2320:
8127:
1932:
8113:
2315:{\displaystyle {\begin{aligned}D((\partial _{i})_{p},q)\ \ &{\stackrel {\mathrm {def} }{=}}\ \ {\tfrac {\partial }{\partial \theta _{p}^{i}}}D(p,q),\\D((\partial _{i}\partial _{j})_{p},(\partial _{k})_{q})\ \ &{\stackrel {\mathrm {def} }{=}}\ \ {\tfrac {\partial }{\partial \theta _{p}^{i}}}{\tfrac {\partial }{\partial \theta _{p}^{j}}}{\tfrac {\partial }{\partial \theta _{q}^{k}}}D(p,q),\ \ \mathrm {etc.} \end{aligned}}}
8151:
8139:
4584:
4294:
2582:
3970:
4921:-divergences, referring to Jeffreys' function as "Jeffreys' measure of divergence" (today "Jeffreys divergence"), and KullbackâLeibler's asymmetric function (in each direction) as "Kullback's and Leibler's measures of discriminatory information" (today "KullbackâLeibler divergence").
4145:
4843:
The use of the term "divergence" â both what functions it refers to, and what various statistical distances are called â has varied significantly over time, but by c. 2000 had settled on the current usage within information geometry, notably in the textbook
4579:{\displaystyle D_{\alpha ,\beta }(p,q)={\frac {2}{(1-\alpha )(1-\beta )}}\int {\Big (}1-{\Big (}{\tfrac {q(x)}{p(x)}}{\Big )}^{\!\!{\frac {1-\alpha }{2}}}{\Big )}{\Big (}1-{\Big (}{\tfrac {q(x)}{p(x)}}{\Big )}^{\!\!{\frac {1-\beta }{2}}}{\Big )}p(x)dx}
2341:
2850:
4945:
The term "divergence" is in contrast to a distance (metric), since the symmetrized divergence does not satisfy the triangle inequality. For example, the term "Bregman distance" is still found, but "Bregman divergence" is now preferred.
3725:
3600:
924:
1347:
3739:
3471:
4275:
4754:
3981:
5080:(continuous with continuous first and second derivatives), since only second derivatives are required. In practice, commonly used statistical manifolds and divergences are infinitely differentiable ("smooth").
3009:, â, â). The converse is also true: every torsion-free dualistic structure on a statistical manifold is induced from some globally defined divergence function (which however need not be unique).
2620:
2346:
1937:
3329:
242:
5043:
2985:
1465:
are probability distributions or other objects under consideration, such that conditions 1, 2 are satisfied. Condition 3 is required for "divergence" as used in information geometry.
1518:, to distinguish them from metric distances, which are notated with a lowercase 'd'. When multiple divergences are in use, they are commonly distinguished with subscripts, as in
1134:
1790:
2577:{\displaystyle {\begin{aligned}D\ &:\ p\mapsto D((\partial _{i})_{p},p),\\D\ &:\ p\mapsto D((\partial _{i})_{p},(\partial _{j})_{p}),\ \ \mathrm {etc.} \end{aligned}}}
1589:
1543:
779:
286:
1830:
986:
460:
5145:
5118:
960:
361:
318:
4831:
5180:
4986:
1669:
1630:
1516:
1437:
750:
614:
588:
416:
4795:
3156:
2615:
1184:
1065:
661:
192:
2991:
1361:, divergences are not required to be symmetric, and the asymmetry is important in applications. Accordingly, one often refers asymmetrically to the divergence "of
1750:
1724:
1698:
1463:
387:
3241:
3129:-divergence and a Bregman divergence is the KullbackâLeibler divergence. The squared Euclidean divergence is a Bregman divergence (corresponding to the function
487:
5331:
Jiao, Jiantao; Courtade, Thomas; No, Albert; Venkat, Kartik; Weissman, Tsachy (December 2014). "Information
Measures: the Curious Case of the Binary Alphabet".
3353:
1231:
1204:
1157:
1026:
1006:
774:
721:
701:
681:
634:
558:
531:
507:
165:
141:
121:
3615:
3488:
4924:
The information geometry definition of divergence (the subject of this article) was initially referred to by alternative terms, including "quasi-distance"
1239:
3965:{\displaystyle D_{JS}(p,q)={\frac {1}{2}}\int p(x)\ln \left(p(x)\right)+q(x)\ln \left(q(x)\right)-(p(x)+q(x))\ln \left({\frac {p(x)+q(x)}{2}}\right)dx}
3046:
4851:
The term "divergence" for a statistical distance was used informally in various contexts from c. 1910 to c. 1940. Its formal use dates at least to
3361:
1469:
7248:
4159:
4140:{\displaystyle D^{(\alpha )}(p,q)={\frac {4}{1-\alpha ^{2}}}{\bigg (}1-\int p(x)^{\frac {1-\alpha }{2}}q(x)^{\frac {1+\alpha }{2}}dx{\bigg )}}
7753:
4648:
3121:; however, other types of divergence functions are also encountered in the literature. The only divergence for probabilities over a finite
4855:, entitled "On a measure of divergence between two statistical populations defined by their probability distributions", which defined the
7903:
7527:
6168:
5595:
Bhattacharyya, A. (1943). "On a measure of divergence between two statistical populations defined by their probability distributions".
4917:
referred generally to such a function as a "coefficient of divergence", and showed that many existing functions could be expressed as
7301:
5816:"Any statistical manifold has a contrast function â on the CÂł-functions taking the minimum at the diagonal of the product manifold"
7740:
5556:
6163:
5863:
6767:
5915:
3246:
4909:
in 1948), referring to the asymmetric function as "the mean information for discrimination ... per observation", while
4768:
of the
Bregman generator of the original divergence. For example, for the squared Euclidean distance, the generator is
28:
7550:
7442:
5808:
5748:
5526:
5689:
Ali, S. M.; Silvey, S. D. (1966). "A General Class of
Coefficients of Divergence of One Distribution from Another".
8155:
7728:
7602:
5274:
Nielsen, F.; Nock, R. (2013). "On the Chi square and higher-order Chi distances for approximating f-divergences".
7786:
7447:
7192:
6563:
6153:
4868:
3480:
3076:
1546:
70:
197:
7837:
7049:
6856:
6745:
6703:
4995:
1850:
to be a statistical manifold, meaning that it can be parametrized with a finite-dimensional coordinate system
6777:
2892:
8177:
8080:
7039:
5942:
3731:
7631:
7580:
7565:
7555:
7424:
7296:
7263:
7089:
7044:
6874:
3092:
2860:
8143:
7975:
7776:
7700:
7001:
6755:
6424:
5888:
3122:
1073:
7860:
7832:
7827:
7575:
7334:
7240:
7220:
7128:
6839:
6657:
6140:
6012:
5610:"Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear Inverse Problems"
3084:
1755:
65:(SED), and divergences can be viewed as generalizations of SED. The other most important divergence is
62:
7592:
7360:
7081:
7006:
6935:
6864:
6784:
6772:
6642:
6630:
6623:
6331:
6052:
3042:
1559:
1552:
Often a different separator between parameters is used, particularly to emphasize the asymmetry. In
8075:
7842:
7705:
7390:
7355:
7319:
7104:
6546:
6455:
6414:
6326:
6017:
5856:
4603:
1521:
51:
1394:
250:
7984:
7597:
7537:
7474:
7112:
7096:
6834:
6686:
6536:
6450:
5637:"A differential geometric approach to statistical inference on the basis of contrast functionals"
5074:
1795:
1592:
1480:
Notation for divergences varies significantly between fields, though there are some conventions.
101:
8182:
8022:
7952:
7745:
7682:
7437:
7324:
6321:
6218:
6125:
6004:
5903:
4863:, entitled "On a Measure of Divergence between Two Multinomial Populations", which defined the
4856:
4151:
3088:
965:
421:
24:
5123:
5096:
2845:{\displaystyle {\begin{aligned}&D=D=0,\\&D=D=-D\ \equiv \ g_{ij}^{(D)},\end{aligned}}}
929:
325:
291:
8047:
7989:
7932:
7758:
7651:
7560:
7286:
7170:
7029:
7021:
6911:
6903:
6718:
6614:
6592:
6551:
6516:
6483:
6429:
6404:
6359:
6298:
6258:
6060:
5883:
4807:
2879:
1404:
In general statistics and probability, "divergence" generally refers to any kind of function
20:
5566:
Bhattacharyya, A. (1946). "On a
Measure of Divergence between Two Multinomial Populations".
5150:
4956:
1639:
1598:
1486:
1407:
729:
593:
563:
395:
7970:
7545:
7494:
7470:
7432:
7350:
7329:
7281:
7160:
7138:
7107:
7016:
6893:
6844:
6762:
6735:
6691:
6647:
6409:
6185:
6065:
5795:
5779:
5054:
4884:
4773:
3134:
1841:
1358:
1162:
1043:
1031:
639:
170:
55:
43:
35:
8:
8117:
8042:
7965:
7646:
7410:
7403:
7365:
7273:
7253:
7225:
6958:
6824:
6819:
6809:
6801:
6619:
6580:
6470:
6460:
6369:
6148:
6104:
6022:
5947:
5849:
4864:
3100:
1729:
1703:
1677:
1442:
1398:
1390:
366:
3720:{\displaystyle H^{2}(p,\,q)=2\int {\Big (}{\sqrt {p(x)}}-{\sqrt {q(x)}}\,{\Big )}^{2}dx}
3181:
469:
8131:
7942:
7796:
7692:
7641:
7517:
7414:
7398:
7375:
7152:
6886:
6869:
6829:
6740:
6635:
6597:
6568:
6528:
6488:
6434:
6351:
6037:
6032:
5800:
5767:
5714:
5702:
5583:
5536:
5514:
5493:
5366:
5340:
5301:
5283:
4621:
4597:
3607:
3595:{\displaystyle D_{\mathrm {KL} }(p,q)=\int p(x)\ln \left({\frac {p(x)}{q(x)}}\right)dx}
3338:
3118:
3080:
1553:
1216:
1189:
1142:
1011:
991:
759:
706:
686:
666:
619:
543:
516:
492:
150:
126:
106:
85:
74:
5464:"Differential Geometry of Curved Exponential Families-Curvatures and Information Loss"
5449:
Adhikari, B. P.; Joshi, D. D. (1956). "Distance, discrimination et résumé exhaustif".
8126:
8037:
8007:
7999:
7819:
7810:
7735:
7666:
7522:
7507:
7482:
7370:
7311:
7177:
7165:
6791:
6708:
6652:
6575:
6419:
6341:
6120:
5994:
5804:
5706:
5575:
5552:
5522:
5485:
5358:
3104:
3021:
2883:
2864:
919:{\displaystyle D(x(p),x(p)+dx)=\textstyle {\frac {1}{2}}dx^{T}g_{p}(x)dx+O(|dx|^{3})}
534:
5543:. Applied Mathematical Sciences. Vol. 194. Springer Japan. pp. XIII, 374.
5370:
77:. There are numerous other specific divergences and classes of divergences, notably
8062:
8017:
7781:
7768:
7661:
7636:
7570:
7502:
7380:
6988:
6881:
6814:
6727:
6674:
6493:
6364:
6158:
6042:
5957:
5924:
5827:
5787:
5757:
5735:
5698:
5675:
5648:
5621:
5544:
5475:
5350:
5305:
5293:
4800:
4760:
3072:
1633:
753:
66:
1342:{\displaystyle D_{S}(p,q)=\textstyle {\frac {1}{2}}{\big (}D(p,q)+D(q,p){\big )}.}
7979:
7723:
7585:
7512:
7187:
7061:
7034:
7011:
6980:
6607:
6602:
6556:
6286:
5937:
5775:
5739:
4906:
1636:; this notation is common for the KL divergence. A colon may be used instead, as
47:
7469:
1233:, its symmetrized version is obtained by averaging it with its dual divergence:
7928:
7923:
6386:
6316:
5962:
4759:
The dual divergence to a
Bregman divergence is the divergence generated by the
463:
5832:
5815:
5762:
5743:
5680:
5663:
5653:
5636:
5548:
1632:, and emphasizes interpreting the divergence as a relative measurement, as in
8171:
8085:
8052:
7915:
7876:
7687:
7656:
7120:
7074:
6679:
6381:
6208:
5972:
5967:
5710:
5626:
5609:
5579:
5489:
5480:
5463:
5362:
5354:
5297:
3466:{\displaystyle D_{f}(p,q)=\int p(x)f{\bigg (}{\frac {q(x)}{p(x)}}{\bigg )}dx}
3096:
1671:; this emphasizes the relative information supporting the two distributions.
1034:
of condition 3 shows that divergence has the dimension of squared distance.
8027:
7960:
7937:
7852:
7182:
6478:
6376:
6311:
6253:
6238:
6175:
6130:
4602:
Bregman divergences correspond to convex functions on convex sets. Given a
3173:
3111:
3017:
78:
16:
Function that measures dissimilarity between two probability distributions
8070:
8032:
7715:
7616:
7478:
7291:
7258:
6750:
6667:
6662:
6306:
6263:
6243:
6223:
6213:
5982:
4270:{\displaystyle D_{\chi ^{2}}(p,q)=\int {\frac {(p(x)-q(x))^{2}}{p(x)}}dx}
3005:(·, ·) generates on a statistical manifold a unique dualistic structure (
5587:
1700:
interprets the parameters as probability distributions, while lowercase
1472:, a commonly used statistical divergence, does not satisfy condition 3.
6916:
6396:
6096:
6027:
5977:
5952:
5872:
5771:
5718:
5497:
4613:
7069:
6921:
6541:
6336:
6248:
6233:
6228:
6193:
5203:
5201:
5199:
4749:{\displaystyle D_{F}(p,q)=F(p)-F(q)-\langle \nabla F(q),p-q\rangle .}
4883:
for statistically distances. Numerous references to earlier uses of
4624:
measures the convexity of: the error of the linear approximation of
6585:
6203:
6080:
6075:
6070:
5691:
Journal of the Royal
Statistical Society. Series B (Methodological)
5218:
5216:
5345:
5288:
5196:
4913:
referred to the asymmetric function as the "directed divergence".
8090:
7791:
5213:
8012:
6993:
6967:
6947:
6198:
5989:
4905:
divergence (this function had already been defined and used by
1483:
Divergences are generally notated with an uppercase 'D', as in
5841:
5509:. Lecture Notes in Statistics. Vol. 28. Springer-Verlag.
5389:
3087:(SED). Minimizing these two divergences is the main way that
1846:
Many properties of divergences can be derived if we restrict
4942:-divergence, and has become standard for the general class.
5932:
1389:
distance, not linear distance, and thus do not satisfy the
2994:
to this connection â* is generated by the dual divergence
1591:; this is similar to, but distinct from, the notation for
1352:
5413:
1752:
interprets them geometrically as points in a space, and
5377:
5093:, p. 80), where the KL divergence between measure
4799:, while for the relative entropy the generator is the
4487:
4395:
3110:
The two most important classes of divergences are the
2239:
2210:
2181:
2019:
1674:
The notation for parameters varies as well. Uppercase
1271:
831:
5568:
SankhyÄ: The Indian
Journal of Statistics (1933-1960)
5425:
5401:
5235:
5233:
5231:
5153:
5126:
5099:
4998:
4959:
4810:
4776:
4651:
4297:
4162:
3984:
3742:
3618:
3491:
3364:
3341:
3249:
3184:
3137:
2895:
2618:
2344:
1935:
1798:
1758:
1732:
1706:
1680:
1642:
1601:
1562:
1524:
1489:
1445:
1410:
1242:
1219:
1192:
1165:
1145:
1076:
1046:
1014:
994:
968:
932:
782:
762:
732:
709:
689:
669:
642:
622:
596:
566:
546:
519:
495:
472:
424:
398:
369:
328:
294:
253:
200:
173:
153:
129:
109:
7754:
Autoregressive conditional heteroskedasticity (ARCH)
5330:
7216:
5312:
5228:
5174:
5139:
5112:
5037:
4980:
4825:
4789:
4748:
4578:
4269:
4139:
3964:
3719:
3594:
3465:
3347:
3324:{\displaystyle f(0)=\lim _{t\to 0^{+}}f(t),f(1)=0}
3323:
3235:
3150:
2979:
2844:
2576:
2314:
1824:
1784:
1744:
1718:
1692:
1663:
1624:
1583:
1537:
1510:
1457:
1431:
1341:
1225:
1198:
1178:
1151:
1128:
1059:
1020:
1000:
980:
954:
918:
768:
744:
715:
695:
675:
655:
628:
608:
582:
552:
525:
501:
481:
454:
410:
381:
355:
312:
280:
236:
186:
159:
135:
115:
4553:
4530:
4529:
4523:
4481:
4468:
4461:
4438:
4437:
4431:
4389:
4376:
4132:
4046:
3700:
3656:
3452:
3413:
8169:
3266:
7302:Multivariate adaptive regression splines (MARS)
5734:
5395:
5090:
4992:denote their functions with a lowercase 'd' as
4950:
4898:
4872:
5507:Differential-Geometrical Methods in Statistics
4879:. The term "divergence" was used generally by
4867:. The term was popularized by its use for the
3020:for some function Æ(·), then it generates the
2325:Now we restrict these functions to a diagonal
560:defines an inner product on the tangent space
535:parametric family of probability distributions
5857:
5594:
5565:
5448:
4888:
4860:
4852:
1330:
1284:
5513:
5207:
4845:
4740:
4710:
513:In applications to statistics, the manifold
5730:(Second ed.). Oxford University Press.
5273:
4901:actually used "divergence" to refer to the
3071:The two most important divergences are the
5902:
5864:
5850:
533:is typically the space of parameters of a
237:{\displaystyle D:M\times M\to [0,\infty )}
50:which establishes the separation from one
6515:
5831:
5761:
5688:
5679:
5652:
5625:
5541:Information Geometry and Its Applications
5479:
5419:
5344:
5287:
5038:{\displaystyle d\left(P_{1},P_{2}\right)}
4989:
4914:
4880:
3696:
3638:
1835:
5813:
5786:
5725:
5431:
5407:
5383:
5262:
4910:
4892:
4876:
2980:{\displaystyle \Gamma _{ij,k}^{(D)}=-D,}
5607:
5333:IEEE Transactions on Information Theory
5318:
4928:, p. 369) and "contrast function"
4606:, continuously differentiable function
988:. It is the Riemannian metric at point
8170:
7828:KaplanâMeier estimator (product limit)
5661:
5634:
5251:
4929:
4591:
3079:, KL divergence), which is central to
1353:Difference from other similar concepts
7901:
7468:
7215:
6514:
6284:
5901:
5845:
5535:
5504:
5461:
5239:
5222:
4953:denoted their asymmetric function as
4933:
4925:
8138:
7838:Accelerated failure time (AFT) model
4636:as an approximation of the value at
1914:, denote the partial derivatives of
1397:) do satisfy generalizations of the
1393:, but some divergences (such as the
1385:". Secondly, divergences generalize
8150:
7433:Analysis of variance (ANOVA, anova)
6285:
5324:
683:, this defines a Riemannian metric
13:
7528:CochranâMantelâHaenszel statistics
6154:Pearson product-moment correlation
5703:10.1111/j.2517-6161.1966.tb00626.x
4932:, though "divergence" was used in
4895:, pp. 6â7, §1.3 Divergence).
4713:
3501:
3498:
3227:
3218:
3203:
2962:
2949:
2939:
2897:
2790:
2777:
2752:
2742:
2708:
2698:
2665:
2631:
2563:
2560:
2557:
2525:
2499:
2458:
2445:
2403:
2356:
2301:
2298:
2295:
2245:
2241:
2216:
2212:
2187:
2183:
2166:
2163:
2160:
2118:
2092:
2082:
2025:
2021:
2004:
2001:
1998:
1950:
1129:{\displaystyle D^{*}(p,q)=D(q,p).}
228:
14:
8194:
5792:Information Theory and Statistics
5749:Annals of Mathematical Statistics
5608:Csiszar, Imre (1 December 1991).
1785:{\displaystyle \mu _{1},\mu _{2}}
1556:, a double bar is commonly used:
8149:
8137:
8125:
8112:
8111:
7902:
5744:"On information and sufficiency"
3167:
466:for infinitesimal displacements
7787:Least-squares spectral analysis
5519:Methods of information geometry
5441:
1584:{\displaystyle D(p\parallel q)}
6768:Mean-unbiased minimum-variance
5871:
5820:Hiroshima Mathematical Journal
5668:Hiroshima Mathematical Journal
5664:"Geometry of minimum contrast"
5641:Hiroshima Mathematical Journal
5276:IEEE Signal Processing Letters
5267:
5256:
5245:
5169:
5157:
5083:
5067:
4975:
4963:
4725:
4719:
4704:
4698:
4689:
4683:
4674:
4662:
4567:
4561:
4513:
4507:
4499:
4493:
4421:
4415:
4407:
4401:
4365:
4353:
4350:
4338:
4326:
4314:
4255:
4249:
4235:
4231:
4225:
4216:
4210:
4204:
4192:
4180:
4102:
4095:
4070:
4063:
4013:
4001:
3996:
3990:
3943:
3937:
3928:
3922:
3903:
3900:
3894:
3885:
3879:
3873:
3862:
3856:
3839:
3833:
3819:
3813:
3796:
3790:
3768:
3756:
3691:
3685:
3672:
3666:
3642:
3629:
3576:
3570:
3562:
3556:
3537:
3531:
3519:
3507:
3444:
3438:
3430:
3424:
3405:
3399:
3387:
3375:
3312:
3306:
3297:
3291:
3273:
3259:
3253:
3230:
3212:
3209:
3206:
3191:
2971:
2935:
2921:
2915:
2830:
2824:
2799:
2773:
2761:
2732:
2723:
2694:
2674:
2655:
2646:
2627:
2544:
2535:
2521:
2509:
2495:
2492:
2486:
2467:
2441:
2428:
2413:
2399:
2396:
2390:
2371:
2352:
2282:
2270:
2137:
2128:
2114:
2102:
2078:
2075:
2062:
2050:
1975:
1960:
1946:
1943:
1658:
1646:
1619:
1612:
1605:
1578:
1566:
1505:
1493:
1426:
1414:
1325:
1313:
1304:
1292:
1265:
1253:
1120:
1108:
1099:
1087:
949:
943:
912:
902:
890:
886:
871:
865:
825:
813:
807:
798:
792:
786:
449:
428:
344:
332:
269:
257:
231:
219:
216:
1:
8081:Geographic information system
7297:Simultaneous equations models
5225:, p. 10, Definition 1.1.
5189:
4951:Kullback & Leibler (1951)
4899:Kullback & Leibler (1951)
4873:Kullback & Leibler (1951)
2878:(·, ·) also defines a unique
1854:, so that for a distribution
1832:interprets them as measures.
1538:{\displaystyle D_{\text{KL}}}
95:
89:
7264:Coefficient of determination
6875:Uniformly most powerful test
5803:in 1968; reprinted in 1978:
5451:Pub. Inst. Stat. Univ. Paris
5091:Kullback & Leibler (1951
5073:Throughout, we only require
4875:and its use in the textbook
3093:principle of maximum entropy
2587:By definition, the function
281:{\displaystyle D(p,q)\geq 0}
7:
7833:Proportional hazards models
7777:Spectral density estimation
7759:Vector autoregression (VAR)
7193:Maximum posterior estimator
6425:Randomized controlled trial
5521:. Oxford University Press.
5517:; Nagaoka, Hiroshi (2000).
5396:Kullback & Leibler 1951
5048:
4889:Adhikari & Joshi (1956)
4869:KullbackâLeibler divergence
3481:KullbackâLeibler divergence
3077:KullbackâLeibler divergence
3066:
1825:{\displaystyle m_{1},m_{2}}
1547:KullbackâLeibler divergence
1475:
752:, we may construct a local
71:KullbackâLeibler divergence
61:The simplest divergence is
10:
8199:
7593:Multivariate distributions
6013:Average absolute deviation
4846:Amari & Nagaoka (2000)
4838:
4595:
3171:
3085:squared Euclidean distance
1839:
63:squared Euclidean distance
18:
8107:
8061:
7998:
7951:
7914:
7910:
7897:
7869:
7851:
7818:
7809:
7767:
7714:
7675:
7624:
7615:
7581:Structural equation model
7536:
7493:
7489:
7464:
7423:
7389:
7343:
7310:
7272:
7239:
7235:
7211:
7151:
7060:
6979:
6943:
6934:
6917:Score/Lagrange multiplier
6902:
6855:
6800:
6726:
6717:
6527:
6523:
6510:
6469:
6443:
6395:
6350:
6332:Sample size determination
6297:
6293:
6280:
6184:
6139:
6113:
6095:
6051:
6003:
5923:
5914:
5910:
5897:
5879:
5726:Jeffreys, Harold (1948).
5549:10.1007/978-4-431-55978-8
5505:Amari, Shun-Ichi (1985).
5462:Amari, Shun-Ichi (1982).
3732:JensenâShannon divergence
3335:-divergence generated by
3043:Fisher information metric
1139:When we wish to contrast
1008:expressed in coordinates
981:{\displaystyle n\times n}
776:, then the divergence is
455:{\displaystyle D(p,p+dp)}
8076:Environmental statistics
7598:Elliptical distributions
7391:Generalized linear model
7320:Simple linear regression
7090:HodgesâLehmann estimator
6547:Probability distribution
6456:Stochastic approximation
6018:Coefficient of variation
5814:Matumoto, Takao (1993).
5614:The Annals of Statistics
5597:Bull. Calcutta Math. Soc
5468:The Annals of Statistics
5355:10.1109/TIT.2014.2360184
5298:10.1109/LSP.2013.2288355
5208:Amari & Nagaoka 2000
5140:{\displaystyle \mu _{2}}
5113:{\displaystyle \mu _{1}}
5060:
3178:Given a convex function
3083:and statistics, and the
1470:total variation distance
1377:", rather than "between
955:{\displaystyle g_{p}(x)}
356:{\displaystyle D(p,q)=0}
313:{\displaystyle p,q\in M}
52:probability distribution
29:Discrepancy (statistics)
19:Not to be confused with
7736:Cross-correlation (XCF)
7344:Non-standard predictors
6778:LehmannâScheffĂ© theorem
6451:Adaptive clinical trial
5833:10.32917/hmj/1206128255
5763:10.1214/aoms/1177729694
5681:10.32917/hmj/1206128508
5662:Eguchi, Shinto (1992).
5654:10.32917/hmj/1206130775
5635:Eguchi, Shinto (1985).
5075:differentiability class
4990:Ali & Silvey (1966)
4915:Ali & Silvey (1966)
4881:Ali & Silvey (1966)
4826:{\displaystyle x\log x}
3089:linear inverse problems
1593:conditional probability
540:Condition 3 means that
462:is a positive-definite
102:differentiable manifold
73:), which is central to
8132:Mathematics portal
7953:Engineering statistics
7861:NelsonâAalen estimator
7438:Analysis of covariance
7325:Ordinary least squares
7249:Pearson product-moment
6653:Statistical functional
6564:Empirical distribution
6397:Controlled experiments
6126:Frequency distribution
5904:Descriptive statistics
5627:10.1214/aos/1176348385
5481:10.1214/aos/1176345779
5176:
5175:{\displaystyle I(1:2)}
5141:
5114:
5039:
4982:
4981:{\displaystyle I(1:2)}
4857:Bhattacharyya distance
4827:
4791:
4750:
4580:
4289:)-product divergence:
4271:
4152:chi-squared divergence
4141:
3966:
3721:
3596:
3467:
3349:
3325:
3237:
3152:
2981:
2861:positive semi-definite
2846:
2578:
2316:
1836:Geometrical properties
1826:
1786:
1746:
1720:
1694:
1665:
1664:{\displaystyle D(p:q)}
1626:
1625:{\displaystyle P(A|B)}
1585:
1539:
1512:
1511:{\displaystyle D(x,y)}
1459:
1433:
1432:{\displaystyle D(p,q)}
1343:
1227:
1200:
1180:
1153:
1130:
1061:
1022:
1002:
982:
956:
920:
770:
746:
745:{\displaystyle p\in M}
717:
697:
677:
657:
630:
610:
609:{\displaystyle p\in M}
584:
583:{\displaystyle T_{p}M}
554:
527:
503:
483:
456:
412:
411:{\displaystyle p\in M}
383:
357:
314:
282:
238:
188:
161:
137:
117:
25:Deviation (statistics)
8048:Population statistics
7990:System identification
7724:Autocorrelation (ACF)
7652:Exponential smoothing
7566:Discriminant analysis
7561:Canonical correlation
7425:Partition of variance
7287:Regression validation
7131:(JonckheereâTerpstra)
7030:Likelihood-ratio test
6719:Frequentist inference
6631:Locationâscale family
6552:Sampling distribution
6517:Statistical inference
6484:Cross-sectional study
6471:Observational studies
6430:Randomized experiment
6259:Stem-and-leaf display
6061:Central limit theorem
5796:John Wiley & Sons
5728:Theory of Probability
5420:Ali & Silvey 1966
5177:
5142:
5115:
5040:
4983:
4885:statistical distances
4828:
4792:
4790:{\displaystyle x^{2}}
4751:
4581:
4272:
4142:
3967:
3722:
3597:
3468:
3350:
3326:
3238:
3153:
3151:{\displaystyle x^{2}}
3061:= 3 + 2ÆâČâČâČ(1)/ÆâČâČ(1)
2982:
2863:and defines a unique
2847:
2579:
2317:
1882:For a pair of points
1840:Further information:
1827:
1787:
1747:
1721:
1695:
1666:
1627:
1586:
1540:
1513:
1460:
1434:
1344:
1228:
1213:Given any divergence
1201:
1181:
1179:{\displaystyle D^{*}}
1154:
1131:
1062:
1060:{\displaystyle D^{*}}
1023:
1003:
983:
957:
921:
771:
747:
718:
698:
678:
658:
656:{\displaystyle C^{2}}
631:
611:
585:
555:
528:
504:
484:
457:
413:
384:
358:
315:
283:
239:
189:
187:{\displaystyle C^{2}}
162:
138:
118:
21:Deviance (statistics)
8178:Statistical distance
7971:Probabilistic design
7556:Principal components
7399:Exponential families
7351:Nonlinear regression
7330:General linear model
7292:Mixed effects models
7282:Errors and residuals
7259:Confounding variable
7161:Bayesian probability
7139:Van der Waerden test
7129:Ordered alternative
6894:Multiple comparisons
6773:RaoâBlackwellization
6736:Estimating equations
6692:Statistical distance
6410:Factorial experiment
5943:Arithmetic-Geometric
5151:
5124:
5097:
5055:Statistical distance
4996:
4957:
4861:Bhattacharyya (1946)
4853:Bhattacharyya (1943)
4808:
4774:
4649:
4295:
4160:
3982:
3740:
3616:
3489:
3362:
3339:
3247:
3182:
3135:
3091:are solved, via the
2893:
2886:â with coefficients
2616:
2342:
1933:
1842:Information geometry
1796:
1756:
1730:
1704:
1678:
1640:
1599:
1560:
1522:
1487:
1443:
1408:
1240:
1217:
1190:
1163:
1143:
1074:
1044:
1032:Dimensional analysis
1012:
992:
966:
962:is a matrix of size
930:
780:
760:
730:
707:
687:
667:
640:
620:
594:
564:
544:
517:
493:
470:
422:
396:
367:
326:
292:
251:
198:
171:
151:
127:
107:
56:statistical manifold
44:statistical distance
36:information geometry
8043:Official statistics
7966:Methods engineering
7647:Seasonal adjustment
7415:Poisson regressions
7335:Bayesian regression
7274:Regression analysis
7254:Partial correlation
7226:Regression analysis
6825:Prediction interval
6820:Likelihood interval
6810:Confidence interval
6802:Interval estimation
6763:Unbiased estimators
6581:Model specification
6461:Up-and-down designs
6149:Partial correlation
6105:Index of dispersion
6023:Interquartile range
5089:A colon is used in
4865:Bhattacharyya angle
4592:Bregman divergences
3119:Bregman divergences
3101:logistic regression
3033:and the connection
3001:Thus, a divergence
2925:
2834:
2262:
2233:
2204:
2042:
1745:{\displaystyle x,y}
1719:{\displaystyle p,q}
1693:{\displaystyle P,Q}
1468:As an example, the
1458:{\displaystyle p,q}
1399:Pythagorean theorem
1391:triangle inequality
382:{\displaystyle p=q}
86:Bregman divergences
8063:Spatial statistics
7943:Medical statistics
7843:First hitting time
7797:Whittle likelihood
7448:Degrees of freedom
7443:Multivariate ANOVA
7376:Heteroscedasticity
7188:Bayesian estimator
7153:Bayesian inference
7002:KolmogorovâSmirnov
6887:Randomization test
6857:Testing hypotheses
6830:Tolerance interval
6741:Maximum likelihood
6636:Exponential family
6569:Density estimation
6529:Statistical theory
6489:Natural experiment
6435:Scientific control
6352:Survey methodology
6038:Standard deviation
5801:Dover Publications
5172:
5137:
5110:
5035:
4978:
4823:
4787:
4746:
4622:Bregman divergence
4598:Bregman divergence
4576:
4518:
4426:
4267:
4137:
3962:
3717:
3608:Hellinger distance
3592:
3463:
3345:
3321:
3287:
3236:{\displaystyle f:}
3233:
3148:
3081:information theory
3012:For example, when
2977:
2896:
2842:
2840:
2811:
2599:) is minimized at
2574:
2572:
2312:
2310:
2265:
2248:
2236:
2219:
2207:
2190:
2045:
2028:
1822:
1782:
1742:
1716:
1690:
1661:
1622:
1581:
1554:information theory
1535:
1508:
1455:
1429:
1395:Bregman divergence
1339:
1338:
1223:
1196:
1176:
1149:
1126:
1057:
1018:
998:
978:
952:
916:
915:
766:
742:
713:
693:
673:
653:
626:
606:
580:
550:
523:
499:
482:{\displaystyle dp}
479:
452:
408:
379:
353:
310:
278:
234:
184:
157:
133:
113:
75:information theory
8165:
8164:
8103:
8102:
8099:
8098:
8038:National accounts
8008:Actuarial science
8000:Social statistics
7893:
7892:
7889:
7888:
7885:
7884:
7820:Survival function
7805:
7804:
7667:Granger causality
7508:Contingency table
7483:Survival analysis
7460:
7459:
7456:
7455:
7312:Linear regression
7207:
7206:
7203:
7202:
7178:Credible interval
7147:
7146:
6930:
6929:
6746:Method of moments
6615:Parametric family
6576:Statistical model
6506:
6505:
6502:
6501:
6420:Random assignment
6342:Statistical power
6276:
6275:
6272:
6271:
6121:Contingency table
6091:
6090:
5958:Generalized/power
5799:. Republished by
5558:978-4-431-55977-1
5339:(12): 7616â7626.
4618:Bregman generator
4589:
4588:
4547:
4517:
4455:
4425:
4369:
4259:
4121:
4089:
4042:
3950:
3782:
3694:
3675:
3580:
3448:
3348:{\displaystyle f}
3265:
3105:linear regression
3041:is the canonical
2884:affine connection
2865:Riemannian metric
2810:
2804:
2555:
2552:
2482:
2472:
2386:
2376:
2293:
2290:
2264:
2235:
2206:
2179:
2176:
2171:
2145:
2142:
2044:
2017:
2014:
2009:
1983:
1980:
1896:with coordinates
1549:(KL divergence).
1532:
1280:
1226:{\displaystyle D}
1208:primal divergence
1199:{\displaystyle D}
1152:{\displaystyle D}
1021:{\displaystyle x}
1001:{\displaystyle p}
840:
769:{\displaystyle x}
756:with coordinates
716:{\displaystyle M}
696:{\displaystyle g}
676:{\displaystyle M}
629:{\displaystyle D}
553:{\displaystyle D}
526:{\displaystyle M}
502:{\displaystyle p}
320:(non-negativity),
160:{\displaystyle M}
136:{\displaystyle n}
116:{\displaystyle M}
8190:
8153:
8152:
8141:
8140:
8130:
8129:
8115:
8114:
8018:Crime statistics
7912:
7911:
7899:
7898:
7816:
7815:
7782:Fourier analysis
7769:Frequency domain
7749:
7696:
7662:Structural break
7622:
7621:
7571:Cluster analysis
7518:Log-linear model
7491:
7490:
7466:
7465:
7407:
7381:Homoscedasticity
7237:
7236:
7213:
7212:
7132:
7124:
7116:
7115:(KruskalâWallis)
7100:
7085:
7040:Cross validation
7025:
7007:AndersonâDarling
6954:
6941:
6940:
6912:Likelihood-ratio
6904:Parametric tests
6882:Permutation test
6865:1- & 2-tails
6756:Minimum distance
6728:Point estimation
6724:
6723:
6675:Optimal decision
6626:
6525:
6524:
6512:
6511:
6494:Quasi-experiment
6444:Adaptive designs
6295:
6294:
6282:
6281:
6159:Rank correlation
5921:
5920:
5912:
5911:
5899:
5898:
5866:
5859:
5852:
5843:
5842:
5837:
5835:
5798:
5783:
5765:
5731:
5722:
5685:
5683:
5658:
5656:
5631:
5629:
5604:
5591:
5562:
5537:Amari, Shun-ichi
5532:
5515:Amari, Shun-ichi
5510:
5501:
5483:
5458:
5435:
5429:
5423:
5417:
5411:
5405:
5399:
5393:
5387:
5381:
5375:
5374:
5348:
5328:
5322:
5316:
5310:
5309:
5291:
5271:
5265:
5260:
5254:
5249:
5243:
5237:
5226:
5220:
5211:
5205:
5183:
5181:
5179:
5178:
5173:
5146:
5144:
5143:
5138:
5136:
5135:
5119:
5117:
5116:
5111:
5109:
5108:
5087:
5081:
5071:
5044:
5042:
5041:
5036:
5034:
5030:
5029:
5028:
5016:
5015:
4987:
4985:
4984:
4979:
4941:
4834:
4832:
4830:
4829:
4824:
4801:negative entropy
4798:
4796:
4794:
4793:
4788:
4786:
4785:
4767:
4761:convex conjugate
4755:
4753:
4752:
4747:
4661:
4660:
4641:
4635:
4629:
4611:
4585:
4583:
4582:
4577:
4557:
4556:
4550:
4549:
4548:
4543:
4532:
4527:
4526:
4519:
4516:
4502:
4488:
4485:
4484:
4472:
4471:
4465:
4464:
4458:
4457:
4456:
4451:
4440:
4435:
4434:
4427:
4424:
4410:
4396:
4393:
4392:
4380:
4379:
4370:
4368:
4333:
4313:
4312:
4276:
4274:
4273:
4268:
4260:
4258:
4244:
4243:
4242:
4202:
4179:
4178:
4177:
4176:
4146:
4144:
4143:
4138:
4136:
4135:
4123:
4122:
4117:
4106:
4091:
4090:
4085:
4074:
4050:
4049:
4043:
4041:
4040:
4039:
4020:
4000:
3999:
3971:
3969:
3968:
3963:
3955:
3951:
3946:
3917:
3869:
3865:
3826:
3822:
3783:
3775:
3755:
3754:
3726:
3724:
3723:
3718:
3710:
3709:
3704:
3703:
3695:
3681:
3676:
3662:
3660:
3659:
3628:
3627:
3601:
3599:
3598:
3593:
3585:
3581:
3579:
3565:
3551:
3506:
3505:
3504:
3477:
3476:
3472:
3470:
3469:
3464:
3456:
3455:
3449:
3447:
3433:
3419:
3417:
3416:
3374:
3373:
3354:
3352:
3351:
3346:
3330:
3328:
3327:
3322:
3286:
3285:
3284:
3242:
3240:
3239:
3234:
3159:
3157:
3155:
3154:
3149:
3147:
3146:
3125:that is both an
3073:relative entropy
3062:
3055:
3036:
3032:
2986:
2984:
2983:
2978:
2970:
2969:
2957:
2956:
2947:
2946:
2924:
2913:
2867:on the manifold
2851:
2849:
2848:
2843:
2841:
2833:
2822:
2808:
2802:
2798:
2797:
2785:
2784:
2760:
2759:
2750:
2749:
2716:
2715:
2706:
2705:
2689:
2673:
2672:
2639:
2638:
2622:
2609:, and therefore
2608:
2583:
2581:
2580:
2575:
2573:
2569:
2553:
2550:
2543:
2542:
2533:
2532:
2517:
2516:
2507:
2506:
2480:
2470:
2466:
2465:
2453:
2452:
2421:
2420:
2411:
2410:
2384:
2374:
2364:
2363:
2334:
2321:
2319:
2318:
2313:
2311:
2307:
2291:
2288:
2266:
2263:
2261:
2256:
2240:
2237:
2234:
2232:
2227:
2211:
2208:
2205:
2203:
2198:
2182:
2177:
2174:
2173:
2172:
2170:
2169:
2157:
2152:
2143:
2140:
2136:
2135:
2126:
2125:
2110:
2109:
2100:
2099:
2090:
2089:
2046:
2043:
2041:
2036:
2020:
2015:
2012:
2011:
2010:
2008:
2007:
1995:
1990:
1981:
1978:
1968:
1967:
1958:
1957:
1895:
1878:
1863:
1831:
1829:
1828:
1823:
1821:
1820:
1808:
1807:
1791:
1789:
1788:
1783:
1781:
1780:
1768:
1767:
1751:
1749:
1748:
1743:
1725:
1723:
1722:
1717:
1699:
1697:
1696:
1691:
1670:
1668:
1667:
1662:
1634:relative entropy
1631:
1629:
1628:
1623:
1615:
1590:
1588:
1587:
1582:
1544:
1542:
1541:
1536:
1534:
1533:
1530:
1517:
1515:
1514:
1509:
1464:
1462:
1461:
1456:
1438:
1436:
1435:
1430:
1348:
1346:
1345:
1340:
1334:
1333:
1288:
1287:
1281:
1273:
1252:
1251:
1232:
1230:
1229:
1224:
1205:
1203:
1202:
1197:
1185:
1183:
1182:
1177:
1175:
1174:
1158:
1156:
1155:
1150:
1135:
1133:
1132:
1127:
1086:
1085:
1066:
1064:
1063:
1058:
1056:
1055:
1027:
1025:
1024:
1019:
1007:
1005:
1004:
999:
987:
985:
984:
979:
961:
959:
958:
953:
942:
941:
925:
923:
922:
917:
911:
910:
905:
893:
864:
863:
854:
853:
841:
833:
775:
773:
772:
767:
754:coordinate chart
751:
749:
748:
743:
722:
720:
719:
714:
702:
700:
699:
694:
682:
680:
679:
674:
662:
660:
659:
654:
652:
651:
635:
633:
632:
627:
615:
613:
612:
607:
589:
587:
586:
581:
576:
575:
559:
557:
556:
551:
532:
530:
529:
524:
508:
506:
505:
500:
488:
486:
485:
480:
461:
459:
458:
453:
417:
415:
414:
409:
388:
386:
385:
380:
362:
360:
359:
354:
319:
317:
316:
311:
287:
285:
284:
279:
243:
241:
240:
235:
193:
191:
190:
185:
183:
182:
166:
164:
163:
158:
142:
140:
139:
134:
122:
120:
119:
114:
67:relative entropy
54:to another on a
8198:
8197:
8193:
8192:
8191:
8189:
8188:
8187:
8168:
8167:
8166:
8161:
8124:
8095:
8057:
7994:
7980:quality control
7947:
7929:Clinical trials
7906:
7881:
7865:
7853:Hazard function
7847:
7801:
7763:
7747:
7710:
7706:BreuschâGodfrey
7694:
7671:
7611:
7586:Factor analysis
7532:
7513:Graphical model
7485:
7452:
7419:
7405:
7385:
7339:
7306:
7268:
7231:
7230:
7199:
7143:
7130:
7122:
7114:
7098:
7083:
7062:Rank statistics
7056:
7035:Model selection
7023:
6981:Goodness of fit
6975:
6952:
6926:
6898:
6851:
6796:
6785:Median unbiased
6713:
6624:
6557:Order statistic
6519:
6498:
6465:
6439:
6391:
6346:
6289:
6287:Data collection
6268:
6180:
6135:
6109:
6087:
6047:
5999:
5916:Continuous data
5906:
5893:
5875:
5870:
5840:
5559:
5529:
5444:
5439:
5438:
5430:
5426:
5418:
5414:
5406:
5402:
5394:
5390:
5382:
5378:
5329:
5325:
5317:
5313:
5272:
5268:
5263:Matumoto (1993)
5261:
5257:
5250:
5246:
5238:
5229:
5221:
5214:
5206:
5197:
5192:
5187:
5186:
5152:
5149:
5148:
5131:
5127:
5125:
5122:
5121:
5104:
5100:
5098:
5095:
5094:
5088:
5084:
5072:
5068:
5063:
5051:
5024:
5020:
5011:
5007:
5006:
5002:
4997:
4994:
4993:
4958:
4955:
4954:
4937:
4911:Kullback (1959)
4907:Harold Jeffreys
4877:Kullback (1959)
4841:
4809:
4806:
4805:
4803:
4781:
4777:
4775:
4772:
4771:
4769:
4763:
4656:
4652:
4650:
4647:
4646:
4637:
4631:
4625:
4616:, known as the
4607:
4604:strictly convex
4600:
4594:
4552:
4551:
4533:
4531:
4528:
4522:
4521:
4520:
4503:
4489:
4486:
4480:
4479:
4467:
4466:
4460:
4459:
4441:
4439:
4436:
4430:
4429:
4428:
4411:
4397:
4394:
4388:
4387:
4375:
4374:
4337:
4332:
4302:
4298:
4296:
4293:
4292:
4245:
4238:
4234:
4203:
4201:
4172:
4168:
4167:
4163:
4161:
4158:
4157:
4131:
4130:
4107:
4105:
4101:
4075:
4073:
4069:
4045:
4044:
4035:
4031:
4024:
4019:
3989:
3985:
3983:
3980:
3979:
3918:
3916:
3912:
3852:
3848:
3809:
3805:
3774:
3747:
3743:
3741:
3738:
3737:
3705:
3699:
3698:
3697:
3680:
3661:
3655:
3654:
3623:
3619:
3617:
3614:
3613:
3566:
3552:
3550:
3546:
3497:
3496:
3492:
3490:
3487:
3486:
3451:
3450:
3434:
3420:
3418:
3412:
3411:
3369:
3365:
3363:
3360:
3359:
3340:
3337:
3336:
3280:
3276:
3269:
3248:
3245:
3244:
3183:
3180:
3179:
3176:
3170:
3142:
3138:
3136:
3133:
3132:
3130:
3069:
3057:
3050:
3034:
3024:
2965:
2961:
2952:
2948:
2942:
2938:
2914:
2900:
2894:
2891:
2890:
2839:
2838:
2823:
2815:
2793:
2789:
2780:
2776:
2755:
2751:
2745:
2741:
2711:
2707:
2701:
2697:
2687:
2686:
2668:
2664:
2634:
2630:
2619:
2617:
2614:
2613:
2600:
2571:
2570:
2556:
2538:
2534:
2528:
2524:
2512:
2508:
2502:
2498:
2473:
2461:
2457:
2448:
2444:
2435:
2434:
2416:
2412:
2406:
2402:
2377:
2359:
2355:
2345:
2343:
2340:
2339:
2326:
2309:
2308:
2294:
2257:
2252:
2244:
2238:
2228:
2223:
2215:
2209:
2199:
2194:
2186:
2180:
2159:
2158:
2153:
2151:
2150:
2146:
2131:
2127:
2121:
2117:
2105:
2101:
2095:
2091:
2085:
2081:
2069:
2068:
2037:
2032:
2024:
2018:
1997:
1996:
1991:
1989:
1988:
1984:
1963:
1959:
1953:
1949:
1936:
1934:
1931:
1930:
1913:
1904:
1883:
1865:
1855:
1844:
1838:
1816:
1812:
1803:
1799:
1797:
1794:
1793:
1776:
1772:
1763:
1759:
1757:
1754:
1753:
1731:
1728:
1727:
1705:
1702:
1701:
1679:
1676:
1675:
1641:
1638:
1637:
1611:
1600:
1597:
1596:
1561:
1558:
1557:
1529:
1525:
1523:
1520:
1519:
1488:
1485:
1484:
1478:
1444:
1441:
1440:
1409:
1406:
1405:
1355:
1329:
1328:
1283:
1282:
1272:
1247:
1243:
1241:
1238:
1237:
1218:
1215:
1214:
1191:
1188:
1187:
1170:
1166:
1164:
1161:
1160:
1144:
1141:
1140:
1081:
1077:
1075:
1072:
1071:
1051:
1047:
1045:
1042:
1041:
1039:dual divergence
1013:
1010:
1009:
993:
990:
989:
967:
964:
963:
937:
933:
931:
928:
927:
906:
901:
900:
889:
859:
855:
849:
845:
832:
781:
778:
777:
761:
758:
757:
731:
728:
727:
708:
705:
704:
688:
685:
684:
668:
665:
664:
647:
643:
641:
638:
637:
621:
618:
617:
595:
592:
591:
571:
567:
565:
562:
561:
545:
542:
541:
518:
515:
514:
494:
491:
490:
471:
468:
467:
423:
420:
419:
397:
394:
393:
392:At every point
368:
365:
364:
363:if and only if
327:
324:
323:
293:
290:
289:
252:
249:
248:
199:
196:
195:
178:
174:
172:
169:
168:
152:
149:
148:
128:
125:
124:
108:
105:
104:
98:
90:§ Examples
48:binary function
32:
17:
12:
11:
5:
8196:
8186:
8185:
8180:
8163:
8162:
8160:
8159:
8147:
8135:
8121:
8108:
8105:
8104:
8101:
8100:
8097:
8096:
8094:
8093:
8088:
8083:
8078:
8073:
8067:
8065:
8059:
8058:
8056:
8055:
8050:
8045:
8040:
8035:
8030:
8025:
8020:
8015:
8010:
8004:
8002:
7996:
7995:
7993:
7992:
7987:
7982:
7973:
7968:
7963:
7957:
7955:
7949:
7948:
7946:
7945:
7940:
7935:
7926:
7924:Bioinformatics
7920:
7918:
7908:
7907:
7895:
7894:
7891:
7890:
7887:
7886:
7883:
7882:
7880:
7879:
7873:
7871:
7867:
7866:
7864:
7863:
7857:
7855:
7849:
7848:
7846:
7845:
7840:
7835:
7830:
7824:
7822:
7813:
7807:
7806:
7803:
7802:
7800:
7799:
7794:
7789:
7784:
7779:
7773:
7771:
7765:
7764:
7762:
7761:
7756:
7751:
7743:
7738:
7733:
7732:
7731:
7729:partial (PACF)
7720:
7718:
7712:
7711:
7709:
7708:
7703:
7698:
7690:
7685:
7679:
7677:
7676:Specific tests
7673:
7672:
7670:
7669:
7664:
7659:
7654:
7649:
7644:
7639:
7634:
7628:
7626:
7619:
7613:
7612:
7610:
7609:
7608:
7607:
7606:
7605:
7590:
7589:
7588:
7578:
7576:Classification
7573:
7568:
7563:
7558:
7553:
7548:
7542:
7540:
7534:
7533:
7531:
7530:
7525:
7523:McNemar's test
7520:
7515:
7510:
7505:
7499:
7497:
7487:
7486:
7462:
7461:
7458:
7457:
7454:
7453:
7451:
7450:
7445:
7440:
7435:
7429:
7427:
7421:
7420:
7418:
7417:
7401:
7395:
7393:
7387:
7386:
7384:
7383:
7378:
7373:
7368:
7363:
7361:Semiparametric
7358:
7353:
7347:
7345:
7341:
7340:
7338:
7337:
7332:
7327:
7322:
7316:
7314:
7308:
7307:
7305:
7304:
7299:
7294:
7289:
7284:
7278:
7276:
7270:
7269:
7267:
7266:
7261:
7256:
7251:
7245:
7243:
7233:
7232:
7229:
7228:
7223:
7217:
7209:
7208:
7205:
7204:
7201:
7200:
7198:
7197:
7196:
7195:
7185:
7180:
7175:
7174:
7173:
7168:
7157:
7155:
7149:
7148:
7145:
7144:
7142:
7141:
7136:
7135:
7134:
7126:
7118:
7102:
7099:(MannâWhitney)
7094:
7093:
7092:
7079:
7078:
7077:
7066:
7064:
7058:
7057:
7055:
7054:
7053:
7052:
7047:
7042:
7032:
7027:
7024:(ShapiroâWilk)
7019:
7014:
7009:
7004:
6999:
6991:
6985:
6983:
6977:
6976:
6974:
6973:
6965:
6956:
6944:
6938:
6936:Specific tests
6932:
6931:
6928:
6927:
6925:
6924:
6919:
6914:
6908:
6906:
6900:
6899:
6897:
6896:
6891:
6890:
6889:
6879:
6878:
6877:
6867:
6861:
6859:
6853:
6852:
6850:
6849:
6848:
6847:
6842:
6832:
6827:
6822:
6817:
6812:
6806:
6804:
6798:
6797:
6795:
6794:
6789:
6788:
6787:
6782:
6781:
6780:
6775:
6760:
6759:
6758:
6753:
6748:
6743:
6732:
6730:
6721:
6715:
6714:
6712:
6711:
6706:
6701:
6700:
6699:
6689:
6684:
6683:
6682:
6672:
6671:
6670:
6665:
6660:
6650:
6645:
6640:
6639:
6638:
6633:
6628:
6612:
6611:
6610:
6605:
6600:
6590:
6589:
6588:
6583:
6573:
6572:
6571:
6561:
6560:
6559:
6549:
6544:
6539:
6533:
6531:
6521:
6520:
6508:
6507:
6504:
6503:
6500:
6499:
6497:
6496:
6491:
6486:
6481:
6475:
6473:
6467:
6466:
6464:
6463:
6458:
6453:
6447:
6445:
6441:
6440:
6438:
6437:
6432:
6427:
6422:
6417:
6412:
6407:
6401:
6399:
6393:
6392:
6390:
6389:
6387:Standard error
6384:
6379:
6374:
6373:
6372:
6367:
6356:
6354:
6348:
6347:
6345:
6344:
6339:
6334:
6329:
6324:
6319:
6317:Optimal design
6314:
6309:
6303:
6301:
6291:
6290:
6278:
6277:
6274:
6273:
6270:
6269:
6267:
6266:
6261:
6256:
6251:
6246:
6241:
6236:
6231:
6226:
6221:
6216:
6211:
6206:
6201:
6196:
6190:
6188:
6182:
6181:
6179:
6178:
6173:
6172:
6171:
6166:
6156:
6151:
6145:
6143:
6137:
6136:
6134:
6133:
6128:
6123:
6117:
6115:
6114:Summary tables
6111:
6110:
6108:
6107:
6101:
6099:
6093:
6092:
6089:
6088:
6086:
6085:
6084:
6083:
6078:
6073:
6063:
6057:
6055:
6049:
6048:
6046:
6045:
6040:
6035:
6030:
6025:
6020:
6015:
6009:
6007:
6001:
6000:
5998:
5997:
5992:
5987:
5986:
5985:
5980:
5975:
5970:
5965:
5960:
5955:
5950:
5948:Contraharmonic
5945:
5940:
5929:
5927:
5918:
5908:
5907:
5895:
5894:
5892:
5891:
5886:
5880:
5877:
5876:
5869:
5868:
5861:
5854:
5846:
5839:
5838:
5826:(2): 327â332.
5811:
5784:
5732:
5723:
5697:(1): 131â142.
5686:
5674:(3): 631â647.
5659:
5647:(2): 341â391.
5632:
5605:
5592:
5574:(4): 401â406.
5563:
5557:
5533:
5527:
5511:
5502:
5474:(2): 357â385.
5459:
5445:
5443:
5440:
5437:
5436:
5424:
5422:, p. 139.
5412:
5400:
5388:
5386:, p. 158.
5376:
5323:
5311:
5266:
5255:
5244:
5227:
5212:
5210:, chapter 3.2.
5194:
5193:
5191:
5188:
5185:
5184:
5171:
5168:
5165:
5162:
5159:
5156:
5147:is written as
5134:
5130:
5107:
5103:
5082:
5065:
5064:
5062:
5059:
5058:
5057:
5050:
5047:
5033:
5027:
5023:
5019:
5014:
5010:
5005:
5001:
4977:
4974:
4971:
4968:
4965:
4962:
4949:Notationally,
4893:Kullback (1959
4840:
4837:
4822:
4819:
4816:
4813:
4784:
4780:
4757:
4756:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4659:
4655:
4596:Main article:
4593:
4590:
4587:
4586:
4575:
4572:
4569:
4566:
4563:
4560:
4555:
4546:
4542:
4539:
4536:
4525:
4515:
4512:
4509:
4506:
4501:
4498:
4495:
4492:
4483:
4478:
4475:
4470:
4463:
4454:
4450:
4447:
4444:
4433:
4423:
4420:
4417:
4414:
4409:
4406:
4403:
4400:
4391:
4386:
4383:
4378:
4373:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4336:
4331:
4328:
4325:
4322:
4319:
4316:
4311:
4308:
4305:
4301:
4290:
4278:
4277:
4266:
4263:
4257:
4254:
4251:
4248:
4241:
4237:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4175:
4171:
4166:
4155:
4148:
4147:
4134:
4129:
4126:
4120:
4116:
4113:
4110:
4104:
4100:
4097:
4094:
4088:
4084:
4081:
4078:
4072:
4068:
4065:
4062:
4059:
4056:
4053:
4048:
4038:
4034:
4030:
4027:
4023:
4018:
4015:
4012:
4009:
4006:
4003:
3998:
3995:
3992:
3988:
3977:
3973:
3972:
3961:
3958:
3954:
3949:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3915:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3868:
3864:
3861:
3858:
3855:
3851:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3825:
3821:
3818:
3815:
3812:
3808:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3781:
3778:
3773:
3770:
3767:
3764:
3761:
3758:
3753:
3750:
3746:
3735:
3728:
3727:
3716:
3713:
3708:
3702:
3693:
3690:
3687:
3684:
3679:
3674:
3671:
3668:
3665:
3658:
3653:
3650:
3647:
3644:
3641:
3637:
3634:
3631:
3626:
3622:
3611:
3603:
3602:
3591:
3588:
3584:
3578:
3575:
3572:
3569:
3564:
3561:
3558:
3555:
3549:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3503:
3500:
3495:
3484:
3475:
3474:
3462:
3459:
3454:
3446:
3443:
3440:
3437:
3432:
3429:
3426:
3423:
3415:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3372:
3368:
3355:is defined as
3344:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3283:
3279:
3275:
3272:
3268:
3264:
3261:
3258:
3255:
3252:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3172:Main article:
3169:
3166:
3145:
3141:
3068:
3065:
2988:
2987:
2976:
2973:
2968:
2964:
2960:
2955:
2951:
2945:
2941:
2937:
2934:
2931:
2928:
2923:
2920:
2917:
2912:
2909:
2906:
2903:
2899:
2853:
2852:
2837:
2832:
2829:
2826:
2821:
2818:
2814:
2807:
2801:
2796:
2792:
2788:
2783:
2779:
2775:
2772:
2769:
2766:
2763:
2758:
2754:
2748:
2744:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2714:
2710:
2704:
2700:
2696:
2693:
2690:
2688:
2685:
2682:
2679:
2676:
2671:
2667:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2637:
2633:
2629:
2626:
2623:
2621:
2585:
2584:
2568:
2565:
2562:
2559:
2549:
2546:
2541:
2537:
2531:
2527:
2523:
2520:
2515:
2511:
2505:
2501:
2497:
2494:
2491:
2488:
2485:
2479:
2476:
2474:
2469:
2464:
2460:
2456:
2451:
2447:
2443:
2440:
2437:
2436:
2433:
2430:
2427:
2424:
2419:
2415:
2409:
2405:
2401:
2398:
2395:
2392:
2389:
2383:
2380:
2378:
2373:
2370:
2367:
2362:
2358:
2354:
2351:
2348:
2347:
2335:, and denote
2323:
2322:
2306:
2303:
2300:
2297:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2260:
2255:
2251:
2247:
2243:
2231:
2226:
2222:
2218:
2214:
2202:
2197:
2193:
2189:
2185:
2168:
2165:
2162:
2156:
2149:
2147:
2139:
2134:
2130:
2124:
2120:
2116:
2113:
2108:
2104:
2098:
2094:
2088:
2084:
2080:
2077:
2074:
2071:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2040:
2035:
2031:
2027:
2023:
2006:
2003:
2000:
1994:
1987:
1985:
1977:
1974:
1971:
1966:
1962:
1956:
1952:
1948:
1945:
1942:
1939:
1938:
1909:
1900:
1837:
1834:
1819:
1815:
1811:
1806:
1802:
1779:
1775:
1771:
1766:
1762:
1741:
1738:
1735:
1715:
1712:
1709:
1689:
1686:
1683:
1660:
1657:
1654:
1651:
1648:
1645:
1621:
1618:
1614:
1610:
1607:
1604:
1580:
1577:
1574:
1571:
1568:
1565:
1528:
1507:
1504:
1501:
1498:
1495:
1492:
1477:
1474:
1454:
1451:
1448:
1428:
1425:
1422:
1419:
1416:
1413:
1354:
1351:
1350:
1349:
1337:
1332:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1286:
1279:
1276:
1270:
1267:
1264:
1261:
1258:
1255:
1250:
1246:
1222:
1195:
1186:, we refer to
1173:
1169:
1148:
1137:
1136:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1084:
1080:
1067:is defined as
1054:
1050:
1017:
997:
977:
974:
971:
951:
948:
945:
940:
936:
914:
909:
904:
899:
896:
892:
888:
885:
882:
879:
876:
873:
870:
867:
862:
858:
852:
848:
844:
839:
836:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
765:
741:
738:
735:
712:
692:
672:
650:
646:
625:
605:
602:
599:
579:
574:
570:
549:
522:
511:
510:
498:
478:
475:
464:quadratic form
451:
448:
445:
442:
439:
436:
433:
430:
427:
407:
404:
401:
390:
378:
375:
372:
352:
349:
346:
343:
340:
337:
334:
331:
321:
309:
306:
303:
300:
297:
277:
274:
271:
268:
265:
262:
259:
256:
233:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
181:
177:
156:
132:
112:
97:
94:
15:
9:
6:
4:
3:
2:
8195:
8184:
8183:F-divergences
8181:
8179:
8176:
8175:
8173:
8158:
8157:
8148:
8146:
8145:
8136:
8134:
8133:
8128:
8122:
8120:
8119:
8110:
8109:
8106:
8092:
8089:
8087:
8086:Geostatistics
8084:
8082:
8079:
8077:
8074:
8072:
8069:
8068:
8066:
8064:
8060:
8054:
8053:Psychometrics
8051:
8049:
8046:
8044:
8041:
8039:
8036:
8034:
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8014:
8011:
8009:
8006:
8005:
8003:
8001:
7997:
7991:
7988:
7986:
7983:
7981:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7958:
7956:
7954:
7950:
7944:
7941:
7939:
7936:
7934:
7930:
7927:
7925:
7922:
7921:
7919:
7917:
7916:Biostatistics
7913:
7909:
7905:
7900:
7896:
7878:
7877:Log-rank test
7875:
7874:
7872:
7868:
7862:
7859:
7858:
7856:
7854:
7850:
7844:
7841:
7839:
7836:
7834:
7831:
7829:
7826:
7825:
7823:
7821:
7817:
7814:
7812:
7808:
7798:
7795:
7793:
7790:
7788:
7785:
7783:
7780:
7778:
7775:
7774:
7772:
7770:
7766:
7760:
7757:
7755:
7752:
7750:
7748:(BoxâJenkins)
7744:
7742:
7739:
7737:
7734:
7730:
7727:
7726:
7725:
7722:
7721:
7719:
7717:
7713:
7707:
7704:
7702:
7701:DurbinâWatson
7699:
7697:
7691:
7689:
7686:
7684:
7683:DickeyâFuller
7681:
7680:
7678:
7674:
7668:
7665:
7663:
7660:
7658:
7657:Cointegration
7655:
7653:
7650:
7648:
7645:
7643:
7640:
7638:
7635:
7633:
7632:Decomposition
7630:
7629:
7627:
7623:
7620:
7618:
7614:
7604:
7601:
7600:
7599:
7596:
7595:
7594:
7591:
7587:
7584:
7583:
7582:
7579:
7577:
7574:
7572:
7569:
7567:
7564:
7562:
7559:
7557:
7554:
7552:
7549:
7547:
7544:
7543:
7541:
7539:
7535:
7529:
7526:
7524:
7521:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7503:Cohen's kappa
7501:
7500:
7498:
7496:
7492:
7488:
7484:
7480:
7476:
7472:
7467:
7463:
7449:
7446:
7444:
7441:
7439:
7436:
7434:
7431:
7430:
7428:
7426:
7422:
7416:
7412:
7408:
7402:
7400:
7397:
7396:
7394:
7392:
7388:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7356:Nonparametric
7354:
7352:
7349:
7348:
7346:
7342:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7317:
7315:
7313:
7309:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7279:
7277:
7275:
7271:
7265:
7262:
7260:
7257:
7255:
7252:
7250:
7247:
7246:
7244:
7242:
7238:
7234:
7227:
7224:
7222:
7219:
7218:
7214:
7210:
7194:
7191:
7190:
7189:
7186:
7184:
7181:
7179:
7176:
7172:
7169:
7167:
7164:
7163:
7162:
7159:
7158:
7156:
7154:
7150:
7140:
7137:
7133:
7127:
7125:
7119:
7117:
7111:
7110:
7109:
7106:
7105:Nonparametric
7103:
7101:
7095:
7091:
7088:
7087:
7086:
7080:
7076:
7075:Sample median
7073:
7072:
7071:
7068:
7067:
7065:
7063:
7059:
7051:
7048:
7046:
7043:
7041:
7038:
7037:
7036:
7033:
7031:
7028:
7026:
7020:
7018:
7015:
7013:
7010:
7008:
7005:
7003:
7000:
6998:
6996:
6992:
6990:
6987:
6986:
6984:
6982:
6978:
6972:
6970:
6966:
6964:
6962:
6957:
6955:
6950:
6946:
6945:
6942:
6939:
6937:
6933:
6923:
6920:
6918:
6915:
6913:
6910:
6909:
6907:
6905:
6901:
6895:
6892:
6888:
6885:
6884:
6883:
6880:
6876:
6873:
6872:
6871:
6868:
6866:
6863:
6862:
6860:
6858:
6854:
6846:
6843:
6841:
6838:
6837:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6811:
6808:
6807:
6805:
6803:
6799:
6793:
6790:
6786:
6783:
6779:
6776:
6774:
6771:
6770:
6769:
6766:
6765:
6764:
6761:
6757:
6754:
6752:
6749:
6747:
6744:
6742:
6739:
6738:
6737:
6734:
6733:
6731:
6729:
6725:
6722:
6720:
6716:
6710:
6707:
6705:
6702:
6698:
6695:
6694:
6693:
6690:
6688:
6685:
6681:
6680:loss function
6678:
6677:
6676:
6673:
6669:
6666:
6664:
6661:
6659:
6656:
6655:
6654:
6651:
6649:
6646:
6644:
6641:
6637:
6634:
6632:
6629:
6627:
6621:
6618:
6617:
6616:
6613:
6609:
6606:
6604:
6601:
6599:
6596:
6595:
6594:
6591:
6587:
6584:
6582:
6579:
6578:
6577:
6574:
6570:
6567:
6566:
6565:
6562:
6558:
6555:
6554:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6534:
6532:
6530:
6526:
6522:
6518:
6513:
6509:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6476:
6474:
6472:
6468:
6462:
6459:
6457:
6454:
6452:
6449:
6448:
6446:
6442:
6436:
6433:
6431:
6428:
6426:
6423:
6421:
6418:
6416:
6413:
6411:
6408:
6406:
6403:
6402:
6400:
6398:
6394:
6388:
6385:
6383:
6382:Questionnaire
6380:
6378:
6375:
6371:
6368:
6366:
6363:
6362:
6361:
6358:
6357:
6355:
6353:
6349:
6343:
6340:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6318:
6315:
6313:
6310:
6308:
6305:
6304:
6302:
6300:
6296:
6292:
6288:
6283:
6279:
6265:
6262:
6260:
6257:
6255:
6252:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6220:
6217:
6215:
6212:
6210:
6209:Control chart
6207:
6205:
6202:
6200:
6197:
6195:
6192:
6191:
6189:
6187:
6183:
6177:
6174:
6170:
6167:
6165:
6162:
6161:
6160:
6157:
6155:
6152:
6150:
6147:
6146:
6144:
6142:
6138:
6132:
6129:
6127:
6124:
6122:
6119:
6118:
6116:
6112:
6106:
6103:
6102:
6100:
6098:
6094:
6082:
6079:
6077:
6074:
6072:
6069:
6068:
6067:
6064:
6062:
6059:
6058:
6056:
6054:
6050:
6044:
6041:
6039:
6036:
6034:
6031:
6029:
6026:
6024:
6021:
6019:
6016:
6014:
6011:
6010:
6008:
6006:
6002:
5996:
5993:
5991:
5988:
5984:
5981:
5979:
5976:
5974:
5971:
5969:
5966:
5964:
5961:
5959:
5956:
5954:
5951:
5949:
5946:
5944:
5941:
5939:
5936:
5935:
5934:
5931:
5930:
5928:
5926:
5922:
5919:
5917:
5913:
5909:
5905:
5900:
5896:
5890:
5887:
5885:
5882:
5881:
5878:
5874:
5867:
5862:
5860:
5855:
5853:
5848:
5847:
5844:
5834:
5829:
5825:
5821:
5817:
5812:
5810:
5809:0-8446-5625-9
5806:
5802:
5797:
5793:
5789:
5785:
5781:
5777:
5773:
5769:
5764:
5759:
5755:
5751:
5750:
5745:
5741:
5740:Leibler, R.A.
5737:
5733:
5729:
5724:
5720:
5716:
5712:
5708:
5704:
5700:
5696:
5692:
5687:
5682:
5677:
5673:
5669:
5665:
5660:
5655:
5650:
5646:
5642:
5638:
5633:
5628:
5623:
5619:
5615:
5611:
5606:
5602:
5598:
5593:
5589:
5585:
5581:
5577:
5573:
5569:
5564:
5560:
5554:
5550:
5546:
5542:
5538:
5534:
5530:
5528:0-8218-0531-2
5524:
5520:
5516:
5512:
5508:
5503:
5499:
5495:
5491:
5487:
5482:
5477:
5473:
5469:
5465:
5460:
5456:
5452:
5447:
5446:
5433:
5432:Kullback 1959
5428:
5421:
5416:
5409:
5408:Kullback 1959
5404:
5398:, p. 80.
5397:
5392:
5385:
5384:Jeffreys 1948
5380:
5372:
5368:
5364:
5360:
5356:
5352:
5347:
5342:
5338:
5334:
5327:
5320:
5315:
5307:
5303:
5299:
5295:
5290:
5285:
5281:
5277:
5270:
5264:
5259:
5253:
5252:Eguchi (1992)
5248:
5242:, p. 10.
5241:
5236:
5234:
5232:
5224:
5219:
5217:
5209:
5204:
5202:
5200:
5195:
5166:
5163:
5160:
5154:
5132:
5128:
5105:
5101:
5092:
5086:
5079:
5076:
5070:
5066:
5056:
5053:
5052:
5046:
5031:
5025:
5021:
5017:
5012:
5008:
5003:
4999:
4991:
4972:
4969:
4966:
4960:
4952:
4947:
4943:
4940:
4935:
4931:
4930:Eguchi (1985)
4927:
4922:
4920:
4916:
4912:
4908:
4904:
4900:
4896:
4894:
4890:
4887:are given in
4886:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4854:
4849:
4847:
4836:
4820:
4817:
4814:
4811:
4802:
4782:
4778:
4766:
4762:
4743:
4737:
4734:
4731:
4728:
4722:
4716:
4707:
4701:
4695:
4692:
4686:
4680:
4677:
4671:
4668:
4665:
4657:
4653:
4645:
4644:
4643:
4640:
4634:
4628:
4623:
4619:
4615:
4610:
4605:
4599:
4573:
4570:
4564:
4558:
4544:
4540:
4537:
4534:
4510:
4504:
4496:
4490:
4476:
4473:
4452:
4448:
4445:
4442:
4418:
4412:
4404:
4398:
4384:
4381:
4371:
4362:
4359:
4356:
4347:
4344:
4341:
4334:
4329:
4323:
4320:
4317:
4309:
4306:
4303:
4299:
4291:
4288:
4284:
4280:
4279:
4264:
4261:
4252:
4246:
4239:
4228:
4222:
4219:
4213:
4207:
4198:
4195:
4189:
4186:
4183:
4173:
4169:
4164:
4156:
4153:
4150:
4149:
4127:
4124:
4118:
4114:
4111:
4108:
4098:
4092:
4086:
4082:
4079:
4076:
4066:
4060:
4057:
4054:
4051:
4036:
4032:
4028:
4025:
4021:
4016:
4010:
4007:
4004:
3993:
3986:
3978:
3976:α-divergence
3975:
3974:
3959:
3956:
3952:
3947:
3940:
3934:
3931:
3925:
3919:
3913:
3909:
3906:
3897:
3891:
3888:
3882:
3876:
3870:
3866:
3859:
3853:
3849:
3845:
3842:
3836:
3830:
3827:
3823:
3816:
3810:
3806:
3802:
3799:
3793:
3787:
3784:
3779:
3776:
3771:
3765:
3762:
3759:
3751:
3748:
3744:
3736:
3733:
3730:
3729:
3714:
3711:
3706:
3688:
3682:
3677:
3669:
3663:
3651:
3648:
3645:
3639:
3635:
3632:
3624:
3620:
3612:
3609:
3605:
3604:
3589:
3586:
3582:
3573:
3567:
3559:
3553:
3547:
3543:
3540:
3534:
3528:
3525:
3522:
3516:
3513:
3510:
3493:
3485:
3482:
3479:
3478:
3460:
3457:
3441:
3435:
3427:
3421:
3408:
3402:
3396:
3393:
3390:
3384:
3381:
3378:
3370:
3366:
3358:
3357:
3356:
3342:
3334:
3318:
3315:
3309:
3303:
3300:
3294:
3288:
3281:
3277:
3270:
3262:
3256:
3250:
3224:
3221:
3215:
3200:
3197:
3194:
3188:
3185:
3175:
3168:f-divergences
3165:
3164:-divergence.
3163:
3160:) but not an
3143:
3139:
3128:
3124:
3120:
3116:
3114:
3108:
3106:
3102:
3099:, notably in
3098:
3097:least squares
3094:
3090:
3086:
3082:
3078:
3074:
3064:
3060:
3053:
3048:
3044:
3040:
3031:
3027:
3023:
3019:
3015:
3010:
3008:
3004:
2999:
2997:
2993:
2974:
2966:
2958:
2953:
2943:
2932:
2929:
2926:
2918:
2910:
2907:
2904:
2901:
2889:
2888:
2887:
2885:
2881:
2877:
2872:
2870:
2866:
2862:
2858:
2855:where matrix
2835:
2827:
2819:
2816:
2812:
2805:
2794:
2786:
2781:
2770:
2767:
2764:
2756:
2746:
2738:
2735:
2729:
2726:
2720:
2717:
2712:
2702:
2691:
2683:
2680:
2677:
2669:
2661:
2658:
2652:
2649:
2643:
2640:
2635:
2624:
2612:
2611:
2610:
2607:
2603:
2598:
2594:
2590:
2566:
2547:
2539:
2529:
2518:
2513:
2503:
2489:
2483:
2477:
2475:
2462:
2454:
2449:
2438:
2431:
2425:
2422:
2417:
2407:
2393:
2387:
2381:
2379:
2368:
2365:
2360:
2349:
2338:
2337:
2336:
2333:
2329:
2304:
2285:
2279:
2276:
2273:
2267:
2258:
2253:
2249:
2229:
2224:
2220:
2200:
2195:
2191:
2154:
2148:
2132:
2122:
2111:
2106:
2096:
2086:
2072:
2065:
2059:
2056:
2053:
2047:
2038:
2033:
2029:
1992:
1986:
1972:
1969:
1964:
1954:
1940:
1929:
1928:
1927:
1925:
1921:
1917:
1912:
1908:
1903:
1899:
1894:
1890:
1886:
1880:
1876:
1872:
1868:
1864:we can write
1862:
1858:
1853:
1849:
1843:
1833:
1817:
1813:
1809:
1804:
1800:
1777:
1773:
1769:
1764:
1760:
1739:
1736:
1733:
1713:
1710:
1707:
1687:
1684:
1681:
1672:
1655:
1652:
1649:
1643:
1635:
1616:
1608:
1602:
1594:
1575:
1572:
1569:
1563:
1555:
1550:
1548:
1526:
1502:
1499:
1496:
1490:
1481:
1473:
1471:
1466:
1452:
1449:
1446:
1423:
1420:
1417:
1411:
1402:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1335:
1322:
1319:
1316:
1310:
1307:
1301:
1298:
1295:
1289:
1277:
1274:
1268:
1262:
1259:
1256:
1248:
1244:
1236:
1235:
1234:
1220:
1211:
1209:
1193:
1171:
1167:
1146:
1123:
1117:
1114:
1111:
1105:
1102:
1096:
1093:
1090:
1082:
1078:
1070:
1069:
1068:
1052:
1048:
1040:
1035:
1033:
1029:
1015:
995:
975:
972:
969:
946:
938:
934:
907:
897:
894:
883:
880:
877:
874:
868:
860:
856:
850:
846:
842:
837:
834:
828:
822:
819:
816:
810:
804:
801:
795:
789:
783:
763:
755:
739:
736:
733:
724:
710:
690:
670:
648:
644:
623:
603:
600:
597:
577:
572:
568:
547:
538:
536:
520:
496:
476:
473:
465:
446:
443:
440:
437:
434:
431:
425:
405:
402:
399:
391:
389:(positivity),
376:
373:
370:
350:
347:
341:
338:
335:
329:
322:
307:
304:
301:
298:
295:
275:
272:
266:
263:
260:
254:
247:
246:
245:
225:
222:
213:
210:
207:
204:
201:
179:
175:
154:
146:
130:
123:of dimension
110:
103:
93:
91:
87:
83:
81:
76:
72:
69:(also called
68:
64:
59:
57:
53:
49:
45:
42:is a kind of
41:
37:
30:
26:
22:
8154:
8142:
8123:
8116:
8028:Econometrics
7978: /
7961:Chemometrics
7938:Epidemiology
7931: /
7904:Applications
7746:ARIMA model
7693:Q-statistic
7642:Stationarity
7538:Multivariate
7481: /
7477: /
7475:Multivariate
7473: /
7413: /
7409: /
7183:Bayes factor
7082:Signed rank
6994:
6968:
6960:
6948:
6696:
6643:Completeness
6479:Cohort study
6377:Opinion poll
6312:Missing data
6299:Study design
6254:Scatter plot
6176:Scatter plot
6169:Spearman's Ï
6131:Grouped data
5823:
5819:
5791:
5788:Kullback, S.
5756:(1): 79â86.
5753:
5747:
5736:Kullback, S.
5727:
5694:
5690:
5671:
5667:
5644:
5640:
5617:
5613:
5600:
5596:
5571:
5567:
5540:
5518:
5506:
5471:
5467:
5454:
5450:
5442:Bibliography
5434:, p. 6.
5427:
5415:
5410:, p. 7.
5403:
5391:
5379:
5336:
5332:
5326:
5319:Csiszar 1991
5314:
5279:
5275:
5269:
5258:
5247:
5085:
5077:
5069:
4948:
4944:
4938:
4934:Amari (1985)
4923:
4918:
4902:
4897:
4850:
4842:
4764:
4758:
4638:
4632:
4626:
4617:
4608:
4601:
4286:
4282:
3332:
3177:
3174:f-divergence
3161:
3126:
3115:-divergences
3112:
3109:
3070:
3058:
3051:
3047:α-connection
3038:
3029:
3025:
3018:f-divergence
3013:
3011:
3006:
3002:
3000:
2995:
2989:
2875:
2873:
2868:
2856:
2854:
2605:
2601:
2596:
2592:
2588:
2586:
2331:
2327:
2324:
1923:
1919:
1915:
1910:
1906:
1901:
1897:
1892:
1888:
1884:
1881:
1874:
1870:
1866:
1860:
1856:
1851:
1847:
1845:
1673:
1551:
1482:
1479:
1467:
1403:
1386:
1382:
1378:
1374:
1370:
1366:
1362:
1356:
1212:
1207:
1138:
1038:
1036:
1030:
725:
539:
512:
244:satisfying:
144:
99:
82:-divergences
79:
60:
39:
33:
8156:WikiProject
8071:Cartography
8033:Jurimetrics
7985:Reliability
7716:Time domain
7695:(LjungâBox)
7617:Time-series
7495:Categorical
7479:Time-series
7471:Categorical
7406:(Bernoulli)
7241:Correlation
7221:Correlation
7017:JarqueâBera
6989:Chi-squared
6751:M-estimator
6704:Asymptotics
6648:Sufficiency
6415:Interaction
6327:Replication
6307:Effect size
6264:Violin plot
6244:Radar chart
6224:Forest plot
6214:Correlogram
6164:Kendall's Ï
4926:Amari (1982
4903:symmetrized
3045:, â is the
2874:Divergence
1369:" or "from
726:Locally at
8172:Categories
8023:Demography
7741:ARMA model
7546:Regression
7123:(Friedman)
7084:(Wilcoxon)
7022:Normality
7012:Lilliefors
6959:Student's
6835:Resampling
6709:Robustness
6697:divergence
6687:Efficiency
6625:(monotone)
6620:Likelihood
6537:Population
6370:Stratified
6322:Population
6141:Dependence
6097:Count data
6028:Percentile
6005:Dispersion
5938:Arithmetic
5873:Statistics
5240:Amari 2016
5223:Amari 2016
5190:References
4614:convex set
3243:such that
590:for every
194:-function
145:divergence
96:Definition
40:divergence
7404:Logistic
7171:posterior
7097:Rank sum
6845:Jackknife
6840:Bootstrap
6658:Bootstrap
6593:Parameter
6542:Statistic
6337:Statistic
6249:Run chart
6234:Pie chart
6229:Histogram
6219:Fan chart
6194:Bar chart
6076:L-moments
5963:Geometric
5711:0035-9246
5603:: 99â109.
5580:0036-4452
5490:0090-5364
5363:0018-9448
5346:1404.6810
5289:1309.3029
5282:: 10â13.
5129:μ
5102:μ
4818:
4741:⟩
4735:−
4714:∇
4711:⟨
4708:−
4693:−
4541:β
4538:−
4477:−
4449:α
4446:−
4385:−
4372:∫
4363:β
4360:−
4348:α
4345:−
4310:β
4304:α
4220:−
4199:∫
4170:χ
4115:α
4083:α
4080:−
4058:∫
4055:−
4033:α
4029:−
3994:α
3910:
3871:−
3846:
3803:
3785:∫
3678:−
3652:∫
3544:
3526:∫
3394:∫
3274:→
3228:∞
3219:∞
3216:−
3210:→
3204:∞
2963:∂
2950:∂
2940:∂
2930:−
2898:Γ
2806:≡
2791:∂
2778:∂
2768:−
2753:∂
2743:∂
2736:⋅
2721:⋅
2709:∂
2699:∂
2666:∂
2659:⋅
2644:⋅
2632:∂
2526:∂
2500:∂
2487:↦
2459:∂
2446:∂
2404:∂
2391:↦
2369:⋅
2357:∂
2250:θ
2246:∂
2242:∂
2221:θ
2217:∂
2213:∂
2192:θ
2188:∂
2184:∂
2119:∂
2093:∂
2083:∂
2030:θ
2026:∂
2022:∂
1951:∂
1774:μ
1761:μ
1573:∥
1172:∗
1083:∗
1053:∗
973:×
737:∈
601:∈
403:∈
305:∈
273:≥
229:∞
217:→
211:×
8118:Category
7811:Survival
7688:Johansen
7411:Binomial
7366:Isotonic
6953:(normal)
6598:location
6405:Blocking
6360:Sampling
6239:QâQ plot
6204:Box plot
6186:Graphics
6081:Skewness
6071:Kurtosis
6043:Variance
5973:Heronian
5968:Harmonic
5790:(1959),
5742:(1951).
5588:25047882
5539:(2016).
5457:: 57â74.
5371:13108908
5049:See also
4988:, while
4936:for the
3606:squared
3123:alphabet
3067:Examples
3054:= ÆâČâČ(1)
3037:, where
2990:and the
1476:Notation
1439:, where
1159:against
616:. Since
288:for all
100:Given a
8144:Commons
8091:Kriging
7976:Process
7933:studies
7792:Wavelet
7625:General
6792:Plug-in
6586:L space
6365:Cluster
6066:Moments
5884:Outline
5780:0039968
5772:2236703
5719:2984279
5498:2240672
5306:4152365
4839:History
4833:
4804:
4797:
4770:
3158:
3131:
2880:torsion
1387:squared
1359:metrics
1357:Unlike
8013:Census
7603:Normal
7551:Manova
7371:Robust
7121:2-way
7113:1-way
6951:-test
6622:
6199:Biplot
5990:Median
5983:Lehmer
5925:Center
5807:
5778:
5770:
5717:
5709:
5586:
5578:
5555:
5525:
5496:
5488:
5369:
5361:
5304:
4859:, and
4620:, the
3331:, the
3056:, and
3022:metric
3016:is an
2882:-free
2809:
2803:
2554:
2551:
2481:
2471:
2385:
2375:
2292:
2289:
2178:
2175:
2144:
2141:
2016:
2013:
1982:
1979:
926:where
7637:Trend
7166:prior
7108:anova
6997:-test
6971:-test
6963:-test
6870:Power
6815:Pivot
6608:shape
6603:scale
6053:Shape
6033:Range
5978:Heinz
5953:Cubic
5889:Index
5768:JSTOR
5715:JSTOR
5620:(4).
5584:JSTOR
5494:JSTOR
5367:S2CID
5341:arXiv
5302:S2CID
5284:arXiv
5061:Notes
4630:from
4612:on a
3035:â = â
1926:) as
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