3471:
3245:
2061:
3158:
3466:{\displaystyle D_{B}(p_{1},p_{2})={1 \over 8}({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{2})^{T}{\boldsymbol {\Sigma }}^{-1}({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{2})+{1 \over 2}\ln \,\left({\det {\boldsymbol {\Sigma }} \over {\sqrt {\det {\boldsymbol {\Sigma }}_{1}\,\det {\boldsymbol {\Sigma }}_{2}}}}\right)}
85:. He has developed this through a series of papers. He developed the method to measure the distance between two non-normal distributions and illustrated this with the classical multinomial populations, this work despite being submitted for publication in 1941, appeared almost five years later in
4273:
is a special case of the
Bhattacharyya distance when the two classes are normally distributed with the same variances. When two classes have similar means but significantly different variances, the Mahalanobis distance would be close to zero, while the Bhattacharyya distance would not be.
1285:
2940:
1690:
2700:
1658:
2212:
3539:
89:. Consequently, Professor Bhattacharyya started working toward developing a distance metric for probability distributions that are absolutely continuous with respect to the Lebesgue measure and published his progress in 1942, at Proceedings of the
3237:
3804:
4181:
1075:
2490:
356:
629:
3874:
3153:{\displaystyle D_{B}(p,q)={\frac {1}{4}}{\frac {(\mu _{p}-\mu _{q})^{2}}{\sigma _{p}^{2}+\sigma _{q}^{2}}}+{\frac {1}{2}}\ln \left({\frac {\sigma _{p}^{2}+\sigma _{q}^{2}}{2\sigma _{p}\sigma _{q}}}\right)}
2835:
2771:
2056:{\displaystyle bc(dx|P,Q)={\sqrt {p(x)q(x)}}\,\lambda (dx)={\sqrt {p(x)q(x)}}\,l(x)\mu (x)={\sqrt {p(x)l(x)q(x)\,l(x)}}\mu (dx)={\sqrt {p'(x)l'(x)q'(x)l'(x)}}\,\mu (dx)={\sqrt {p'(x)q'(x)}}\,\lambda '(dx)}
4738:
Djouadi, A.; Snorrason, O.; Garber, F. (1990). "The quality of
Training-Sample estimates of the Bhattacharyya coefficient". IEEE Transactions on Pattern Analysis and Machine Intelligence. 12 (1): 92â97.
261:
2881:
1067:
699:
2485:
1459:
1027:
3934:
3624:
911:
849:
3708:
2324:
2073:
3480:
1411:
2386:
3727:
3585:
1373:
173:
476:
423:
2932:
2274:
1348:
1308:
977:
957:
787:
719:
4260:
4222:
2254:
4085:
4058:
3653:
2905:
1467:
1328:
534:
505:
3971:
3171:
937:
659:
4887:
4488:
4031:
4011:
3991:
2406:
1682:
763:
743:
146:
126:
4622:
François
Goudail, Philippe Réfrégier, Guillaume Delyon, "Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images",
4735:
Kailath, T. (1967). "The
Divergence and Bhattacharyya Distance Measures in Signal Selection". IEEE Transactions on Communication Technology. 15 (1): 52â60.
4093:
4565:
Arıkan, Erdal (July 2009). "Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels".
1280:{\displaystyle bc(dx|P,Q)={\sqrt {p(x)q(x)}}\,\lambda (dx)={\sqrt {{\frac {P(dx)}{\lambda (dx)}}(x){\frac {Q(dx)}{\lambda (dx)}}(x)}}\lambda (dx).}
4363:
272:
4732:
Nielsen, F.; Boltz, S. (2010). "The BurbeaâRao and
Bhattacharyya centroids". IEEE Transactions on Information Theory. 57 (8): 5455â5466.
546:
90:
3811:
2776:
2712:
2695:{\displaystyle BC(P,Q)=\int _{\mathcal {X}}{\sqrt {p(x)}}Q(dx)=\int _{\mathcal {X}}{\sqrt {\frac {P(dx)}{Q(dx)}}}Q(dx)=E_{Q}\left}
4892:
4302:
4555:
Devroye, L., Gyorfi, L. & Lugosi, G. A Probabilistic Theory of
Pattern Recognition. Discrete Appl Math 73, 192â194 (1997).
4882:
4715:
181:
94:
82:
4710:
2840:
1032:
664:
2411:
4332:
4278:
4266:
982:
4635:
Chang Huai You, "An SVM Kernel With GMM-Supervector Based on the
Bhattacharyya Distance for Speaker Recognition",
3879:
766:
74:
43:
3590:
4705:
4652:, 1996. ICSLP 96. Proceedings., Fourth International Conference on, Vol 4, pp. 2005â2008 vol.4, 3â6 Oct 1996
854:
792:
4484:"On a measure of divergence between two statistical populations defined by their probability distributions"
2207:{\displaystyle BC(P,Q)=\int _{\mathcal {X}}bc(dx|P,Q)=\int _{\mathcal {X}}{\sqrt {p(x)q(x)}}\,\lambda (dx)}
3534:{\displaystyle {\boldsymbol {\Sigma }}={{\boldsymbol {\Sigma }}_{1}+{\boldsymbol {\Sigma }}_{2} \over 2}.}
3665:
2279:
1416:
1378:
4517:"The Perfect Marriage and Much More: Combining Dimension Reduction, Distance Measures and Covariance"
4312:
4270:
3542:
2329:
722:
149:
58:
3555:
154:
4297:
4285:
3656:
3165:
2884:
537:
62:
4483:
428:
375:
4789:
4763:
4592:
4574:
4528:
4439:
4403:
4307:
4224:. The quality of estimation depends on the choice of buckets; too few buckets would overestimate
3660:
2910:
39:
4461:
1353:
4828:
4781:
4683:
4359:
2259:
1333:
1293:
962:
942:
772:
704:
86:
4227:
4189:
2221:
1653:{\displaystyle P(dx)=p(x)\lambda (dx)=p'(x)\lambda '(dx)=p(x)l(x)\mu (dx)=p'(x)l'(x)\mu (dx)}
4859:
4820:
4773:
4675:
4584:
4538:
4395:
4383:
4322:
3944:
The
Bhattacharyya coefficient quantifies the "closeness" of two random statistical samples.
3719:
3232:{\displaystyle p_{i}={\mathcal {N}}({\boldsymbol {\mu }}_{i},\,{\boldsymbol {\Sigma }}_{i})}
4501:
4063:
4036:
3631:
4497:
4284:
The
Bhattacharyya distance is used in feature extraction and selection, image processing,
2890:
1313:
510:
481:
3950:
979:-almost everywhere. Such a measure, even such a probability measure, always exists, e.g.
916:
638:
17:
4317:
4016:
3996:
3976:
3799:{\displaystyle {\frac {1}{2}}-{\frac {1}{2}}{\sqrt {1-4\rho ^{2}}}\leq L^{*}\leq \rho }
2391:
1667:
748:
728:
131:
111:
4269:
is estimating the separability of classes. Up to a multiplicative factor, the squared
4876:
4407:
4793:
4609:
Euisun Choi, Chulhee Lee, "Feature extraction based on the
Bhattacharyya distance",
4327:
78:
4727:
4596:
4176:{\displaystyle BC(\mathbf {p} ,\mathbf {q} )=\sum _{i=1}^{n}{\sqrt {p_{i}q_{i}}},}
73:
Both the Bhattacharyya distance and the Bhattacharyya coefficient are named after
4662:
Chattopadhyay, Aparna; Chattopadhyay, Asis Kumar; B-Rao, Chandrika (2004-06-01).
4353:
4808:
4751:
4542:
4847:
4824:
4664:"Bhattacharyya's distance measure as a precursor of genetic distance measures"
51:
31:
4832:
4785:
4777:
4687:
4588:
4399:
4848:"The quality of training sample estimates of the Bhattacharyya coefficient"
4722:
4809:"The Divergence and Bhattacharyya Distance Measures in Signal Selection"
4443:
4427:
4679:
4863:
4663:
351:{\displaystyle BC(P,Q)=\sum _{x\in {\mathcal {X}}}{\sqrt {P(x)Q(x)}}}
3718:
The Bhattacharyya distance can be used to upper and lower bound the
4533:
4768:
4579:
4516:
624:{\displaystyle BC(P,Q)=\int _{\mathcal {X}}{\sqrt {p(x)q(x)}}\,dx}
4428:"On a Measure of Divergence between Two Multinomial Populations"
3869:{\displaystyle \rho =\mathbb {E} {\sqrt {\eta (X)(1-\eta (X))}}}
93:
and the final work has appeared in 1943 in the Bulletin of the
61:, despite being named a "distance", since it does not obey the
4852:
IEEE Transactions on Pattern Analysis and Machine Intelligence
4648:
Mak, B., "Phone clustering using the Bhattacharyya distance",
4384:"Anil Kumar Bhattacharyya (1915-1996): A Reverent Remembrance"
2830:{\displaystyle q\sim {\mathcal {N}}(\mu _{q},\sigma _{q}^{2})}
2766:{\displaystyle p\sim {\mathcal {N}}(\mu _{p},\sigma _{p}^{2})}
4277:
The Bhattacharyya coefficient is used in the construction of
3190:
2846:
2788:
2724:
2570:
2524:
2154:
2107:
1051:
1041:
683:
673:
580:
313:
160:
50:, which is a measure of the amount of overlap between two
540:
functions, the Bhattacharyya coefficient is defined as
993:
4521:
Physica A: Statistical Mechanics and its Applications
4230:
4192:
4096:
4066:
4039:
4019:
3999:
3979:
3953:
3882:
3814:
3730:
3668:
3634:
3593:
3558:
3483:
3248:
3174:
2943:
2913:
2893:
2843:
2779:
2715:
2493:
2414:
2394:
2332:
2282:
2262:
2224:
2076:
1693:
1670:
1470:
1419:
1381:
1356:
1336:
1316:
1296:
1078:
1035:
985:
965:
945:
919:
857:
795:
775:
751:
731:
707:
667:
641:
549:
513:
484:
431:
378:
275:
184:
157:
134:
114:
256:{\displaystyle D_{B}(P,Q)=-\ln \left(BC(P,Q)\right)}
4613:, Volume 36, Issue 8, August 2003, Pages 1703â1709
4254:
4216:
4175:
4079:
4052:
4025:
4005:
3985:
3965:
3928:
3868:
3798:
3702:
3647:
3618:
3579:
3533:
3465:
3231:
3152:
2926:
2899:
2875:
2829:
2765:
2694:
2479:
2400:
2380:
2318:
2268:
2248:
2206:
2055:
1676:
1652:
1453:
1405:
1367:
1342:
1322:
1302:
1279:
1061:
1021:
971:
951:
931:
905:
843:
781:
757:
737:
713:
693:
653:
623:
528:
499:
470:
417:
350:
255:
167:
140:
120:
4846:Djouadi, A.; Snorrason, O.; Garber, F.D. (1990).
4723:Statistical Intuition of Bhattacharyya's distance
4728:Some of the properties of Bhattacharyya Distance
3440:
3424:
3413:
2876:{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}
2067:We finally define the Bhattacharyya coefficient
4087:, then the sample Bhattacharyya coefficient is
3993:buckets, and let the frequency of samples from
1062:{\displaystyle ({\mathcal {X}},{\mathcal {B}})}
694:{\displaystyle ({\mathcal {X}},{\mathcal {B}})}
635:More generally, given two probability measures
2480:{\displaystyle p(x)={\frac {P(dx)}{Q(dx)}}(x)}
4489:Bulletin of the Calcutta Mathematical Society
1022:{\displaystyle \lambda ={\tfrac {1}{2}}(P+Q)}
8:
4752:"The Burbea-Rao and Bhattacharyya Centroids"
3929:{\displaystyle \eta (X)=\mathbb {P} (Y=1|X)}
1029:. Then define the Bhattacharyya measure on
175:, the Bhattacharyya distance is defined as
38:is a quantity which represents a notion of
27:Similarity of two probability distributions
4466:Proceedings of the Indian Science Congress
4355:The Oxford Dictionary of Statistical Terms
4767:
4578:
4532:
4388:Calcutta Statistical Association Bulletin
4229:
4191:
4162:
4152:
4146:
4140:
4129:
4114:
4106:
4095:
4071:
4065:
4044:
4038:
4018:
3998:
3978:
3952:
3915:
3899:
3898:
3881:
3826:
3822:
3821:
3813:
3784:
3769:
3754:
3744:
3731:
3729:
3669:
3667:
3639:
3633:
3604:
3592:
3557:
3516:
3511:
3501:
3496:
3492:
3484:
3482:
3449:
3444:
3439:
3433:
3428:
3422:
3416:
3410:
3405:
3392:
3380:
3375:
3365:
3360:
3347:
3342:
3335:
3325:
3320:
3310:
3305:
3291:
3279:
3266:
3253:
3247:
3220:
3215:
3213:
3204:
3199:
3189:
3188:
3179:
3173:
3137:
3127:
3112:
3107:
3094:
3089:
3082:
3062:
3050:
3045:
3032:
3027:
3015:
3005:
2992:
2982:
2972:
2948:
2942:
2918:
2912:
2892:
2864:
2845:
2844:
2842:
2818:
2813:
2800:
2787:
2786:
2778:
2754:
2749:
2736:
2723:
2722:
2714:
2648:
2638:
2576:
2569:
2568:
2530:
2523:
2522:
2492:
2430:
2413:
2393:
2331:
2281:
2261:
2223:
2188:
2160:
2153:
2152:
2128:
2106:
2105:
2075:
2032:
1994:
1975:
1903:
1870:
1832:
1804:
1776:
1757:
1729:
1709:
1692:
1669:
1469:
1418:
1380:
1355:
1335:
1315:
1295:
1210:
1163:
1161:
1142:
1114:
1094:
1077:
1050:
1049:
1040:
1039:
1034:
992:
984:
964:
944:
918:
856:
794:
774:
750:
730:
706:
682:
681:
672:
671:
666:
640:
614:
586:
579:
578:
548:
512:
483:
430:
377:
320:
312:
311:
304:
274:
189:
183:
159:
158:
156:
133:
113:
4626:, Vol. 21, Issue 7, pp. 1231â1240 (2004)
4756:IEEE Transactions on Information Theory
4750:Nielsen, Frank; Boltz, Sylvain (2011).
4742:
4567:IEEE Transactions on Information Theory
4344:
3947:Given two sequences from distributions
3619:{\displaystyle 0\leq D_{B}\leq \infty }
3376:
3361:
3321:
3306:
3200:
4262:, while too many would underestimate.
3541:Note that the first term is a squared
906:{\displaystyle Q(dx)=q(x)\lambda (dx)}
844:{\displaystyle P(dx)=p(x)\lambda (dx)}
4288:, phone clustering, and in genetics.
361:is the Bhattacharyya coefficient for
7:
4888:Statistical deviation and dispersion
4477:
4475:
4455:
4453:
4421:
4419:
4417:
4377:
4375:
370:continuous probability distributions
4813:IEEE Transactions on Communications
3703:{\displaystyle {\sqrt {1-BC(p,q)}}}
2319:{\displaystyle 0\leq BC(P,Q)\leq 1}
1454:{\displaystyle \lambda '=l'(x)\mu }
4462:"On discrimination and divergence"
3613:
1290:It does not depend on the measure
913:for probability density functions
363:discrete probability distributions
25:
1406:{\displaystyle \lambda =l(x)\mu }
4482:Bhattacharyya, A. (March 1943).
4115:
4107:
3512:
3497:
3485:
3445:
3429:
3417:
3343:
3216:
2388:is absolutely continuous wrt to
4639:, IEEE, Vol 16, Is 1, pp. 49-52
2381:{\displaystyle P(dx)=p(x)Q(dx)}
2276:, and by the Cauchy inequality
1375:are absolutely continuous i.e.
81:who worked in the 1930s at the
46:. It is closely related to the
4249:
4237:
4211:
4199:
4119:
4103:
3936:is the posterior probability.
3923:
3916:
3903:
3892:
3886:
3861:
3858:
3852:
3840:
3837:
3831:
3695:
3683:
3580:{\displaystyle 0\leq BC\leq 1}
3386:
3356:
3332:
3301:
3285:
3259:
3226:
3195:
3012:
2985:
2966:
2954:
2870:
2851:
2824:
2793:
2760:
2729:
2681:
2672:
2664:
2655:
2628:
2619:
2609:
2600:
2592:
2583:
2558:
2549:
2541:
2535:
2512:
2500:
2474:
2468:
2462:
2453:
2445:
2436:
2424:
2418:
2408:with Radon Nikodym derivative
2375:
2366:
2360:
2354:
2345:
2336:
2307:
2295:
2243:
2231:
2201:
2192:
2183:
2177:
2171:
2165:
2142:
2129:
2119:
2095:
2083:
2050:
2041:
2027:
2021:
2010:
2004:
1988:
1979:
1970:
1964:
1953:
1947:
1936:
1930:
1919:
1913:
1897:
1888:
1880:
1874:
1867:
1861:
1855:
1849:
1843:
1837:
1826:
1820:
1814:
1808:
1799:
1793:
1787:
1781:
1770:
1761:
1752:
1746:
1740:
1734:
1723:
1710:
1700:
1647:
1638:
1632:
1626:
1615:
1609:
1595:
1586:
1580:
1574:
1568:
1562:
1553:
1544:
1533:
1527:
1513:
1504:
1498:
1492:
1483:
1474:
1445:
1439:
1397:
1391:
1271:
1262:
1254:
1248:
1242:
1233:
1225:
1216:
1207:
1201:
1195:
1186:
1178:
1169:
1155:
1146:
1137:
1131:
1125:
1119:
1108:
1095:
1085:
1056:
1036:
1016:
1004:
900:
891:
885:
879:
870:
861:
838:
829:
823:
817:
808:
799:
688:
668:
609:
603:
597:
591:
568:
556:
523:
517:
494:
488:
459:
453:
444:
435:
406:
400:
391:
382:
343:
337:
331:
325:
294:
282:
245:
233:
207:
195:
168:{\displaystyle {\mathcal {X}}}
1:
1310:, for if we choose a measure
95:Calcutta Mathematical Society
4468:. Asiatic Society of Bengal.
1350:and an other measure choice
471:{\displaystyle Q(dx)=q(x)dx}
418:{\displaystyle P(dx)=p(x)dx}
83:Indian Statistical Institute
4711:Encyclopedia of Mathematics
4543:10.1016/j.physa.2019.04.174
4358:. Oxford University Press.
4303:KullbackâLeibler divergence
2927:{\displaystyle \sigma ^{2}}
2218:By the above, the quantity
4909:
4426:Bhattacharyya, A. (1946).
4382:Sen, Pranab Kumar (1996).
4333:Fidelity of quantum states
3164:And in general, given two
4825:10.1109/TCOM.1967.1089532
4637:Signal Processing Letters
4460:Bhattacharyya, A (1942).
4186:which is an estimator of
1368:{\displaystyle \lambda '}
107:probability distributions
48:Bhattacharyya coefficient
44:probability distributions
18:Bhattacharyya coefficient
4778:10.1109/TIT.2011.2159046
4706:"Bhattacharyya distance"
4589:10.1109/TIT.2009.2021379
4400:10.1177/0008068319960301
2269:{\displaystyle \lambda }
1343:{\displaystyle \lambda }
1303:{\displaystyle \lambda }
972:{\displaystyle \lambda }
952:{\displaystyle \lambda }
782:{\displaystyle \lambda }
714:{\displaystyle \lambda }
75:Anil Kumar Bhattacharyya
54:samples or populations.
4893:Anil Kumar Bhattacharya
4352:Dodge, Yadolah (2003).
4255:{\displaystyle BC(P,Q)}
4217:{\displaystyle BC(P,Q)}
2249:{\displaystyle BC(P,Q)}
91:Indian Science Congress
4668:Journal of Biosciences
4515:Kashyap, Ravi (2019).
4256:
4218:
4177:
4145:
4081:
4054:
4027:
4007:
3987:
3967:
3930:
3870:
3800:
3704:
3649:
3620:
3581:
3535:
3467:
3233:
3154:
2928:
2901:
2877:
2831:
2767:
2696:
2481:
2402:
2382:
2320:
2270:
2250:
2208:
2057:
1678:
1654:
1455:
1407:
1369:
1344:
1324:
1304:
1281:
1063:
1023:
973:
953:
933:
907:
845:
783:
759:
739:
715:
695:
661:on a measurable space
655:
625:
530:
501:
472:
419:
352:
257:
169:
142:
122:
36:Bhattacharyya distance
4257:
4219:
4178:
4125:
4082:
4080:{\displaystyle q_{i}}
4055:
4053:{\displaystyle p_{i}}
4028:
4008:
3988:
3968:
3931:
3871:
3801:
3714:Bounds on Bayes error
3705:
3650:
3648:{\displaystyle D_{B}}
3621:
3582:
3536:
3468:
3234:
3155:
2929:
2902:
2878:
2832:
2768:
2697:
2482:
2403:
2383:
2321:
2271:
2251:
2209:
2058:
1679:
1655:
1456:
1408:
1370:
1345:
1325:
1305:
1282:
1064:
1024:
974:
954:
934:
908:
846:
784:
767:absolutely continuous
760:
740:
716:
696:
656:
626:
531:
502:
473:
420:
353:
258:
170:
143:
123:
4883:Statistical distance
4807:Kailath, T. (1967).
4313:Mahalanobis distance
4271:Mahalanobis distance
4228:
4190:
4094:
4064:
4060:, and similarly for
4037:
4017:
3997:
3977:
3951:
3880:
3812:
3728:
3666:
3632:
3591:
3556:
3543:Mahalanobis distance
3481:
3246:
3172:
2941:
2911:
2900:{\displaystyle \mu }
2891:
2841:
2777:
2713:
2491:
2412:
2392:
2330:
2280:
2260:
2222:
2074:
1691:
1668:
1468:
1417:
1379:
1354:
1334:
1323:{\displaystyle \mu }
1314:
1294:
1076:
1033:
983:
963:
943:
917:
855:
793:
773:
749:
729:
725:) measure such that
705:
665:
639:
547:
529:{\displaystyle q(x)}
511:
500:{\displaystyle p(x)}
482:
429:
376:
273:
182:
155:
132:
112:
4611:Pattern Recognition
4298:Bhattacharyya angle
4286:speaker recognition
3966:{\displaystyle P,Q}
3657:triangle inequality
3166:multivariate normal
3117:
3099:
3055:
3037:
2885:normal distribution
2823:
2759:
2326:. In particular if
2256:does not depend on
932:{\displaystyle p,q}
654:{\displaystyle P,Q}
538:probability density
63:triangle inequality
4680:10.1007/BF02703410
4308:Hellinger distance
4252:
4214:
4173:
4077:
4050:
4023:
4003:
3983:
3963:
3926:
3866:
3796:
3700:
3661:Hellinger distance
3655:does not obey the
3645:
3616:
3577:
3531:
3463:
3229:
3150:
3103:
3085:
3041:
3023:
2924:
2897:
2873:
2827:
2809:
2763:
2745:
2692:
2477:
2398:
2378:
2316:
2266:
2246:
2204:
2053:
1674:
1664:and similarly for
1650:
1451:
1403:
1365:
1340:
1320:
1300:
1277:
1059:
1019:
1002:
969:
949:
929:
903:
841:
779:
755:
735:
711:
691:
651:
621:
526:
497:
468:
415:
348:
319:
253:
165:
138:
118:
4365:978-0-19-920613-1
4265:A common task in
4168:
4026:{\displaystyle i}
4006:{\displaystyle P}
3986:{\displaystyle n}
3864:
3775:
3752:
3739:
3698:
3526:
3457:
3455:
3400:
3299:
3144:
3070:
3057:
2980:
2686:
2685:
2614:
2613:
2544:
2466:
2401:{\displaystyle Q}
2186:
2030:
1973:
1883:
1802:
1755:
1677:{\displaystyle Q}
1257:
1246:
1199:
1140:
1001:
758:{\displaystyle Q}
738:{\displaystyle P}
612:
346:
300:
141:{\displaystyle Q}
121:{\displaystyle P}
16:(Redirected from
4900:
4868:
4867:
4864:10.1109/34.41388
4843:
4837:
4836:
4804:
4798:
4797:
4771:
4762:(8): 5455â5466.
4747:
4719:
4692:
4691:
4659:
4653:
4646:
4640:
4633:
4627:
4620:
4614:
4607:
4601:
4600:
4582:
4573:(7): 3051â3073.
4562:
4556:
4553:
4547:
4546:
4536:
4512:
4506:
4505:
4479:
4470:
4469:
4457:
4448:
4447:
4423:
4412:
4411:
4394:(3â4): 151â158.
4379:
4370:
4369:
4349:
4261:
4259:
4258:
4253:
4223:
4221:
4220:
4215:
4182:
4180:
4179:
4174:
4169:
4167:
4166:
4157:
4156:
4147:
4144:
4139:
4118:
4110:
4086:
4084:
4083:
4078:
4076:
4075:
4059:
4057:
4056:
4051:
4049:
4048:
4032:
4030:
4029:
4024:
4012:
4010:
4009:
4004:
3992:
3990:
3989:
3984:
3973:, bin them into
3972:
3970:
3969:
3964:
3935:
3933:
3932:
3927:
3919:
3902:
3875:
3873:
3872:
3867:
3865:
3827:
3825:
3805:
3803:
3802:
3797:
3789:
3788:
3776:
3774:
3773:
3755:
3753:
3745:
3740:
3732:
3720:Bayes error rate
3709:
3707:
3706:
3701:
3699:
3670:
3654:
3652:
3651:
3646:
3644:
3643:
3625:
3623:
3622:
3617:
3609:
3608:
3586:
3584:
3583:
3578:
3540:
3538:
3537:
3532:
3527:
3522:
3521:
3520:
3515:
3506:
3505:
3500:
3493:
3488:
3472:
3470:
3469:
3464:
3462:
3458:
3456:
3454:
3453:
3448:
3438:
3437:
3432:
3423:
3421:
3420:
3411:
3401:
3393:
3385:
3384:
3379:
3370:
3369:
3364:
3355:
3354:
3346:
3340:
3339:
3330:
3329:
3324:
3315:
3314:
3309:
3300:
3292:
3284:
3283:
3271:
3270:
3258:
3257:
3238:
3236:
3235:
3230:
3225:
3224:
3219:
3209:
3208:
3203:
3194:
3193:
3184:
3183:
3159:
3157:
3156:
3151:
3149:
3145:
3143:
3142:
3141:
3132:
3131:
3118:
3116:
3111:
3098:
3093:
3083:
3071:
3063:
3058:
3056:
3054:
3049:
3036:
3031:
3021:
3020:
3019:
3010:
3009:
2997:
2996:
2983:
2981:
2973:
2953:
2952:
2933:
2931:
2930:
2925:
2923:
2922:
2906:
2904:
2903:
2898:
2882:
2880:
2879:
2874:
2869:
2868:
2850:
2849:
2836:
2834:
2833:
2828:
2822:
2817:
2805:
2804:
2792:
2791:
2772:
2770:
2769:
2764:
2758:
2753:
2741:
2740:
2728:
2727:
2701:
2699:
2698:
2693:
2691:
2687:
2684:
2667:
2650:
2649:
2643:
2642:
2615:
2612:
2595:
2578:
2577:
2575:
2574:
2573:
2545:
2531:
2529:
2528:
2527:
2486:
2484:
2483:
2478:
2467:
2465:
2448:
2431:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2325:
2323:
2322:
2317:
2275:
2273:
2272:
2267:
2255:
2253:
2252:
2247:
2213:
2211:
2210:
2205:
2187:
2161:
2159:
2158:
2157:
2132:
2112:
2111:
2110:
2062:
2060:
2059:
2054:
2040:
2031:
2020:
2003:
1995:
1974:
1963:
1946:
1929:
1912:
1904:
1884:
1833:
1803:
1777:
1756:
1730:
1713:
1683:
1681:
1680:
1675:
1659:
1657:
1656:
1651:
1625:
1608:
1543:
1526:
1460:
1458:
1457:
1452:
1438:
1427:
1412:
1410:
1409:
1404:
1374:
1372:
1371:
1366:
1364:
1349:
1347:
1346:
1341:
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1286:
1284:
1283:
1278:
1258:
1247:
1245:
1228:
1211:
1200:
1198:
1181:
1164:
1162:
1141:
1115:
1098:
1068:
1066:
1065:
1060:
1055:
1054:
1045:
1044:
1028:
1026:
1025:
1020:
1003:
994:
978:
976:
975:
970:
958:
956:
955:
950:
939:with respect to
938:
936:
935:
930:
912:
910:
909:
904:
850:
848:
847:
842:
788:
786:
785:
780:
769:with respect to
764:
762:
761:
756:
744:
742:
741:
736:
720:
718:
717:
712:
700:
698:
697:
692:
687:
686:
677:
676:
660:
658:
657:
652:
630:
628:
627:
622:
613:
587:
585:
584:
583:
535:
533:
532:
527:
506:
504:
503:
498:
477:
475:
474:
469:
424:
422:
421:
416:
357:
355:
354:
349:
347:
321:
318:
317:
316:
262:
260:
259:
254:
252:
248:
194:
193:
174:
172:
171:
166:
164:
163:
147:
145:
144:
139:
127:
125:
124:
119:
21:
4908:
4907:
4903:
4902:
4901:
4899:
4898:
4897:
4873:
4872:
4871:
4845:
4844:
4840:
4806:
4805:
4801:
4749:
4748:
4744:
4704:
4701:
4696:
4695:
4661:
4660:
4656:
4650:Spoken Language
4647:
4643:
4634:
4630:
4621:
4617:
4608:
4604:
4564:
4563:
4559:
4554:
4550:
4514:
4513:
4509:
4481:
4480:
4473:
4459:
4458:
4451:
4425:
4424:
4415:
4381:
4380:
4373:
4366:
4351:
4350:
4346:
4341:
4294:
4226:
4225:
4188:
4187:
4158:
4148:
4092:
4091:
4067:
4062:
4061:
4040:
4035:
4034:
4015:
4014:
3995:
3994:
3975:
3974:
3949:
3948:
3942:
3878:
3877:
3810:
3809:
3780:
3765:
3726:
3725:
3716:
3664:
3663:
3635:
3630:
3629:
3600:
3589:
3588:
3554:
3553:
3551:
3510:
3495:
3494:
3479:
3478:
3443:
3427:
3412:
3406:
3374:
3359:
3341:
3331:
3319:
3304:
3275:
3262:
3249:
3244:
3243:
3214:
3198:
3175:
3170:
3169:
3133:
3123:
3119:
3084:
3078:
3022:
3011:
3001:
2988:
2984:
2944:
2939:
2938:
2914:
2909:
2908:
2889:
2888:
2860:
2839:
2838:
2796:
2775:
2774:
2732:
2711:
2710:
2707:
2668:
2651:
2644:
2634:
2596:
2579:
2564:
2518:
2489:
2488:
2449:
2432:
2410:
2409:
2390:
2389:
2328:
2327:
2278:
2277:
2258:
2257:
2220:
2219:
2148:
2101:
2072:
2071:
2033:
2013:
1996:
1956:
1939:
1922:
1905:
1689:
1688:
1684:. We then have
1666:
1665:
1618:
1601:
1536:
1519:
1466:
1465:
1431:
1420:
1415:
1414:
1377:
1376:
1357:
1352:
1351:
1332:
1331:
1312:
1311:
1292:
1291:
1229:
1212:
1182:
1165:
1074:
1073:
1031:
1030:
981:
980:
961:
960:
941:
940:
915:
914:
853:
852:
791:
790:
789:i.e. such that
771:
770:
747:
746:
727:
726:
703:
702:
663:
662:
637:
636:
574:
545:
544:
509:
508:
480:
479:
427:
426:
374:
373:
271:
270:
226:
222:
185:
180:
179:
153:
152:
130:
129:
110:
109:
103:
71:
28:
23:
22:
15:
12:
11:
5:
4906:
4904:
4896:
4895:
4890:
4885:
4875:
4874:
4870:
4869:
4838:
4799:
4741:
4740:
4739:
4736:
4733:
4730:
4725:
4720:
4700:
4699:External links
4697:
4694:
4693:
4674:(2): 135â138.
4654:
4641:
4628:
4615:
4602:
4557:
4548:
4507:
4471:
4449:
4438:(4): 401â406.
4413:
4371:
4364:
4343:
4342:
4340:
4337:
4336:
4335:
4330:
4325:
4320:
4318:Chernoff bound
4315:
4310:
4305:
4300:
4293:
4290:
4267:classification
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4184:
4183:
4172:
4165:
4161:
4155:
4151:
4143:
4138:
4135:
4132:
4128:
4124:
4121:
4117:
4113:
4109:
4105:
4102:
4099:
4074:
4070:
4047:
4043:
4022:
4002:
3982:
3962:
3959:
3956:
3941:
3938:
3925:
3922:
3918:
3914:
3911:
3908:
3905:
3901:
3897:
3894:
3891:
3888:
3885:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3824:
3820:
3817:
3795:
3792:
3787:
3783:
3779:
3772:
3768:
3764:
3761:
3758:
3751:
3748:
3743:
3738:
3735:
3715:
3712:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3642:
3638:
3615:
3612:
3607:
3603:
3599:
3596:
3576:
3573:
3570:
3567:
3564:
3561:
3550:
3547:
3530:
3525:
3519:
3514:
3509:
3504:
3499:
3491:
3487:
3475:
3474:
3461:
3452:
3447:
3442:
3436:
3431:
3426:
3419:
3415:
3409:
3404:
3399:
3396:
3391:
3388:
3383:
3378:
3373:
3368:
3363:
3358:
3353:
3350:
3345:
3338:
3334:
3328:
3323:
3318:
3313:
3308:
3303:
3298:
3295:
3290:
3287:
3282:
3278:
3274:
3269:
3265:
3261:
3256:
3252:
3228:
3223:
3218:
3212:
3207:
3202:
3197:
3192:
3187:
3182:
3178:
3168:distributions
3162:
3161:
3148:
3140:
3136:
3130:
3126:
3122:
3115:
3110:
3106:
3102:
3097:
3092:
3088:
3081:
3077:
3074:
3069:
3066:
3061:
3053:
3048:
3044:
3040:
3035:
3030:
3026:
3018:
3014:
3008:
3004:
3000:
2995:
2991:
2987:
2979:
2976:
2971:
2968:
2965:
2962:
2959:
2956:
2951:
2947:
2921:
2917:
2896:
2872:
2867:
2863:
2859:
2856:
2853:
2848:
2826:
2821:
2816:
2812:
2808:
2803:
2799:
2795:
2790:
2785:
2782:
2762:
2757:
2752:
2748:
2744:
2739:
2735:
2731:
2726:
2721:
2718:
2706:
2703:
2690:
2683:
2680:
2677:
2674:
2671:
2666:
2663:
2660:
2657:
2654:
2647:
2641:
2637:
2633:
2630:
2627:
2624:
2621:
2618:
2611:
2608:
2605:
2602:
2599:
2594:
2591:
2588:
2585:
2582:
2572:
2567:
2563:
2560:
2557:
2554:
2551:
2548:
2543:
2540:
2537:
2534:
2526:
2521:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2476:
2473:
2470:
2464:
2461:
2458:
2455:
2452:
2447:
2444:
2441:
2438:
2435:
2429:
2426:
2423:
2420:
2417:
2397:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2265:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2216:
2215:
2203:
2200:
2197:
2194:
2191:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2156:
2151:
2147:
2144:
2141:
2138:
2135:
2131:
2127:
2124:
2121:
2118:
2115:
2109:
2104:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2065:
2064:
2052:
2049:
2046:
2043:
2039:
2036:
2029:
2026:
2023:
2019:
2016:
2012:
2009:
2006:
2002:
1999:
1993:
1990:
1987:
1984:
1981:
1978:
1972:
1969:
1966:
1962:
1959:
1955:
1952:
1949:
1945:
1942:
1938:
1935:
1932:
1928:
1925:
1921:
1918:
1915:
1911:
1908:
1902:
1899:
1896:
1893:
1890:
1887:
1882:
1879:
1876:
1873:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1775:
1772:
1769:
1766:
1763:
1760:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1728:
1725:
1722:
1719:
1716:
1712:
1708:
1705:
1702:
1699:
1696:
1673:
1662:
1661:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1624:
1621:
1617:
1614:
1611:
1607:
1604:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1542:
1539:
1535:
1532:
1529:
1525:
1522:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1450:
1447:
1444:
1441:
1437:
1434:
1430:
1426:
1423:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1363:
1360:
1339:
1319:
1299:
1288:
1287:
1276:
1273:
1270:
1267:
1264:
1261:
1256:
1253:
1250:
1244:
1241:
1238:
1235:
1232:
1227:
1224:
1221:
1218:
1215:
1209:
1206:
1203:
1197:
1194:
1191:
1188:
1185:
1180:
1177:
1174:
1171:
1168:
1160:
1157:
1154:
1151:
1148:
1145:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1113:
1110:
1107:
1104:
1101:
1097:
1093:
1090:
1087:
1084:
1081:
1058:
1053:
1048:
1043:
1038:
1018:
1015:
1012:
1009:
1006:
1000:
997:
991:
988:
968:
948:
928:
925:
922:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
778:
754:
734:
710:
690:
685:
680:
675:
670:
650:
647:
644:
633:
632:
620:
617:
611:
608:
605:
602:
599:
596:
593:
590:
582:
577:
573:
570:
567:
564:
561:
558:
555:
552:
525:
522:
519:
516:
496:
493:
490:
487:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
359:
358:
345:
342:
339:
336:
333:
330:
327:
324:
315:
310:
307:
303:
299:
296:
293:
290:
287:
284:
281:
278:
264:
263:
251:
247:
244:
241:
238:
235:
232:
229:
225:
221:
218:
215:
212:
209:
206:
203:
200:
197:
192:
188:
162:
137:
117:
102:
99:
70:
67:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4905:
4894:
4891:
4889:
4886:
4884:
4881:
4880:
4878:
4865:
4861:
4857:
4853:
4849:
4842:
4839:
4834:
4830:
4826:
4822:
4818:
4814:
4810:
4803:
4800:
4795:
4791:
4787:
4783:
4779:
4775:
4770:
4765:
4761:
4757:
4753:
4746:
4743:
4737:
4734:
4731:
4729:
4726:
4724:
4721:
4717:
4713:
4712:
4707:
4703:
4702:
4698:
4689:
4685:
4681:
4677:
4673:
4669:
4665:
4658:
4655:
4651:
4645:
4642:
4638:
4632:
4629:
4625:
4619:
4616:
4612:
4606:
4603:
4598:
4594:
4590:
4586:
4581:
4576:
4572:
4568:
4561:
4558:
4552:
4549:
4544:
4540:
4535:
4530:
4526:
4522:
4518:
4511:
4508:
4503:
4499:
4495:
4491:
4490:
4485:
4478:
4476:
4472:
4467:
4463:
4456:
4454:
4450:
4445:
4441:
4437:
4433:
4429:
4422:
4420:
4418:
4414:
4409:
4405:
4401:
4397:
4393:
4389:
4385:
4378:
4376:
4372:
4367:
4361:
4357:
4356:
4348:
4345:
4338:
4334:
4331:
4329:
4326:
4324:
4323:RĂ©nyi entropy
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4295:
4291:
4289:
4287:
4282:
4280:
4275:
4272:
4268:
4263:
4246:
4243:
4240:
4234:
4231:
4208:
4205:
4202:
4196:
4193:
4170:
4163:
4159:
4153:
4149:
4141:
4136:
4133:
4130:
4126:
4122:
4111:
4100:
4097:
4090:
4089:
4088:
4072:
4068:
4045:
4041:
4020:
4000:
3980:
3960:
3957:
3954:
3945:
3939:
3937:
3920:
3912:
3909:
3906:
3895:
3889:
3883:
3855:
3849:
3846:
3843:
3834:
3828:
3818:
3815:
3806:
3793:
3790:
3785:
3781:
3777:
3770:
3766:
3762:
3759:
3756:
3749:
3746:
3741:
3736:
3733:
3723:
3721:
3713:
3711:
3692:
3689:
3686:
3680:
3677:
3674:
3671:
3662:
3659:, though the
3658:
3640:
3636:
3627:
3610:
3605:
3601:
3597:
3594:
3574:
3571:
3568:
3565:
3562:
3559:
3548:
3546:
3544:
3528:
3523:
3517:
3507:
3502:
3489:
3459:
3450:
3434:
3407:
3402:
3397:
3394:
3389:
3381:
3371:
3366:
3351:
3348:
3336:
3326:
3316:
3311:
3296:
3293:
3288:
3280:
3276:
3272:
3267:
3263:
3254:
3250:
3242:
3241:
3240:
3221:
3210:
3205:
3185:
3180:
3176:
3167:
3146:
3138:
3134:
3128:
3124:
3120:
3113:
3108:
3104:
3100:
3095:
3090:
3086:
3079:
3075:
3072:
3067:
3064:
3059:
3051:
3046:
3042:
3038:
3033:
3028:
3024:
3016:
3006:
3002:
2998:
2993:
2989:
2977:
2974:
2969:
2963:
2960:
2957:
2949:
2945:
2937:
2936:
2935:
2919:
2915:
2907:and variance
2894:
2886:
2865:
2861:
2857:
2854:
2819:
2814:
2810:
2806:
2801:
2797:
2783:
2780:
2755:
2750:
2746:
2742:
2737:
2733:
2719:
2716:
2705:Gaussian case
2704:
2702:
2688:
2678:
2675:
2669:
2661:
2658:
2652:
2645:
2639:
2635:
2631:
2625:
2622:
2616:
2606:
2603:
2597:
2589:
2586:
2580:
2565:
2561:
2555:
2552:
2546:
2538:
2532:
2519:
2515:
2509:
2506:
2503:
2497:
2494:
2471:
2459:
2456:
2450:
2442:
2439:
2433:
2427:
2421:
2415:
2395:
2372:
2369:
2363:
2357:
2351:
2348:
2342:
2339:
2333:
2313:
2310:
2304:
2301:
2298:
2292:
2289:
2286:
2283:
2263:
2240:
2237:
2234:
2228:
2225:
2198:
2195:
2189:
2180:
2174:
2168:
2162:
2149:
2145:
2139:
2136:
2133:
2125:
2122:
2116:
2113:
2102:
2098:
2092:
2089:
2086:
2080:
2077:
2070:
2069:
2068:
2047:
2044:
2037:
2034:
2024:
2017:
2014:
2007:
2000:
1997:
1991:
1985:
1982:
1976:
1967:
1960:
1957:
1950:
1943:
1940:
1933:
1926:
1923:
1916:
1909:
1906:
1900:
1894:
1891:
1885:
1877:
1871:
1864:
1858:
1852:
1846:
1840:
1834:
1829:
1823:
1817:
1811:
1805:
1796:
1790:
1784:
1778:
1773:
1767:
1764:
1758:
1749:
1743:
1737:
1731:
1726:
1720:
1717:
1714:
1706:
1703:
1697:
1694:
1687:
1686:
1685:
1671:
1644:
1641:
1635:
1629:
1622:
1619:
1612:
1605:
1602:
1598:
1592:
1589:
1583:
1577:
1571:
1565:
1559:
1556:
1550:
1547:
1540:
1537:
1530:
1523:
1520:
1516:
1510:
1507:
1501:
1495:
1489:
1486:
1480:
1477:
1471:
1464:
1463:
1462:
1448:
1442:
1435:
1432:
1428:
1424:
1421:
1400:
1394:
1388:
1385:
1382:
1361:
1358:
1337:
1317:
1297:
1274:
1268:
1265:
1259:
1251:
1239:
1236:
1230:
1222:
1219:
1213:
1204:
1192:
1189:
1183:
1175:
1172:
1166:
1158:
1152:
1149:
1143:
1134:
1128:
1122:
1116:
1111:
1105:
1102:
1099:
1091:
1088:
1082:
1079:
1072:
1071:
1070:
1046:
1013:
1010:
1007:
998:
995:
989:
986:
966:
946:
926:
923:
920:
897:
894:
888:
882:
876:
873:
867:
864:
858:
835:
832:
826:
820:
814:
811:
805:
802:
796:
776:
768:
752:
732:
724:
708:
678:
648:
645:
642:
618:
615:
606:
600:
594:
588:
575:
571:
565:
562:
559:
553:
550:
543:
542:
541:
539:
520:
514:
491:
485:
465:
462:
456:
450:
447:
441:
438:
432:
412:
409:
403:
397:
394:
388:
385:
379:
371:
366:
364:
340:
334:
328:
322:
308:
305:
301:
297:
291:
288:
285:
279:
276:
269:
268:
267:
249:
242:
239:
236:
230:
227:
223:
219:
216:
213:
210:
204:
201:
198:
190:
186:
178:
177:
176:
151:
135:
115:
108:
100:
98:
96:
92:
88:
84:
80:
76:
68:
66:
64:
60:
55:
53:
49:
45:
41:
37:
33:
19:
4858:(1): 92â97.
4855:
4851:
4841:
4819:(1): 52â60.
4816:
4812:
4802:
4759:
4755:
4745:
4709:
4671:
4667:
4657:
4649:
4644:
4636:
4631:
4623:
4618:
4610:
4605:
4570:
4566:
4560:
4551:
4524:
4520:
4510:
4493:
4487:
4465:
4435:
4431:
4391:
4387:
4354:
4347:
4328:F-divergence
4283:
4276:
4264:
4185:
3946:
3943:
3940:Applications
3807:
3724:
3717:
3628:
3552:
3476:
3163:
2708:
2217:
2066:
1663:
1289:
723:sigma finite
634:
369:
367:
362:
360:
265:
148:on the same
106:
104:
79:statistician
72:
57:It is not a
56:
47:
42:between two
35:
29:
4279:polar codes
52:statistical
4877:Categories
4534:1603.09060
4527:: 120938.
4496:: 99â109.
4339:References
4013:in bucket
3549:Properties
2887:with mean
1330:such that
101:Definition
40:similarity
32:statistics
4833:0096-2244
4786:0018-9448
4769:1004.5049
4716:EMS Press
4688:0973-7138
4580:0807.3917
4408:164326977
4127:∑
3884:η
3850:η
3847:−
3829:η
3816:ρ
3794:ρ
3791:≤
3786:∗
3778:≤
3767:ρ
3760:−
3742:−
3675:−
3614:∞
3611:≤
3598:≤
3572:≤
3563:≤
3513:Σ
3498:Σ
3486:Σ
3446:Σ
3430:Σ
3418:Σ
3377:μ
3372:−
3362:μ
3349:−
3344:Σ
3322:μ
3317:−
3307:μ
3217:Σ
3201:μ
3135:σ
3125:σ
3105:σ
3087:σ
3076:
3043:σ
3025:σ
3003:μ
2999:−
2990:μ
2916:σ
2895:μ
2862:σ
2855:μ
2811:σ
2798:μ
2784:∼
2747:σ
2734:μ
2720:∼
2566:∫
2520:∫
2311:≤
2287:≤
2264:λ
2190:λ
2150:∫
2103:∫
2035:λ
1977:μ
1886:μ
1818:μ
1759:λ
1636:μ
1584:μ
1538:λ
1502:λ
1449:μ
1422:λ
1401:μ
1383:λ
1359:λ
1338:λ
1318:μ
1298:λ
1260:λ
1231:λ
1184:λ
1144:λ
987:λ
967:λ
947:λ
889:λ
827:λ
777:λ
709:λ
576:∫
309:∈
302:∑
220:
214:−
4794:14238708
4444:25047882
4292:See also
2837:, where
2038:′
2018:′
2001:′
1961:′
1944:′
1927:′
1910:′
1623:′
1606:′
1541:′
1524:′
1461:, then
1436:′
1425:′
1362:′
959:defined
536:are the
4718:, 2001
4502:0010358
4432:SankhyÄ
2934:; then
2883:is the
2487:, then
372:, with
87:Sankhya
69:History
4831:
4792:
4784:
4686:
4624:JOSA A
4597:889822
4595:
4500:
4442:
4406:
4362:
3808:where
3710:does.
3477:where
851:, and
721:be a (
701:, let
478:where
266:where
150:domain
59:metric
34:, the
4790:S2CID
4764:arXiv
4593:S2CID
4575:arXiv
4529:arXiv
4440:JSTOR
4404:S2CID
4829:ISSN
4782:ISSN
4684:ISSN
4360:ISBN
3876:and
3587:and
2709:Let
1413:and
765:are
745:and
507:and
425:and
368:For
128:and
105:For
77:, a
4860:doi
4821:doi
4774:doi
4676:doi
4585:doi
4539:doi
4525:536
4396:doi
4033:be
3441:det
3425:det
3414:det
1069:by
30:In
4879::
4856:12
4854:.
4850:.
4827:.
4817:15
4815:.
4811:.
4788:.
4780:.
4772:.
4760:57
4758:.
4754:.
4714:,
4708:,
4682:.
4672:29
4670:.
4666:.
4591:.
4583:.
4571:55
4569:.
4537:.
4523:.
4519:.
4498:MR
4494:35
4492:.
4486:.
4474:^
4464:.
4452:^
4434:.
4430:.
4416:^
4402:.
4392:46
4390:.
4386:.
4374:^
4281:.
3722::
3626:.
3545:.
3403:ln
3239:,
3073:ln
2773:,
365:.
217:ln
97:.
65:.
4866:.
4862::
4835:.
4823::
4796:.
4776::
4766::
4690:.
4678::
4599:.
4587::
4577::
4545:.
4541::
4531::
4504:.
4446:.
4436:7
4410:.
4398::
4368:.
4250:)
4247:Q
4244:,
4241:P
4238:(
4235:C
4232:B
4212:)
4209:Q
4206:,
4203:P
4200:(
4197:C
4194:B
4171:,
4164:i
4160:q
4154:i
4150:p
4142:n
4137:1
4134:=
4131:i
4123:=
4120:)
4116:q
4112:,
4108:p
4104:(
4101:C
4098:B
4073:i
4069:q
4046:i
4042:p
4021:i
4001:P
3981:n
3961:Q
3958:,
3955:P
3924:)
3921:X
3917:|
3913:1
3910:=
3907:Y
3904:(
3900:P
3896:=
3893:)
3890:X
3887:(
3862:)
3859:)
3856:X
3853:(
3844:1
3841:(
3838:)
3835:X
3832:(
3823:E
3819:=
3782:L
3771:2
3763:4
3757:1
3750:2
3747:1
3737:2
3734:1
3696:)
3693:q
3690:,
3687:p
3684:(
3681:C
3678:B
3672:1
3641:B
3637:D
3606:B
3602:D
3595:0
3575:1
3569:C
3566:B
3560:0
3529:.
3524:2
3518:2
3508:+
3503:1
3490:=
3473:,
3460:)
3451:2
3435:1
3408:(
3398:2
3395:1
3390:+
3387:)
3382:2
3367:1
3357:(
3352:1
3337:T
3333:)
3327:2
3312:1
3302:(
3297:8
3294:1
3289:=
3286:)
3281:2
3277:p
3273:,
3268:1
3264:p
3260:(
3255:B
3251:D
3227:)
3222:i
3211:,
3206:i
3196:(
3191:N
3186:=
3181:i
3177:p
3160:.
3147:)
3139:q
3129:p
3121:2
3114:2
3109:q
3101:+
3096:2
3091:p
3080:(
3068:2
3065:1
3060:+
3052:2
3047:q
3039:+
3034:2
3029:p
3017:2
3013:)
3007:q
2994:p
2986:(
2978:4
2975:1
2970:=
2967:)
2964:q
2961:,
2958:p
2955:(
2950:B
2946:D
2920:2
2871:)
2866:2
2858:,
2852:(
2847:N
2825:)
2820:2
2815:q
2807:,
2802:q
2794:(
2789:N
2781:q
2761:)
2756:2
2751:p
2743:,
2738:p
2730:(
2725:N
2717:p
2689:]
2682:)
2679:x
2676:d
2673:(
2670:Q
2665:)
2662:x
2659:d
2656:(
2653:P
2646:[
2640:Q
2636:E
2632:=
2629:)
2626:x
2623:d
2620:(
2617:Q
2610:)
2607:x
2604:d
2601:(
2598:Q
2593:)
2590:x
2587:d
2584:(
2581:P
2571:X
2562:=
2559:)
2556:x
2553:d
2550:(
2547:Q
2542:)
2539:x
2536:(
2533:p
2525:X
2516:=
2513:)
2510:Q
2507:,
2504:P
2501:(
2498:C
2495:B
2475:)
2472:x
2469:(
2463:)
2460:x
2457:d
2454:(
2451:Q
2446:)
2443:x
2440:d
2437:(
2434:P
2428:=
2425:)
2422:x
2419:(
2416:p
2396:Q
2376:)
2373:x
2370:d
2367:(
2364:Q
2361:)
2358:x
2355:(
2352:p
2349:=
2346:)
2343:x
2340:d
2337:(
2334:P
2314:1
2308:)
2305:Q
2302:,
2299:P
2296:(
2293:C
2290:B
2284:0
2244:)
2241:Q
2238:,
2235:P
2232:(
2229:C
2226:B
2214:.
2202:)
2199:x
2196:d
2193:(
2184:)
2181:x
2178:(
2175:q
2172:)
2169:x
2166:(
2163:p
2155:X
2146:=
2143:)
2140:Q
2137:,
2134:P
2130:|
2126:x
2123:d
2120:(
2117:c
2114:b
2108:X
2099:=
2096:)
2093:Q
2090:,
2087:P
2084:(
2081:C
2078:B
2063:.
2051:)
2048:x
2045:d
2042:(
2028:)
2025:x
2022:(
2015:q
2011:)
2008:x
2005:(
1998:p
1992:=
1989:)
1986:x
1983:d
1980:(
1971:)
1968:x
1965:(
1958:l
1954:)
1951:x
1948:(
1941:q
1937:)
1934:x
1931:(
1924:l
1920:)
1917:x
1914:(
1907:p
1901:=
1898:)
1895:x
1892:d
1889:(
1881:)
1878:x
1875:(
1872:l
1868:)
1865:x
1862:(
1859:q
1856:)
1853:x
1850:(
1847:l
1844:)
1841:x
1838:(
1835:p
1830:=
1827:)
1824:x
1821:(
1815:)
1812:x
1809:(
1806:l
1800:)
1797:x
1794:(
1791:q
1788:)
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1782:(
1779:p
1774:=
1771:)
1768:x
1765:d
1762:(
1753:)
1750:x
1747:(
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1741:)
1738:x
1735:(
1732:p
1727:=
1724:)
1721:Q
1718:,
1715:P
1711:|
1707:x
1704:d
1701:(
1698:c
1695:b
1672:Q
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1648:)
1645:x
1642:d
1639:(
1633:)
1630:x
1627:(
1620:l
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1610:(
1603:p
1599:=
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1593:x
1590:d
1587:(
1581:)
1578:x
1575:(
1572:l
1569:)
1566:x
1563:(
1560:p
1557:=
1554:)
1551:x
1548:d
1545:(
1534:)
1531:x
1528:(
1521:p
1517:=
1514:)
1511:x
1508:d
1505:(
1499:)
1496:x
1493:(
1490:p
1487:=
1484:)
1481:x
1478:d
1475:(
1472:P
1446:)
1443:x
1440:(
1433:l
1429:=
1398:)
1395:x
1392:(
1389:l
1386:=
1275:.
1272:)
1269:x
1266:d
1263:(
1255:)
1252:x
1249:(
1243:)
1240:x
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1234:(
1226:)
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1220:d
1217:(
1214:Q
1208:)
1205:x
1202:(
1196:)
1193:x
1190:d
1187:(
1179:)
1176:x
1173:d
1170:(
1167:P
1159:=
1156:)
1153:x
1150:d
1147:(
1138:)
1135:x
1132:(
1129:q
1126:)
1123:x
1120:(
1117:p
1112:=
1109:)
1106:Q
1103:,
1100:P
1096:|
1092:x
1089:d
1086:(
1083:c
1080:b
1057:)
1052:B
1047:,
1042:X
1037:(
1017:)
1014:Q
1011:+
1008:P
1005:(
999:2
996:1
990:=
927:q
924:,
921:p
901:)
898:x
895:d
892:(
886:)
883:x
880:(
877:q
874:=
871:)
868:x
865:d
862:(
859:Q
839:)
836:x
833:d
830:(
824:)
821:x
818:(
815:p
812:=
809:)
806:x
803:d
800:(
797:P
753:Q
733:P
689:)
684:B
679:,
674:X
669:(
649:Q
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643:P
631:.
619:x
616:d
610:)
607:x
604:(
601:q
598:)
595:x
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589:p
581:X
572:=
569:)
566:Q
563:,
560:P
557:(
554:C
551:B
524:)
521:x
518:(
515:q
495:)
492:x
489:(
486:p
466:x
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460:)
457:x
454:(
451:q
448:=
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442:x
439:d
436:(
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404:x
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398:p
395:=
392:)
389:x
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344:)
341:x
338:(
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332:)
329:x
326:(
323:P
314:X
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298:=
295:)
292:Q
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286:P
283:(
280:C
277:B
250:)
246:)
243:Q
240:,
237:P
234:(
231:C
228:B
224:(
211:=
208:)
205:Q
202:,
199:P
196:(
191:B
187:D
161:X
136:Q
116:P
20:)
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