00:41:14.050
many times is Y equal 1 divided by all
00:41:14.050 --> 00:41:16.850
my data is the probability of Y equal
00:41:16.850 --> 00:41:17.070
1.
00:41:17.790 --> 00:41:19.450
If it's continuous, you can Model it as
00:41:19.450 --> 00:41:21.660
a Gaussian or chop it up into bins and
00:41:21.660 --> 00:41:23.000
then turn it into a classification
00:41:23.000 --> 00:41:23.360
problem.
00:41:25.690 --> 00:41:26.050
Right.
00:41:28.290 --> 00:41:31.550
So I'll give you your minute or two,
00:41:31.550 --> 00:41:32.230
Stretch break.
00:41:32.230 --> 00:41:33.650
But I want you to think about this
00:41:33.650 --> 00:41:34.370
while you do that.
00:41:35.390 --> 00:41:38.100
So suppose I want to classify a fruit
00:41:38.100 --> 00:41:40.230
based on description and my Features
00:41:40.230 --> 00:41:42.389
are weight, color, shape and whether
00:41:42.390 --> 00:41:44.190
it's a hard whether the outside is
00:41:44.190 --> 00:41:44.470
hard.
00:41:45.330 --> 00:41:47.960
And so first, here's some examples of
00:41:47.960 --> 00:41:49.100
those Features.
00:41:49.100 --> 00:41:50.750
See if you can figure out which fruit
00:41:50.750 --> 00:41:51.990
correspond to these Features.
00:41:52.630 --> 00:41:56.150
And second, what might be a good set of
00:41:56.150 --> 00:41:58.080
models to use for probability of XI
00:41:58.080 --> 00:41:59.730
given fruit for those four Features?
00:42:01.210 --> 00:42:03.620
So you have two minutes to think about
00:42:03.620 --> 00:42:05.630
it and Oregon Stretch or use the
00:42:05.630 --> 00:42:07.240
bathroom or check your e-mail or
00:42:07.240 --> 00:42:07.620
whatever.
00:44:24.040 --> 00:44:24.730
Alright.
00:44:26.640 --> 00:44:31.100
So first, what is the top 1.5 pounds
00:44:31.100 --> 00:44:31.640
red round?
00:44:31.640 --> 00:44:33.750
Yes, OK, good.
00:44:33.750 --> 00:44:34.870
That's what I was thinking.
00:44:34.870 --> 00:44:37.930
What's the 2nd 115 pounds?
00:44:39.070 --> 00:44:39.810
Avocado.
00:44:39.810 --> 00:44:41.260
That's a huge avocado.
00:44:43.770 --> 00:44:44.660
What is it?
00:44:46.290 --> 00:44:48.090
Watermelon watermelons, what I was
00:44:48.090 --> 00:44:48.450
thinking.
00:44:49.170 --> 00:44:52.140
.1 pounds purple round and not hard.
00:44:53.330 --> 00:44:54.980
I was thinking of a Grape.
00:44:54.980 --> 00:44:55.980
OK, good.
00:44:57.480 --> 00:44:58.900
There wasn't really, there wasn't
00:44:58.900 --> 00:45:00.160
necessarily a right answer.
00:45:00.160 --> 00:45:01.790
It's just kind of what I was thinking.
00:45:02.800 --> 00:45:05.642
Alright, and then how do you Model the
00:45:05.642 --> 00:45:07.700
probability of the feature given the
00:45:07.700 --> 00:45:08.450
fruit for each of these?
00:45:08.450 --> 00:45:09.550
So let's say the weight.
00:45:09.550 --> 00:45:11.172
What would be a good model for
00:45:11.172 --> 00:45:13.270
probability of XI given the label?
00:45:15.080 --> 00:45:17.420
Gaussian would, Gaussian would probably
00:45:17.420 --> 00:45:18.006
be a good choice.
00:45:18.006 --> 00:45:19.820
It has each of these probably has some
00:45:19.820 --> 00:45:21.250
expectation, maybe a Gaussian
00:45:21.250 --> 00:45:22.130
distribution around it.
00:45:24.000 --> 00:45:26.490
Alright, what about the color red,
00:45:26.490 --> 00:45:27.315
green, purple?
00:45:27.315 --> 00:45:28.440
What could I do for that?
00:45:31.440 --> 00:45:35.610
So I could use a multinomial so I can
00:45:35.610 --> 00:45:37.210
just turn it into discrete very
00:45:37.210 --> 00:45:39.410
discrete numbers, integer numbers and
00:45:39.410 --> 00:45:41.480
then count and the shape.
00:45:50.470 --> 00:45:52.470
So if there's assuming that there's
00:45:52.470 --> 00:45:54.470
other shapes, I don't know if there are
00:45:54.470 --> 00:45:55.880
star fruit for example.
00:45:56.790 --> 00:45:58.940
And then multinomial.
00:45:58.940 --> 00:46:00.640
But either way I'll turn it in discrete
00:46:00.640 --> 00:46:04.090
variables and count and the yes nodes.
00:46:05.540 --> 00:46:07.010
So that will be Binomial.
00:46:08.240 --> 00:46:08.540
OK.
00:46:14.840 --> 00:46:18.500
All right, so now we know how to
00:46:18.500 --> 00:46:20.770
Estimate probability of X given Y.
00:46:20.770 --> 00:46:23.065
Now after I go through all that work on
00:46:23.065 --> 00:46:25.178
the training data and I get new test
00:46:25.178 --> 00:46:25.512
sample.
00:46:25.512 --> 00:46:27.900
Now I want to know what's the most
00:46:27.900 --> 00:46:29.620
likely label of that test sample.
00:46:31.200 --> 00:46:31.660
So.
00:46:32.370 --> 00:46:33.860
I can write this in two ways.
00:46:33.860 --> 00:46:36.615
One is I can write Y is the argmax over
00:46:36.615 --> 00:46:38.735
the product of probability of XI given
00:46:38.735 --> 00:46:39.959
Y times probability of Y.
00:46:40.990 --> 00:46:44.334
Or I can write it as the argmax of the
00:46:44.334 --> 00:46:46.718
log of that, which is just the argmax
00:46:46.718 --> 00:46:48.970
of Y of the sum over I of log of
00:46:48.970 --> 00:46:50.904
probability of XI given Yi plus log of
00:46:50.904 --> 00:46:51.599
probability of Y.
00:46:52.570 --> 00:46:55.130
And I can do that because the thing
00:46:55.130 --> 00:46:57.798
that maximizes X also maximizes log of
00:46:57.798 --> 00:46:59.280
X and vice versa.
00:46:59.280 --> 00:47:01.910
And that's actually a really useful
00:47:01.910 --> 00:47:04.270
property because often the logs are
00:47:04.270 --> 00:47:05.745
probabilities are a lot simpler.
00:47:05.745 --> 00:47:08.790
And for example, if I took for example
00:47:08.790 --> 00:47:10.434
at the Gaussian, if I take the log of
00:47:10.434 --> 00:47:11.950
the Gaussian, then it just becomes a
00:47:11.950 --> 00:47:12.760
squared term.
00:47:13.640 --> 00:47:16.400
And the other thing is that these
00:47:16.400 --> 00:47:18.350
probability of Xis might be.
00:47:18.470 --> 00:47:21.553
If I have a lot of them, if I have like
00:47:21.553 --> 00:47:23.723
500 of them and they're on average like
00:47:23.723 --> 00:47:26.320
.1, that would be like .1 to the 500,
00:47:26.320 --> 00:47:27.530
which is going to go outside in
00:47:27.530 --> 00:47:28.690
numerical precision.
00:47:28.690 --> 00:47:30.740
So if you try to Compute this product
00:47:30.740 --> 00:47:32.290
directly, you're probably going to get
00:47:32.290 --> 00:47:34.470
0 or some kind of wonky value.
00:47:35.190 --> 00:47:37.320
And so it's much better to take the sum
00:47:37.320 --> 00:47:39.265
of the logs than to take the product of
00:47:39.265 --> 00:47:40.060
the probabilities.
00:47:42.650 --> 00:47:44.290
Right, so, but I can compute the
00:47:44.290 --> 00:47:45.830
probability of X&Y or the log
00:47:45.830 --> 00:47:48.004
probability of X&Y for each value of Y
00:47:48.004 --> 00:47:49.630
and then choose the value with maximum
00:47:49.630 --> 00:47:50.240
likelihood.
00:47:50.240 --> 00:47:51.686
That will work in the case of the
00:47:51.686 --> 00:47:53.409
digits because I only have 10 digits.
00:47:54.420 --> 00:47:56.940
And so I can check for each possible
00:47:56.940 --> 00:48:00.365
Digit, how likely is the sum of log
00:48:00.365 --> 00:48:01.958
probability of XI given Yi plus
00:48:01.958 --> 00:48:03.770
probability log probability of Y.
00:48:03.770 --> 00:48:06.980
And then I choose the Digit Digit label
00:48:06.980 --> 00:48:08.570
that makes this most likely.
00:48:11.240 --> 00:48:12.580
That's pretty simple.
00:48:12.580 --> 00:48:14.110
In the case of Y is discrete.
00:48:14.900 --> 00:48:16.415
And again, I just want to emphasize
00:48:16.415 --> 00:48:18.983
that this thing of turning product of
00:48:18.983 --> 00:48:21.070
probabilities into a sum of log
00:48:21.070 --> 00:48:23.250
probabilities is really, really widely
00:48:23.250 --> 00:48:23.760
used.
00:48:23.760 --> 00:48:27.610
Almost anytime you Solve for anything
00:48:27.610 --> 00:48:29.140
with probabilities, it involves that
00:48:29.140 --> 00:48:29.380
step.
00:48:31.840 --> 00:48:34.420
Now if Y is continuous, it's a bit more
00:48:34.420 --> 00:48:36.610
complicated and I.
00:48:37.440 --> 00:48:39.890
So I have the derivation here for you.
00:48:39.890 --> 00:48:42.166
So this is for the case.
00:48:42.166 --> 00:48:44.859
I'm going to use as an example the case
00:48:44.860 --> 00:48:47.470
where I'm modeling probability of Y
00:48:47.470 --> 00:48:51.400
minus XI of 1 dimensional Gaussian.
00:48:53.280 --> 00:48:56.260
And anytime you solve this kind of
00:48:56.260 --> 00:48:58.320
thing you're going to go through, you
00:48:58.320 --> 00:48:59.580
would go through the same derivation.
00:48:59.580 --> 00:49:00.280
If it's not.
00:49:00.280 --> 00:49:03.180
Just like a simple matter of if you
00:49:03.180 --> 00:49:05.000
don't have discrete wise, if you have
00:49:05.000 --> 00:49:06.360
continuous wise, then you have to find
00:49:06.360 --> 00:49:08.320
the Y that actually maximizes this
00:49:08.320 --> 00:49:10.760
because you can't check all possible
00:49:10.760 --> 00:49:12.310
values of a continuous variable.
00:49:14.180 --> 00:49:15.390
So it's not.
00:49:16.540 --> 00:49:17.451
It's a lot.
00:49:17.451 --> 00:49:18.362
It's a lot.
00:49:18.362 --> 00:49:20.350
It's a fair number of equations, but
00:49:20.350 --> 00:49:23.420
it's not anything super complicated.
00:49:23.420 --> 00:49:24.940
Let me see if I can get my cursor up
00:49:24.940 --> 00:49:25.960
there again, OK?
00:49:26.710 --> 00:49:29.560
Alright, so first I take the partial
00:49:29.560 --> 00:49:32.526
derivative of the log probability of
00:49:32.526 --> 00:49:34.780
X&Y with respect to Y and set it equal
00:49:34.780 --> 00:49:35.190
to 0.
00:49:35.190 --> 00:49:36.890
So you might remember from calculus
00:49:36.890 --> 00:49:38.720
like if you want to find the min or Max
00:49:38.720 --> 00:49:39.580
of some value.
00:49:40.290 --> 00:49:43.109
Then take the partial with respect to
00:49:43.110 --> 00:49:44.750
some variable.
00:49:44.750 --> 00:49:47.340
You take the partial derivative with
00:49:47.340 --> 00:49:48.800
respect to that variable and set it
00:49:48.800 --> 00:49:49.539
equal to 0.
00:49:50.680 --> 00:49:51.360
And.
00:49:53.080 --> 00:49:55.020
So here I did that.
00:49:55.020 --> 00:49:58.100
Now I've plugged in this Gaussian
00:49:58.100 --> 00:50:00.200
distribution and taken the log.
00:50:01.050 --> 00:50:02.510
And I kind of like there's some
00:50:02.510 --> 00:50:04.020
invisible steps here, because there's
00:50:04.020 --> 00:50:06.410
some terms like the log of one over
00:50:06.410 --> 00:50:07.940
square of 2π Sigma.
00:50:08.580 --> 00:50:10.069
That just don't.
00:50:10.069 --> 00:50:12.290
Those terms don't matter because they
00:50:12.290 --> 00:50:13.080
don't involve Y.
00:50:13.080 --> 00:50:14.743
So the partial derivative of those
00:50:14.743 --> 00:50:16.215
terms with respect to Y is 0.
00:50:16.215 --> 00:50:19.090
So I just didn't include them.
00:50:19.750 --> 00:50:21.815
So these are the terms that include Y
00:50:21.815 --> 00:50:23.590
and I've already taken the log.
00:50:23.590 --> 00:50:25.550
This was originally east to the -, 1
00:50:25.550 --> 00:50:27.839
half whatever is shown here, and the
00:50:27.839 --> 00:50:30.360
log of X of X is equal to X.
00:50:31.840 --> 00:50:33.490
And so I get this guy.
00:50:34.450 --> 00:50:36.530
Now I broke it out into different
00:50:36.530 --> 00:50:39.320
terms, so I did the quadratic of Y
00:50:39.320 --> 00:50:41.190
minus XI minus MU I ^2.
00:50:42.420 --> 00:50:44.100
Mainly so that I don't have to use the
00:50:44.100 --> 00:50:45.620
chain rule and I can keep my
00:50:45.620 --> 00:50:46.740
derivatives really Simple.
00:50:47.830 --> 00:50:51.959
So here I just broke that out to y ^2 y
00:50:51.960 --> 00:50:54.130
axis YMUI.
00:50:54.130 --> 00:50:55.530
And again, I don't need to worry about
00:50:55.530 --> 00:50:57.779
the MU I squared over Sigma I squared
00:50:57.780 --> 00:50:59.750
because it doesn't involve Y so I just
00:50:59.750 --> 00:51:00.230
left it out.
00:51:02.140 --> 00:51:03.990
I.
00:51:04.100 --> 00:51:07.021
Take the derivative with respect to Y.
00:51:07.021 --> 00:51:09.468
So the derivative of y ^2 is 2 Y.
00:51:09.468 --> 00:51:10.976
So this half goes away.
00:51:10.976 --> 00:51:14.080
Derivative of YX is just X.
00:51:15.070 --> 00:51:18.000
So this should be a subscript I.
00:51:18.730 --> 00:51:21.120
And then I did the same for these guys
00:51:21.120 --> 00:51:21.330
here.
00:51:22.500 --> 00:51:25.740
It's just basic algebra, so I just try
00:51:25.740 --> 00:51:27.610
to group the terms that involve Y and
00:51:27.610 --> 00:51:29.480
the terms that don't involve Yi, put
00:51:29.480 --> 00:51:30.840
the terms that don't involve Y and the
00:51:30.840 --> 00:51:33.370
right side, and then finally I divide
00:51:33.370 --> 00:51:36.830
the coefficient of Y and I get this guy
00:51:36.830 --> 00:51:37.150
here.
00:51:38.030 --> 00:51:41.269
So at the end Y is equal to 1 over the
00:51:41.270 --> 00:51:44.408
sum over all the features of 1 / sqrt.
00:51:44.408 --> 00:51:46.690
I mean one over Sigma I ^2.
00:51:47.420 --> 00:51:50.580
Plus one over Sigma y ^2 which is the
00:51:50.580 --> 00:51:52.160
standard deviation of the Prior of Y.
00:51:52.160 --> 00:51:53.906
Or if I just assumed uniform likelihood
00:51:53.906 --> 00:51:55.520
of Yi wouldn't need that term.
00:51:56.610 --> 00:51:59.400
And then that's times the sum over all
00:51:59.400 --> 00:52:02.700
the features of that feature value.
00:52:02.700 --> 00:52:03.930
This should be subscript I.
00:52:04.940 --> 00:52:10.430
Plus MU I divided by Sigma I ^2 plus mu
00:52:10.430 --> 00:52:13.811
Y, the Prior mean of Y divided by Sigma
00:52:13.811 --> 00:52:14.539
y ^2.
00:52:16.150 --> 00:52:18.940
And so this is just a, it's actually
00:52:18.940 --> 00:52:19.849
just a weighted.
00:52:19.850 --> 00:52:22.823
If you say that one over Sigma I
00:52:22.823 --> 00:52:26.035
squared is Wei, it's like a weight for
00:52:26.035 --> 00:52:27.565
that prediction of the ith feature.
00:52:27.565 --> 00:52:29.830
This is just a weighted average of the
00:52:29.830 --> 00:52:31.720
predictions from all the Features
00:52:31.720 --> 00:52:33.250
that's weighted by one over the
00:52:33.250 --> 00:52:35.573
steering deviation squared or one over
00:52:35.573 --> 00:52:36.190
the variance.
00:52:37.590 --> 00:52:40.421
And so I have one over the sum over I
00:52:40.421 --> 00:52:45.683
of WI plus WY times, the sum X plus mu
00:52:45.683 --> 00:52:49.722
I XI plus MU I times, Wei plus mu Y
00:52:49.722 --> 00:52:50.100
times.
00:52:50.100 --> 00:52:50.670
Why?
00:52:51.630 --> 00:52:53.240
Amy sounds similar, unfortunately.
00:52:54.780 --> 00:52:56.430
So it's just the weighted average of
00:52:56.430 --> 00:52:57.910
all the predictions of the individual
00:52:57.910 --> 00:52:58.174
features.
00:52:58.174 --> 00:53:00.093
And it makes sense that it kind of
00:53:00.093 --> 00:53:01.624
makes sense intuitively that the weight
00:53:01.624 --> 00:53:02.650
is 1 over the variance.
00:53:02.650 --> 00:53:04.490
So if you have really high variance,
00:53:04.490 --> 00:53:05.790
then the weight is small.
00:53:05.790 --> 00:53:08.155
So if, for example, maybe the
00:53:08.155 --> 00:53:09.839
temperature in Sacramento is a really
00:53:09.840 --> 00:53:11.513
bad predictor for the temperature in
00:53:11.513 --> 00:53:12.984
Cleveland, so it will have high
00:53:12.984 --> 00:53:14.840
variance and it gets a little weight,
00:53:14.840 --> 00:53:16.460
while the temperature in Cleveland the
00:53:16.460 --> 00:53:19.130
previous day is much more highly
00:53:19.130 --> 00:53:20.849
predictive, has lower variance, so
00:53:20.850 --> 00:53:21.639
it'll get more weight.
00:53:32.280 --> 00:53:35.380
So let me pause here.
00:53:35.380 --> 00:53:38.690
So any questions about?
00:53:39.670 --> 00:53:43.255
Estimating the likelihoods P of X given
00:53:43.255 --> 00:53:47.970
Y, or solving for the Y that makes.
00:53:47.970 --> 00:53:49.880
That's most likely given your
00:53:49.880 --> 00:53:50.500
likelihoods.
00:53:52.460 --> 00:53:54.470
And obviously if I'm happy to work
00:53:54.470 --> 00:53:56.610
through this in office hours as well in
00:53:56.610 --> 00:53:59.940
the TAS should also if you want to like
00:53:59.940 --> 00:54:01.100
spend more time working through the
00:54:01.100 --> 00:54:01.530
equations.
00:54:03.920 --> 00:54:04.930
I just want to pause.
00:54:04.930 --> 00:54:07.830
I know it's a lot of math to soak up.
00:54:09.870 --> 00:54:13.260
And really, it's not that memorizing
00:54:13.260 --> 00:54:14.370
these things isn't important.
00:54:14.370 --> 00:54:15.860
It's really the process that you just
00:54:15.860 --> 00:54:17.385
set the partial derivative with respect
00:54:17.385 --> 00:54:20.140
to Y, set it to zero, and then you do
00:54:20.140 --> 00:54:20.540
the.
00:54:21.250 --> 00:54:23.120
Do the partial derivative and solve the
00:54:23.120 --> 00:54:23.510
algebra.
00:54:26.700 --> 00:54:28.050
All right, I'll go on then.
00:54:28.050 --> 00:54:31.990
So far, this is pure maximum likelihood
00:54:31.990 --> 00:54:32.530
estimation.
00:54:32.530 --> 00:54:34.920
I'm not, I'm not imposing any kinds of
00:54:34.920 --> 00:54:36.470
Priors over my parameters.
00:54:37.570 --> 00:54:39.600
In practice, you do want to impose a
00:54:39.600 --> 00:54:41.010
Prior in your parameters to make sure
00:54:41.010 --> 00:54:42.220
you don't have any zeros.
00:54:43.750 --> 00:54:46.380
Otherwise, like if some in the digits
00:54:46.380 --> 00:54:48.809
case for example the test sample had a
00:54:48.810 --> 00:54:50.470
dot in an unlikely place.
00:54:50.470 --> 00:54:52.662
If I had just had like a one and some
00:54:52.662 --> 00:54:54.030
unlikely pixel, all the probabilities
00:54:54.030 --> 00:54:55.630
would be 0 and you wouldn't know what
00:54:55.630 --> 00:54:57.620
the label is because of that one stupid
00:54:57.620 --> 00:54:57.970
pixel.
00:54:58.730 --> 00:55:01.040
So you want to have some kind of Prior?
00:55:01.730 --> 00:55:03.425
To avoid these zero probabilities.
00:55:03.425 --> 00:55:06.260
So the most common case if you're
00:55:06.260 --> 00:55:08.760
estimating a distribution of discrete
00:55:08.760 --> 00:55:10.430
variables like a multinomial or
00:55:10.430 --> 00:55:13.010
Binomial, is to just initialize with
00:55:13.010 --> 00:55:13.645
some count.
00:55:13.645 --> 00:55:16.180
So you just say for example alpha
00:55:16.180 --> 00:55:16.880
equals one.
00:55:17.610 --> 00:55:20.110
And now I say the probability of X I =
00:55:20.110 --> 00:55:21.620
V given y = K.
00:55:22.400 --> 00:55:24.950
Is Alpha plus the count of how many
00:55:24.950 --> 00:55:27.740
times XI equals V and y = K.
00:55:28.690 --> 00:55:31.865
Divided by the all the different values
00:55:31.865 --> 00:55:35.300
of alpha plus account of XI equals that
00:55:35.300 --> 00:55:37.610
value in y = K probably for clarity I
00:55:37.610 --> 00:55:39.700
should have used something other than B
00:55:39.700 --> 00:55:41.630
in the denominator, but hopefully
00:55:41.630 --> 00:55:42.230
that's clear enough.
00:55:43.060 --> 00:55:46.170
Here's the and then here's the Python
00:55:46.170 --> 00:55:47.070
for that, so it's just.
00:55:47.880 --> 00:55:50.350
Sum of all the values where XI equals V
00:55:50.350 --> 00:55:52.470
and y = K Plus some alpha.
00:55:53.300 --> 00:55:54.980
So if alpha equals zero, then I don't
00:55:54.980 --> 00:55:55.710
have any Prior.
00:55:56.840 --> 00:56:00.450
And then I'm just dividing by the sum
00:56:00.450 --> 00:56:04.270
of times at y = K and there will be.
00:56:04.850 --> 00:56:06.540
The number of alphas will be equal to
00:56:06.540 --> 00:56:08.150
the number of different values, so this
00:56:08.150 --> 00:56:10.510
is like a little bit of a shortcut, but
00:56:10.510 --> 00:56:11.330
it's the same thing.
00:56:12.860 --> 00:56:14.760
If I have a continuous variable and
00:56:14.760 --> 00:56:15.060
I've.
00:56:15.730 --> 00:56:17.010
Modeled it with the Gaussian.
00:56:17.010 --> 00:56:18.470
Then the usual thing to do is just to
00:56:18.470 --> 00:56:20.180
add a small value to your steering
00:56:20.180 --> 00:56:21.420
deviation or your variance.
00:56:22.110 --> 00:56:24.320
And you might want to make that value
00:56:24.320 --> 00:56:27.650
if N is unknown, then make it dependent
00:56:27.650 --> 00:56:29.300
on north so that if you have a huge
00:56:29.300 --> 00:56:31.395
number of samples then the effect of
00:56:31.395 --> 00:56:33.880
the Prior will go down, which is what
00:56:33.880 --> 00:56:34.170
you want.
00:56:36.140 --> 00:56:39.513
So for example, you can say that the
00:56:39.513 --> 00:56:41.990
stern deviation is whatever this
00:56:41.990 --> 00:56:44.770
whatever the MLE estimate of the stern
00:56:44.770 --> 00:56:47.340
deviation is, plus some small value
00:56:47.340 --> 00:56:49.730
sqrt 1 over the length of north.
00:56:50.420 --> 00:56:51.350
Of X, sorry.
00:57:00.440 --> 00:57:02.670
So what the Prior does is it.
00:57:02.810 --> 00:57:05.995
In the case of the discrete variables,
00:57:05.995 --> 00:57:09.110
the Prior is trying to push your
00:57:09.110 --> 00:57:11.152
Estimate towards a uniform likelihood.
00:57:11.152 --> 00:57:13.000
In fact, in both cases it's pushing it
00:57:13.000 --> 00:57:14.280
towards a uniform likelihood.
00:57:15.400 --> 00:57:18.670
So if you had a really large alpha,
00:57:18.670 --> 00:57:20.550
then let's say.
00:57:22.090 --> 00:57:23.440
Let's say that.
00:57:24.620 --> 00:57:25.850
Or I don't know if I can think of
00:57:25.850 --> 00:57:26.170
something.
00:57:28.140 --> 00:57:29.550
Let's say you have a population of
00:57:29.550 --> 00:57:30.900
students and you're trying to estimate
00:57:30.900 --> 00:57:32.510
the probability that a student is male.
00:57:33.520 --> 00:57:36.570
If I say alpha equals 1000, then I'm
00:57:36.570 --> 00:57:37.860
going to need like an awful lot of
00:57:37.860 --> 00:57:40.156
students before I budge very far from a
00:57:40.156 --> 00:57:42.070
5050 chance that a student is male or
00:57:42.070 --> 00:57:42.620
female.
00:57:42.620 --> 00:57:44.057
Because I'll start with saying there's
00:57:44.057 --> 00:57:46.213
1000 males and 1000 females, and then
00:57:46.213 --> 00:57:48.676
I'll count all the males and add them
00:57:48.676 --> 00:57:50.832
to 1000, count all the females, add
00:57:50.832 --> 00:57:53.370
them to 1000, and then I would take the
00:57:53.370 --> 00:57:55.210
male plus 1000 count and divide it by
00:57:55.210 --> 00:57:57.660
2000 plus the total population.
00:57:59.130 --> 00:58:00.860
If Alpha is 0, then I'm going to get
00:58:00.860 --> 00:58:03.410
just my raw empirical Estimate.
00:58:03.410 --> 00:58:06.810
So if I had like 3 students and I say
00:58:06.810 --> 00:58:09.090
alpha equals zero, and I have two males
00:58:09.090 --> 00:58:11.140
and a female, then I'll say 2/3 of them
00:58:11.140 --> 00:58:11.550
are male.
00:58:12.410 --> 00:58:14.670
If I say alpha is 1 and I have two
00:58:14.670 --> 00:58:17.110
males and a female, then I would say
00:58:17.110 --> 00:58:20.490
that my probability of male is 3 / 5
00:58:20.490 --> 00:58:24.100
because it's 2 + 1 / 3 + 2.
00:58:27.060 --> 00:58:28.330
Their deviation it's the same.
00:58:28.330 --> 00:58:30.240
It's like trying to just broaden your
00:58:30.240 --> 00:58:32.600
variance from what you would Estimate
00:58:32.600 --> 00:58:33.580
directly from the data.
00:58:36.500 --> 00:58:39.260
So I think I will not ask you all these
00:58:39.260 --> 00:58:41.210
probabilities because they're kind of
00:58:41.210 --> 00:58:43.220
you've shown the ability to count
00:58:43.220 --> 00:58:44.810
before mostly.
00:58:46.550 --> 00:58:47.640
And.
00:58:47.850 --> 00:58:50.060
So here's for example, the probability
00:58:50.060 --> 00:58:54.509
of X 1 = 0 and y = 0 is 2 out of four.
00:58:54.510 --> 00:58:56.050
I can get that just by looking down
00:58:56.050 --> 00:58:56.670
these rows.
00:58:56.670 --> 00:58:58.870
It takes a little bit of time, but
00:58:58.870 --> 00:59:02.786
there's four times that y = 0 and out
00:59:02.786 --> 00:59:06.660
of those two times X 1 = 0 and so this
00:59:06.660 --> 00:59:07.440
is 2 out of four.
00:59:08.090 --> 00:59:08.930
And the same.
00:59:08.930 --> 00:59:11.260
I can use the same counting method to
00:59:11.260 --> 00:59:13.120
get all of these other probabilities
00:59:13.120 --> 00:59:13.410
here.
00:59:15.770 --> 00:59:19.450
So just to check that everyone's awake,
00:59:19.450 --> 00:59:22.970
if I, what is the probability of Y?
00:59:23.840 --> 00:59:27.370
And X 1 = 1 and X 2 = 1.
00:59:28.500 --> 00:59:30.019
So can you get it from?
00:59:30.019 --> 00:59:32.560
Can you get it from this guy under an
00:59:32.560 --> 00:59:33.450
independence?
00:59:33.450 --> 00:59:35.670
So get it from this under an under an I
00:59:35.670 --> 00:59:36.540
Bayes assumption.
00:59:41.350 --> 00:59:43.240
Let's say I should say probability of Y
00:59:43.240 --> 00:59:43.860
equal 1.
00:59:45.380 --> 00:59:47.910
Probability of y = 1 given X 1 = 1 and
00:59:47.910 --> 00:59:48.930
X 2 = 1.
00:59:57.500 --> 01:00:00.560
And you don't worry about simplifying
01:00:00.560 --> 01:00:02.610
your numerator and denominator.
01:00:03.530 --> 01:00:05.110
What are the things that get multiplied
01:00:05.110 --> 01:00:05.610
together?
01:00:10.460 --> 01:00:14.350
Not sort of, partly that's in there.
01:00:15.220 --> 01:00:17.880
Raise your hand if you think the
01:00:17.880 --> 01:00:18.560
answer.
01:00:19.550 --> 01:00:21.130
I just want to give everyone time.
01:00:24.650 --> 01:00:27.962
But I mean probability of y = 1 given X
01:00:27.962 --> 01:00:29.960
1 = 1 and X 2 = 1.
01:00:39.830 --> 01:00:41.220
A Naive Bayes assumption.
01:01:24.310 --> 01:01:25.800
The raise your hand if you.
01:01:26.490 --> 01:01:27.030
Finished.
01:01:56.450 --> 01:01:57.740
But don't tell me the answer yet.
01:02:18.470 --> 01:02:19.260
Equals one.
01:02:23.210 --> 01:02:23.420
Alright.
01:02:23.420 --> 01:02:24.830
Did anybody get it yet?
01:02:24.830 --> 01:02:25.950
Raise your hand if you did.
01:02:25.950 --> 01:02:26.910
I just don't want to.
01:02:28.170 --> 01:02:29.110
Give it too early.
01:03:46.370 --> 01:03:46.960
Alright.
01:03:48.170 --> 01:03:52.029
Example, some people have gotten it, so
01:03:52.030 --> 01:03:53.950
let me I'll start going through it.
01:03:53.950 --> 01:03:55.480
All right, so the Naive Bayes
01:03:55.480 --> 01:03:56.005
assumption.
01:03:56.005 --> 01:03:57.760
So this would be.
01:03:58.060 --> 01:03:58.250
OK.
01:04:00.690 --> 01:04:02.960
OK, probability it's actually my touch
01:04:02.960 --> 01:04:03.230
screen.
01:04:03.230 --> 01:04:04.400
I think is kind of broken.
01:04:05.250 --> 01:04:09.560
Probability of X1 given Y times
01:04:09.560 --> 01:04:14.815
probability X2 given Y sorry equals
01:04:14.815 --> 01:04:15.200
one.
01:04:16.630 --> 01:04:19.050
Times probability of Y equal 1.
01:04:19.910 --> 01:04:21.950
Right, so it's the product of the
01:04:21.950 --> 01:04:23.180
probabilities of the Features.
01:04:23.180 --> 01:04:24.730
Give them label times the probability
01:04:24.730 --> 01:04:25.240
of the label.
01:04:26.500 --> 01:04:29.990
And so that will be probability of XYX.
01:04:31.030 --> 01:04:32.819
1 = 1.
01:04:33.850 --> 01:04:37.317
Given probability of Yi mean given y =
01:04:37.317 --> 01:04:38.260
1 is 3/4.
01:04:42.110 --> 01:04:46.010
And probably the X 2 = 1 given y = 1 is
01:04:46.010 --> 01:04:46.750
3/4.
01:04:49.250 --> 01:04:52.550
And the probability that y = 1 is two
01:04:52.550 --> 01:04:53.940
quarters or 1/2.
01:04:58.570 --> 01:05:00.180
So it's 930 seconds.
01:05:01.120 --> 01:05:01.390
Right.
01:05:02.580 --> 01:05:05.846
And the probability that y = 0 given X
01:05:05.846 --> 01:05:08.059
1 = 1 and Y 1 = 1.
01:05:09.800 --> 01:05:11.840
I mean sorry, the probability of y = 0
01:05:11.840 --> 01:05:14.480
given the X is equal equal 1.
01:05:15.620 --> 01:05:16.770
Is.
01:05:18.600 --> 01:05:19.190
Let's see.
01:05:20.250 --> 01:05:23.780
So that would be 2 fourths times 2
01:05:23.780 --> 01:05:24.300
fourths.
01:05:25.180 --> 01:05:26.320
Times 2 fourths.
01:05:27.260 --> 01:05:31.300
So if X 1 = 1 and X2 equal 1, then it's
01:05:31.300 --> 01:05:33.540
more likely that Y is equal to 1 than
01:05:33.540 --> 01:05:35.070
that Y is equal to 0.
01:05:41.720 --> 01:05:46.750
If I had if I use my Prior, this is how
01:05:46.750 --> 01:05:48.055
the probabilities would change.
01:05:48.055 --> 01:05:51.060
So if I say alpha equals one, you can
01:05:51.060 --> 01:05:52.900
see that the probabilities get less
01:05:52.900 --> 01:05:53.510
Peaky.
01:05:53.510 --> 01:05:56.422
So I went from 1/4 to 261 quarter and
01:05:56.422 --> 01:05:58.951
3/4 to 2/6 and four six for example.
01:05:58.951 --> 01:06:02.316
So 1/3 and 2/3 is more uniform than 1/4
01:06:02.316 --> 01:06:03.129
and 3/4.
01:06:05.050 --> 01:06:07.040
And then if the initial estimate was
01:06:07.040 --> 01:06:09.020
1/2, the final Estimate will still be
01:06:09.020 --> 01:06:11.620
1/2 because it's because this Prior is
01:06:11.620 --> 01:06:13.650
just trying to push things towards 1/2.
01:06:20.780 --> 01:06:24.220
So I want to give one example of a use
01:06:24.220 --> 01:06:24.550
case.
01:06:24.550 --> 01:06:25.685
So I've actually.
01:06:25.685 --> 01:06:28.360
I mean I want to say like I used Naive
01:06:28.360 --> 01:06:30.630
Bayes, but I use that assumption pretty
01:06:30.630 --> 01:06:31.440
often.
01:06:31.440 --> 01:06:33.480
For example if I wanted to Estimate a
01:06:33.480 --> 01:06:35.210
distribution of RGB colors.
01:06:36.740 --> 01:06:38.410
I would first convert it to a different
01:06:38.410 --> 01:06:39.860
color space, but let's just say I want
01:06:39.860 --> 01:06:41.780
to Estimate distribution of LGBT RGB
01:06:41.780 --> 01:06:42.390
colors.
01:06:42.390 --> 01:06:45.055
Then even though it's 3 dimensions, is
01:06:45.055 --> 01:06:45.690
a pretty.
01:06:45.690 --> 01:06:47.920
You need like a lot of data to estimate
01:06:47.920 --> 01:06:48.610
that distribution.
01:06:48.610 --> 01:06:50.700
And So what I might do is I'll say,
01:06:50.700 --> 01:06:52.820
well, I'm going to assume that RG and B
01:06:52.820 --> 01:06:54.645
are independent and so the probability
01:06:54.645 --> 01:06:57.350
of RGB is just the probability of R
01:06:57.350 --> 01:06:58.808
times probability of G times
01:06:58.808 --> 01:06:59.524
probability B.
01:06:59.524 --> 01:07:01.600
And I compute a histogram for each of
01:07:01.600 --> 01:07:04.940
those, and I use that to get my as my
01:07:04.940 --> 01:07:06.230
likelihood Estimate.
01:07:06.560 --> 01:07:08.520
So it's like really commonly used in
01:07:08.520 --> 01:07:10.120
that kind of setting where you want to
01:07:10.120 --> 01:07:11.770
Estimate the distribution of multiple
01:07:11.770 --> 01:07:13.380
variables and there's just no way to
01:07:13.380 --> 01:07:13.810
get a Joint.
01:07:13.810 --> 01:07:17.100
The only options you really have are to
01:07:17.100 --> 01:07:18.410
make something the Naive Bayes
01:07:18.410 --> 01:07:21.330
assumption or to do a mixture of
01:07:21.330 --> 01:07:23.416
Gaussians, which we'll talk about later
01:07:23.416 --> 01:07:24.320
in the semester.
01:07:26.380 --> 01:07:27.940
Right, But here's the case where it's
01:07:27.940 --> 01:07:29.450
used for object detection.
01:07:29.450 --> 01:07:32.280
So this was by Schneiderman Kanadi and
01:07:32.280 --> 01:07:35.500
it was the most accurate face and car
01:07:35.500 --> 01:07:36.520
detector for a while.
01:07:37.450 --> 01:07:39.720
They detector is based on wavelet
01:07:39.720 --> 01:07:41.420
coefficients which are just like local
01:07:41.420 --> 01:07:42.610
intensity differences.
01:07:43.320 --> 01:07:46.010
And the.
01:07:46.090 --> 01:07:48.880
The It's a Probabilistic framework, so
01:07:48.880 --> 01:07:51.070
they're trying to say whether if you
01:07:51.070 --> 01:07:54.107
Extract a window of Features from the
01:07:54.107 --> 01:07:56.386
image, some Features over some part of
01:07:56.386 --> 01:07:56.839
the image.
01:07:57.450 --> 01:07:59.020
And Extract all the wavelet
01:07:59.020 --> 01:08:00.330
coefficients.
01:08:00.330 --> 01:08:02.390
Then you want to say that it's a face
01:08:02.390 --> 01:08:03.950
if the probability of those
01:08:03.950 --> 01:08:05.853
coefficients is greater given that it's
01:08:05.853 --> 01:08:08.390
a face, than given that's not a face
01:08:08.390 --> 01:08:10.330
times the probability that's a face
01:08:10.330 --> 01:08:11.730
over the probability that's not a face.
01:08:12.430 --> 01:08:14.680
So it's this basic Probabilistic Model.
01:08:14.680 --> 01:08:16.740
And again, the probability modeling.
01:08:16.740 --> 01:08:17.920
The probability of all those
01:08:17.920 --> 01:08:19.370
coefficients is way too hard.
01:08:20.330 --> 01:08:23.290
On the other hand, modeling all the
01:08:23.290 --> 01:08:25.560
Features as independent given the label
01:08:25.560 --> 01:08:26.950
is a little bit too much of a
01:08:26.950 --> 01:08:28.410
simplifying assumption.
01:08:28.410 --> 01:08:30.270
So they use this algorithm that they
01:08:30.270 --> 01:08:33.220
call semi Naive Bayes which is proposed
01:08:33.220 --> 01:08:34.040
earlier.
01:08:35.220 --> 01:08:37.946
Where you just you Model the
01:08:37.946 --> 01:08:39.803
probabilities of little groups of
01:08:39.803 --> 01:08:41.380
features and then you say that the
01:08:41.380 --> 01:08:43.166
total probability is the probability
01:08:43.166 --> 01:08:44.830
the product or the probabilities of
01:08:44.830 --> 01:08:45.849
these groups of Features.
01:08:46.710 --> 01:08:47.845
So they call these patterns.
01:08:47.845 --> 01:08:50.160
So first you do some look at the mutual
01:08:50.160 --> 01:08:51.870
information, you have ways of measuring
01:08:51.870 --> 01:08:54.050
the dependence of different variables,
01:08:54.050 --> 01:08:56.470
and you cluster the Features together
01:08:56.470 --> 01:08:58.280
based on their dependencies.
01:08:58.920 --> 01:09:00.430
And then for little clusters of
01:09:00.430 --> 01:09:02.149
Features, 3 Features.
01:09:03.060 --> 01:09:05.800
You Estimate the probability of the
01:09:05.800 --> 01:09:08.500
Joint combination of these features and
01:09:08.500 --> 01:09:11.230
then the total probability of all the
01:09:11.230 --> 01:09:11.620
Features.
01:09:11.620 --> 01:09:12.920
I'm glad this isn't worker.
01:09:12.920 --> 01:09:14.788
The total probability of all the
01:09:14.788 --> 01:09:16.660
features is the product of the
01:09:16.660 --> 01:09:18.270
probabilities of each of these groups
01:09:18.270 --> 01:09:18.840
of Features.
01:09:19.890 --> 01:09:21.140
And so you Model.
01:09:21.140 --> 01:09:23.616
Likely a set of features are given that
01:09:23.616 --> 01:09:25.270
it's a face, and how likely they are
01:09:25.270 --> 01:09:27.790
given that it's not a face or given a
01:09:27.790 --> 01:09:29.280
random patch from an image.
01:09:29.930 --> 01:09:32.260
And then that can be used to classify
01:09:32.260 --> 01:09:33.060
images as face.
01:09:33.060 --> 01:09:33.896
You're not face.
01:09:33.896 --> 01:09:35.560
And you would Estimate this separately
01:09:35.560 --> 01:09:37.120
for cars and for each orientation of
01:09:37.120 --> 01:09:38.110
car question.
01:09:43.310 --> 01:09:45.399
So the question was what beat the 2005
01:09:45.400 --> 01:09:45.840
model?
01:09:45.840 --> 01:09:47.750
I'm not really sure that there was
01:09:47.750 --> 01:09:50.180
something that beat it in 2006, but
01:09:50.180 --> 01:09:53.820
that when Dalal Triggs SVM based
01:09:53.820 --> 01:09:55.570
detector came out.
01:09:56.200 --> 01:09:57.680
And I think it might have been, I
01:09:57.680 --> 01:10:00.617
didn't look it up so I'm not sure, but
01:10:00.617 --> 01:10:02.930
I was, I'm pretty confident it was the
01:10:02.930 --> 01:10:04.947
most accurate up to 2005, but not
01:10:04.947 --> 01:10:06.070
confident after that.
01:10:07.250 --> 01:10:10.430
And now it took a while for face
01:10:10.430 --> 01:10:12.650
detection to get more accurate than
01:10:12.650 --> 01:10:15.630
most famous face detector was actually
01:10:15.630 --> 01:10:18.330
the Viola joins detector, which was
01:10:18.330 --> 01:10:20.515
popular because it was really fast.
01:10:20.515 --> 01:10:24.046
This thing man at a couple frames per
01:10:24.046 --> 01:10:26.414
second, but Viola Jones ran at 15
01:10:26.414 --> 01:10:28.560
frames per second in 2001.
01:10:30.310 --> 01:10:31.960
But Viola Jones wasn't quite as
01:10:31.960 --> 01:10:32.460
accurate.
01:10:35.210 --> 01:10:37.840
Alright, so Summary of Naive bees.
01:10:38.180 --> 01:10:38.790
And.
01:10:39.940 --> 01:10:41.740
So the key assumption is that the
01:10:41.740 --> 01:10:43.460
Features are independent given the
01:10:43.460 --> 01:10:43.870
labels.
01:10:46.730 --> 01:10:48.110
The parameters are just the
01:10:48.110 --> 01:10:50.173
probabilities, are the parameters of
01:10:50.173 --> 01:10:51.990
each of these probability functions,
01:10:51.990 --> 01:10:53.908
the probability of each feature given Y
01:10:53.908 --> 01:10:55.750
and probability of Y and justice.
01:10:55.750 --> 01:10:57.250
Like in the Simple fruit example I
01:10:57.250 --> 01:10:59.405
gave, you can use different models for
01:10:59.405 --> 01:10:59.976
different features.
01:10:59.976 --> 01:11:02.340
Some of the features could be discrete
01:11:02.340 --> 01:11:04.120
values and some could be continuous
01:11:04.120 --> 01:11:04.560
values.
01:11:04.560 --> 01:11:05.520
That's not a problem.
01:11:08.520 --> 01:11:10.150
You have to choose which probability
01:11:10.150 --> 01:11:11.510
function you're going to use for each
01:11:11.510 --> 01:11:11.940
feature.
01:11:14.450 --> 01:11:16.250
Nine days can be useful if you have
01:11:16.250 --> 01:11:18.080
limited training data, because you only
01:11:18.080 --> 01:11:19.560
have to Estimate these one-dimensional
01:11:19.560 --> 01:11:21.150
distributions, which you can do from
01:11:21.150 --> 01:11:22.370
relatively few Samples.
01:11:23.000 --> 01:11:24.420
And if the features are not highly
01:11:24.420 --> 01:11:26.540
interdependent, and it can also be
01:11:26.540 --> 01:11:27.970
useful as a baseline if you want
01:11:27.970 --> 01:11:29.766
something that's fast to code, train
01:11:29.766 --> 01:11:30.579
and test.
01:11:30.580 --> 01:11:32.900
So as you do your homework, I think out
01:11:32.900 --> 01:11:34.860
of the methods, Naive Bayes has the
01:11:34.860 --> 01:11:37.140
lowest training plus test time.
01:11:37.140 --> 01:11:40.139
Logistic regression is going to be
01:11:40.140 --> 01:11:42.618
roughly tied for test time, but it
01:11:42.618 --> 01:11:43.680
takes an awful lot.
01:11:43.680 --> 01:11:45.980
Well, it takes longer to train.
01:11:45.980 --> 01:11:48.379
KNN takes no time to train, but takes a
01:11:48.380 --> 01:11:49.570
whole lot longer to test.
01:11:54.630 --> 01:11:56.830
So when not to use?
01:11:56.830 --> 01:11:58.760
Usually Logistic or linear regression
01:11:58.760 --> 01:12:01.070
will work better if you have enough
01:12:01.070 --> 01:12:01.440
data.
01:12:02.230 --> 01:12:05.510
And the reason is that under most
01:12:05.510 --> 01:12:07.860
probability the exponential
01:12:07.860 --> 01:12:09.790
distribution of probability models
01:12:09.790 --> 01:12:11.940
which include Binomial, multinomial and
01:12:11.940 --> 01:12:12.530
Gaussian.
01:12:13.640 --> 01:12:15.657
You can rewrite Naive Bayes as a linear
01:12:15.657 --> 01:12:18.993
function of the input features, but the
01:12:18.993 --> 01:12:21.740
linear function is highly constrained
01:12:21.740 --> 01:12:23.750
based on this, estimating likelihoods
01:12:23.750 --> 01:12:25.650
for each feature separately.
01:12:25.650 --> 01:12:27.500
Where linear and logistic regression,
01:12:27.500 --> 01:12:28.970
which we'll talk about next Thursday,
01:12:28.970 --> 01:12:30.815
are not constrained, you can solve for
01:12:30.815 --> 01:12:32.300
the full range of coefficients.
01:12:33.440 --> 01:12:35.050
The other issue is that it doesn't
01:12:35.050 --> 01:12:37.890
provide a very good confidence Estimate
01:12:37.890 --> 01:12:39.720
because it over counts the influence of
01:12:39.720 --> 01:12:40.880
dependent variables.
01:12:40.880 --> 01:12:42.860
If you repeat a feature of many times,
01:12:42.860 --> 01:12:44.680
it's going to count it every time, and
01:12:44.680 --> 01:12:47.215
so it will tend to have too much weight
01:12:47.215 --> 01:12:48.930
and give you bad confidence estimates.
01:12:51.010 --> 01:12:55.100
9 Bayes is easy and fast to train, Fast
01:12:55.100 --> 01:12:56.130
for inference.
01:12:56.130 --> 01:12:57.400
You can use it with different kinds of
01:12:57.400 --> 01:12:58.040
variables.
01:12:58.040 --> 01:12:59.220
It doesn't account for feature
01:12:59.220 --> 01:13:00.730
interaction, doesn't provide good
01:13:00.730 --> 01:13:01.670
confidence estimates.
01:13:02.390 --> 01:13:04.210
And it's best when used with discrete
01:13:04.210 --> 01:13:06.270
variables, those that can be fit well
01:13:06.270 --> 01:13:08.830
by a Gaussian, or if you use kernel
01:13:08.830 --> 01:13:10.690
density estimation, which is something
01:13:10.690 --> 01:13:11.840
that we'll talk about later in this
01:13:11.840 --> 01:13:13.580
semester, a more general like
01:13:13.580 --> 01:13:15.080
continuous distribution function.
01:13:17.210 --> 01:13:19.560
And justice, as a reminder, don't pack
01:13:19.560 --> 01:13:21.730
up until I'm done, but this will be the
01:13:21.730 --> 01:13:22.570
second to last slide.
01:13:24.220 --> 01:13:25.890
So things remember.
01:13:27.140 --> 01:13:28.950
So Probabilistic models are really
01:13:28.950 --> 01:13:30.837
large class of machine learning
01:13:30.837 --> 01:13:31.160
methods.
01:13:31.160 --> 01:13:32.590
There are many different kinds of
01:13:32.590 --> 01:13:34.690
machine learning methods that are based
01:13:34.690 --> 01:13:36.480
on estimating the likelihoods of the
01:13:36.480 --> 01:13:38.170
label given the data or the data given
01:13:38.170 --> 01:13:38.730
the label.
01:13:39.580 --> 01:13:41.630
Naive Bayes assumes that Features are
01:13:41.630 --> 01:13:45.430
independent given the label, and it's
01:13:45.430 --> 01:13:46.860
easy and fast to estimate the
01:13:46.860 --> 01:13:48.920
parameters and reduces the risk of
01:13:48.920 --> 01:13:50.480
overfitting when you have limited data.
01:13:52.270 --> 01:13:52.590
It's.
01:13:52.590 --> 01:13:55.190
You don't usually have to derive how to
01:13:55.190 --> 01:13:57.910
solve for the likelihood parameters,
01:13:57.910 --> 01:13:59.660
but you can do it if you want to by
01:13:59.660 --> 01:14:00.954
taking the partial derivative.
01:14:00.954 --> 01:14:03.540
Usually it's usually you would be using
01:14:03.540 --> 01:14:06.140
a common a common kind of Model and you
01:14:06.140 --> 01:14:07.290
can just look up the Emily.
01:14:09.490 --> 01:14:11.160
The Prediction involves finding the way
01:14:11.160 --> 01:14:13.190
that maximizes the probability of the
01:14:13.190 --> 01:14:15.150
data and the label, either by trying
01:14:15.150 --> 01:14:17.250
all the possible values of Y or solving
01:14:17.250 --> 01:14:18.230
the partial derivative.
01:14:19.270 --> 01:14:21.535
And finally, Maximizing log probability
01:14:21.535 --> 01:14:24.060
of I is equivalent to Maximizing
01:14:24.060 --> 01:14:25.360
probability of.
01:14:25.520 --> 01:14:27.310
Sorry, Maximizing log probability of
01:14:27.310 --> 01:14:30.270
X&Y is equivalent to maximizing the
01:14:30.270 --> 01:14:32.250
probability of X&Y, and it's usually
01:14:32.250 --> 01:14:34.000
much easier, so it's important to
01:14:34.000 --> 01:14:34.390
remember that.
01:14:35.970 --> 01:14:36.180
Right.
01:14:36.180 --> 01:14:37.840
And then next class I'm going to talk
01:14:37.840 --> 01:14:40.030
about logistic regression and linear
01:14:40.030 --> 01:14:40.700
regression.
01:14:41.530 --> 01:14:44.870
And one more thing is I posted a review
01:14:44.870 --> 01:14:49.310
questions and answers to the 1st 2
01:14:49.310 --> 01:14:51.440
cannon and this lecture on the web
01:14:51.440 --> 01:14:52.050
page.
01:14:52.050 --> 01:14:53.690
You don't have to do them but they're
01:14:53.690 --> 01:14:55.410
good review for the exam or just the
01:14:55.410 --> 01:14:56.820
check your knowledge after each
01:14:56.820 --> 01:14:57.200
lecture.
01:14:57.890 --> 01:14:58.750
Thank you.
01:15:11.030 --> 01:15:11.320
I.
|