1272:
375:
41:
653:
370:{\displaystyle p\left(x_{1},\ldots ,x_{k}\right)={\frac {\Gamma \left(\nu _{1}+\cdots +\nu _{k+1}\right)}{\prod _{j=1}^{k+1}\Gamma \left(\nu _{j}\right)}}x_{1}^{\nu _{1}-1}\cdots x_{k}^{\nu _{k}-1}\times \left(1+\sum _{i=1}^{k}x_{i}\right)^{-\sum _{j=1}^{k+1}\nu _{j}},\qquad x_{i}>0.}
398:
648:{\displaystyle E\left={\frac {\Gamma \left(\nu _{k+1}-\sum _{j=1}^{k}q_{j}\right)}{\Gamma \left(\nu _{k+1}\right)}}\prod _{j=1}^{k}{\frac {\Gamma \left(\nu _{j}+q_{j}\right)}{\Gamma \left(\nu _{j}\right)}}}
717:
885:
782:
952:
957:
T. Bdiri et al. have developed several models that use the inverted
Dirichlet distribution to represent and model non-Gaussian data. They have introduced finite and infinite
1198:
Bdiri, Taoufik; Bouguila, Nizar; Ziou, Djemel (2014). "Object clustering and recognition using multi-finite mixtures for semantic classes and hierarchy modeling".
1332:
813:
1225:
Bdiri, Taoufik; Bouguila, Nizar; Ziou, Djemel (2013). "Visual Scenes
Categorization Using a Flexible Hierarchical Mixture Model Supporting Users Ontology".
1342:
1313:
1337:
1242:
1140:
1107:
1347:
788:
385:
969:
to model infinite mixtures. T. Bdiri et al. have also used the inverted
Dirichlet distribution to propose an approach to generate
661:
818:
1306:
722:
1063:
Bdiri, Taoufik; Nizar, Bouguila (2012). "Positive vectors clustering using inverted
Dirichlet finite mixture models".
890:
1352:
1299:
1090:
Bdiri, Taoufik; Bouguila, Nizar (2011). "Learning
Inverted Dirichlet Mixtures for Positive Data Clustering".
978:
25:
795:
is used instead of the categories' probabilities- if the negative multinomial parameter vector is given by
970:
381:
29:
1271:
389:
1123:
Bdiri, Taoufik; Bouguila, Nizar (2011). "An
Infinite Mixture of Inverted Dirichlet Distributions".
1157:
962:
1248:
1180:
1045:
998:
974:
1238:
1136:
1103:
966:
1230:
1207:
1172:
1128:
1095:
1072:
1037:
1010:
1283:
958:
798:
1326:
1184:
1049:
1252:
1014:
1099:
1132:
1211:
1076:
1279:
1176:
1041:
792:
17:
1227:
2013 IEEE 25th
International Conference on Tools with Artificial Intelligence
1158:"Bayesian learning of inverted Dirichlet mixtures for SVM kernels generation"
1234:
1094:. Lecture Notes in Computer Science. Vol. 6743. pp. 265–272.
1127:. Lecture Notes in Computer Science. Vol. 7063. pp. 71–78.
1001:(1965). "The inverted Dirichlet distribution with applications".
1028:
Ghorbel, M. (2010). "On the inverted
Dirichlet distribution".
1092:
Rough Sets, Fuzzy Sets, Data Mining and
Granular Computing
712:{\displaystyle q_{j}>-\nu _{j},1\leqslant j\leqslant k}
880:{\displaystyle x_{i}={\frac {p_{i}}{p_{0}}},i=1\ldots k}
815:, by changing parameters of the negative multinomial to
787:
The inverted
Dirichlet distribution is conjugate to the
1287:
32:. It was first described by Tiao and Cuttman in 1965.
893:
821:
801:
725:
664:
401:
44:
946:
879:
807:
776:
711:
647:
369:
1030:Communications in Statistics - Theory and Methods
777:{\displaystyle \nu _{k+1}>q_{1}+\ldots +q_{k}}
35:The distribution has a density function given by
1003:Journal of the American Statistical Association
961:of inverted Dirichlet distributions using the
1307:
965:technique to estimate the parameters and the
8:
947:{\displaystyle p_{0}=1-\sum _{i=1}^{k}p_{i}}
1314:
1300:
384:and arises naturally when considering the
938:
928:
917:
898:
892:
851:
841:
835:
826:
820:
800:
768:
749:
730:
724:
685:
669:
663:
632:
608:
595:
580:
574:
563:
540:
516:
506:
495:
476:
461:
445:
440:
435:
425:
414:
400:
355:
339:
323:
312:
304:
293:
283:
272:
239:
234:
229:
208:
203:
198:
181:
158:
147:
124:
105:
90:
76:
57:
43:
1156:Bdiri, Taoufik; Nizar, Bouguila (2013).
24:is a multivariate generalization of the
990:
1333:Multivariate continuous distributions
380:The distribution has applications in
7:
1268:
1266:
977:and another approach to establish
621:
583:
529:
464:
388:. It can be characterized by its
170:
93:
14:
1165:Neural Computing and Applications
789:negative multinomial distribution
386:multivariate Student distribution
1343:Exponential family distributions
1270:
1200:Expert Systems with Applications
1065:Expert Systems with Applications
350:
22:inverted Dirichlet distribution
1015:10.1080/01621459.1965.10480828
1:
1338:Conjugate prior distributions
1125:Neural Information Processing
1286:. You can help Knowledge by
1100:10.1007/978-3-642-21881-1_42
1133:10.1007/978-3-642-24958-7_9
1369:
1265:
1212:10.1016/j.eswa.2013.08.005
1077:10.1016/j.eswa.2011.08.063
1177:10.1007/s00521-012-1094-z
1042:10.1080/03610920802627062
791:if a generalized form of
1348:Continuous distributions
28:, and is related to the
979:hierarchical clustering
26:beta prime distribution
1282:-related article is a
971:Support Vector Machine
948:
933:
881:
809:
778:
713:
649:
579:
511:
430:
382:statistical regression
371:
334:
288:
169:
30:Dirichlet distribution
1235:10.1109/ICTAI.2013.48
949:
913:
882:
810:
779:
714:
650:
559:
491:
410:
372:
308:
268:
143:
1229:. pp. 262–267.
891:
819:
799:
723:
662:
399:
42:
973:kernels basing on
452:
252:
221:
975:Bayesian inference
944:
877:
805:
774:
709:
645:
431:
367:
225:
194:
1295:
1294:
1244:978-1-4799-2972-6
1142:978-3-642-24957-0
1109:978-3-642-21880-4
967:Dirichlet process
857:
808:{\displaystyle p}
643:
557:
192:
1360:
1353:Statistics stubs
1316:
1309:
1302:
1274:
1267:
1257:
1256:
1222:
1216:
1215:
1206:(4): 1218–1235.
1195:
1189:
1188:
1171:(5): 1443–1458.
1162:
1153:
1147:
1146:
1120:
1114:
1113:
1087:
1081:
1080:
1071:(2): 1869–1882.
1060:
1054:
1053:
1025:
1019:
1018:
1009:(311): 793–805.
995:
953:
951:
950:
945:
943:
942:
932:
927:
903:
902:
886:
884:
883:
878:
858:
856:
855:
846:
845:
836:
831:
830:
814:
812:
811:
806:
783:
781:
780:
775:
773:
772:
754:
753:
741:
740:
718:
716:
715:
710:
690:
689:
674:
673:
654:
652:
651:
646:
644:
642:
641:
637:
636:
619:
618:
614:
613:
612:
600:
599:
581:
578:
573:
558:
556:
555:
551:
550:
527:
526:
522:
521:
520:
510:
505:
487:
486:
462:
457:
453:
451:
450:
449:
439:
429:
424:
376:
374:
373:
368:
360:
359:
346:
345:
344:
343:
333:
322:
303:
299:
298:
297:
287:
282:
251:
244:
243:
233:
220:
213:
212:
202:
193:
191:
190:
186:
185:
168:
157:
141:
140:
136:
135:
134:
110:
109:
91:
86:
82:
81:
80:
62:
61:
1368:
1367:
1363:
1362:
1361:
1359:
1358:
1357:
1323:
1322:
1321:
1320:
1263:
1261:
1260:
1245:
1224:
1223:
1219:
1197:
1196:
1192:
1160:
1155:
1154:
1150:
1143:
1122:
1121:
1117:
1110:
1089:
1088:
1084:
1062:
1061:
1057:
1027:
1026:
1022:
997:
996:
992:
987:
934:
894:
889:
888:
847:
837:
822:
817:
816:
797:
796:
764:
745:
726:
721:
720:
681:
665:
660:
659:
628:
624:
620:
604:
591:
590:
586:
582:
536:
532:
528:
512:
472:
471:
467:
463:
441:
409:
405:
397:
396:
351:
335:
289:
261:
257:
256:
235:
204:
177:
173:
142:
120:
101:
100:
96:
92:
72:
53:
52:
48:
40:
39:
12:
11:
5:
1366:
1364:
1356:
1355:
1350:
1345:
1340:
1335:
1325:
1324:
1319:
1318:
1311:
1304:
1296:
1293:
1292:
1275:
1259:
1258:
1243:
1217:
1190:
1148:
1141:
1115:
1108:
1082:
1055:
1020:
989:
988:
986:
983:
963:Newton–Raphson
959:mixture models
941:
937:
931:
926:
923:
920:
916:
912:
909:
906:
901:
897:
876:
873:
870:
867:
864:
861:
854:
850:
844:
840:
834:
829:
825:
804:
771:
767:
763:
760:
757:
752:
748:
744:
739:
736:
733:
729:
708:
705:
702:
699:
696:
693:
688:
684:
680:
677:
672:
668:
658:provided that
656:
655:
640:
635:
631:
627:
623:
617:
611:
607:
603:
598:
594:
589:
585:
577:
572:
569:
566:
562:
554:
549:
546:
543:
539:
535:
531:
525:
519:
515:
509:
504:
501:
498:
494:
490:
485:
482:
479:
475:
470:
466:
460:
456:
448:
444:
438:
434:
428:
423:
420:
417:
413:
408:
404:
378:
377:
366:
363:
358:
354:
349:
342:
338:
332:
329:
326:
321:
318:
315:
311:
307:
302:
296:
292:
286:
281:
278:
275:
271:
267:
264:
260:
255:
250:
247:
242:
238:
232:
228:
224:
219:
216:
211:
207:
201:
197:
189:
184:
180:
176:
172:
167:
164:
161:
156:
153:
150:
146:
139:
133:
130:
127:
123:
119:
116:
113:
108:
104:
99:
95:
89:
85:
79:
75:
71:
68:
65:
60:
56:
51:
47:
13:
10:
9:
6:
4:
3:
2:
1365:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1330:
1328:
1317:
1312:
1310:
1305:
1303:
1298:
1297:
1291:
1289:
1285:
1281:
1276:
1273:
1269:
1264:
1254:
1250:
1246:
1240:
1236:
1232:
1228:
1221:
1218:
1213:
1209:
1205:
1201:
1194:
1191:
1186:
1182:
1178:
1174:
1170:
1166:
1159:
1152:
1149:
1144:
1138:
1134:
1130:
1126:
1119:
1116:
1111:
1105:
1101:
1097:
1093:
1086:
1083:
1078:
1074:
1070:
1066:
1059:
1056:
1051:
1047:
1043:
1039:
1035:
1031:
1024:
1021:
1016:
1012:
1008:
1004:
1000:
994:
991:
984:
982:
980:
976:
972:
968:
964:
960:
955:
939:
935:
929:
924:
921:
918:
914:
910:
907:
904:
899:
895:
874:
871:
868:
865:
862:
859:
852:
848:
842:
838:
832:
827:
823:
802:
794:
790:
785:
769:
765:
761:
758:
755:
750:
746:
742:
737:
734:
731:
727:
706:
703:
700:
697:
694:
691:
686:
682:
678:
675:
670:
666:
638:
633:
629:
625:
615:
609:
605:
601:
596:
592:
587:
575:
570:
567:
564:
560:
552:
547:
544:
541:
537:
533:
523:
517:
513:
507:
502:
499:
496:
492:
488:
483:
480:
477:
473:
468:
458:
454:
446:
442:
436:
432:
426:
421:
418:
415:
411:
406:
402:
395:
394:
393:
391:
390:mixed moments
387:
383:
364:
361:
356:
352:
347:
340:
336:
330:
327:
324:
319:
316:
313:
309:
305:
300:
294:
290:
284:
279:
276:
273:
269:
265:
262:
258:
253:
248:
245:
240:
236:
230:
226:
222:
217:
214:
209:
205:
199:
195:
187:
182:
178:
174:
165:
162:
159:
154:
151:
148:
144:
137:
131:
128:
125:
121:
117:
114:
111:
106:
102:
97:
87:
83:
77:
73:
69:
66:
63:
58:
54:
49:
45:
38:
37:
36:
33:
31:
27:
23:
19:
1288:expanding it
1277:
1262:
1226:
1220:
1203:
1199:
1193:
1168:
1164:
1151:
1124:
1118:
1091:
1085:
1068:
1064:
1058:
1033:
1029:
1023:
1006:
1002:
999:Tiao, George
993:
956:
786:
657:
379:
34:
21:
15:
1327:Categories
1280:statistics
985:References
793:odds ratio
18:statistics
1185:254025619
1050:122956752
1036:: 21–37.
915:∑
911:−
872:…
759:…
728:ν
704:⩽
698:⩽
683:ν
679:−
630:ν
622:Γ
593:ν
584:Γ
561:∏
538:ν
530:Γ
493:∑
489:−
474:ν
465:Γ
412:∏
337:ν
310:∑
306:−
270:∑
254:×
246:−
237:ν
223:⋯
215:−
206:ν
179:ν
171:Γ
145:∏
122:ν
115:⋯
103:ν
94:Γ
67:…
1253:1236111
20:, the
1251:
1241:
1183:
1139:
1106:
1048:
887:where
1278:This
1249:S2CID
1181:S2CID
1161:(PDF)
1046:S2CID
1284:stub
1239:ISBN
1137:ISBN
1104:ISBN
743:>
719:and
676:>
362:>
1231:doi
1208:doi
1173:doi
1129:doi
1096:doi
1073:doi
1038:doi
1011:doi
16:In
1329::
1247:.
1237:.
1204:41
1202:.
1179:.
1169:23
1167:.
1163:.
1135:.
1102:.
1069:39
1067:.
1044:.
1034:39
1032:.
1007:60
1005:.
981:.
954:.
784:.
392::
365:0.
1315:e
1308:t
1301:v
1290:.
1255:.
1233::
1214:.
1210::
1187:.
1175::
1145:.
1131::
1112:.
1098::
1079:.
1075::
1052:.
1040::
1017:.
1013::
940:i
936:p
930:k
925:1
922:=
919:i
908:1
905:=
900:0
896:p
875:k
869:1
866:=
863:i
860:,
853:0
849:p
843:i
839:p
833:=
828:i
824:x
803:p
770:k
766:q
762:+
756:+
751:1
747:q
738:1
735:+
732:k
707:k
701:j
695:1
692:,
687:j
671:j
667:q
639:)
634:j
626:(
616:)
610:j
606:q
602:+
597:j
588:(
576:k
571:1
568:=
565:j
553:)
548:1
545:+
542:k
534:(
524:)
518:j
514:q
508:k
503:1
500:=
497:j
484:1
481:+
478:k
469:(
459:=
455:]
447:i
443:q
437:i
433:x
427:k
422:1
419:=
416:i
407:[
403:E
357:i
353:x
348:,
341:j
331:1
328:+
325:k
320:1
317:=
314:j
301:)
295:i
291:x
285:k
280:1
277:=
274:i
266:+
263:1
259:(
249:1
241:k
231:k
227:x
218:1
210:1
200:1
196:x
188:)
183:j
175:(
166:1
163:+
160:k
155:1
152:=
149:j
138:)
132:1
129:+
126:k
118:+
112:+
107:1
98:(
88:=
84:)
78:k
74:x
70:,
64:,
59:1
55:x
50:(
46:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.