4538:
2976:
4387:
2284:
3343:
3723:
2971:{\displaystyle {\begin{aligned}{\underline {\mathcal {E}}}&=\inf \limits _{G_{0}\in \mathbb {P} }\int f\,dG_{n}={\frac {s}{s+n}}\inf f+\int f(X){\frac {1}{s+n}}\sum \limits _{i=1}^{n}\delta _{X_{i}}(dX)\\&={\frac {s}{s+n}}\inf f+{\frac {n}{s+n}}{\frac {\sum \limits _{i=1}^{n}f(X_{i})}{n}},\\{\overline {\mathcal {E}}}&=\sup \limits _{G_{0}\in \mathbb {P} }\int f\,dG_{n}={\frac {s}{s+n}}\sup f+\int f(X){\frac {1}{s+n}}\sum \limits _{i=1}^{n}\delta _{X_{i}}(dX)\\&={\frac {s}{s+n}}\sup f+{\frac {n}{s+n}}{\frac {\sum \limits _{i=1}^{n}f(X_{i})}{n}}.\end{aligned}}}
6540:
than X) and that, given the available data, the IDP test is indeterminate. In such a situation the frequentist test always issues a determinate response (for instance I can tell that Y is better than X), but it turns out that its response is completely random, like if we were tossing of a coin. On the other side, the IDP test acknowledges the impossibility of making a decision in these cases. Thus, by saying "I do not know", the IDP test provides a richer information to the analyst. The analyst could for instance use this information to collect more data.
4382:{\displaystyle {\begin{aligned}&{\underline {\mathcal {E}}}\left={\underline {\mathcal {E}}}})\mid X_{1},\dots ,X_{n}]\\={}&{\frac {n}{s+n}}{\frac {\sum \limits _{i=1}^{n}\mathbb {I} _{(\infty ,x]}(X_{i})}{n}}={\frac {n}{s+n}}{\hat {F}}(x),\\&{\overline {\mathcal {E}}}\left={\overline {\mathcal {E}}}\left})\mid X_{1},\dots ,X_{n}\right]\\={}&{\frac {s}{s+n}}+{\frac {n}{s+n}}{\frac {\sum \limits _{i=1}^{n}\mathbb {I} _{(\infty ,x]}(X_{i})}{n}}={\frac {s}{s+n}}+{\frac {n}{s+n}}{\hat {F}}(x).\end{aligned}}}
1963:
1381:
6539:
It has been empirically verified that when the IDP test is indeterminate, the frequentist tests are virtually behaving as random guessers. This surprising result has practical consequences in hypothesis testing. Assume that we are trying to compare the effects of two medical treatments (Y is better
5323:
5573:
3032:
in prior near-ignorance, because the IDP requires by the modeller the elicitation of a parameter. However, this is a simple elicitation problem for a nonparametric prior, since we only have to choose the value of a positive scalar (there are not infinitely many parameters left in the IDP model).
6535:
Although the IDP test shares several similarities with a standard
Bayesian approach, at the same time it embodies a significant change of paradigm when it comes to take decisions. In fact the IDP based tests have the advantage of producing an indeterminate outcome when the decision is
6513:
Dirichlet processes are frequently used in
Bayesian nonparametric statistics. The Imprecise Dirichlet Process can be employed instead of the Dirichlet processes in any application in which prior information is lacking (it is therefore important to model this state of prior ignorance).
1759:
4541:
Beta distributions for the lower (red) and upper (blue) probability corresponding to the observations {-1.17, 0.44, 1.17, 3.28, 1.44, 1.98}. The area in gives the lower (0.891) and the upper (0.9375) probability of the hypothesis "the median is greater than
498:
3234:
6723:
Benavoli, Alessio; Mangili, Francesca; Corani, Giorgio; Ruggeri, Fabrizio; Zaffalon, Marco (2014). "A Bayesian
Wilcoxon signed-rank test based on the Dirichlet process". Proceedings of the 30th International Conference on Machine Learning (ICML
1155:
5814:
6528:
The
Bayesian approach allows us to formulate the hypothesis test as a decision problem. This means that we can verify the evidence in favor of the null hypothesis and not only rejecting it and take decisions which minimize the expected
987:
3337:
4906:
5081:
5331:
6521:. Based on the Imprecise Dirichlet Process, Bayesian nonparametric near-ignorance versions of the following classical nonparametric estimators have been derived: the Wilcoxon rank sum test and the Wilcoxon signed-rank test.
6206:
3728:
2289:
6532:
Because of the nonparametric prior near-ignorance, IDP based tests allows us to start the hypothesis test with very weak prior assumptions, much in the direction of letting data speak for themselves.
3454:
can also be chosen to have some desirable frequentist properties (e.g., credible intervals to be calibrated frequentist intervals, hypothesis tests to be calibrated for the Type I error, etc.), see
5006:
5070:
6463:
1958:{\displaystyle {\underline {\mathcal {E}}}=\inf \limits _{G_{0}\in \mathbb {P} }\int f\,dG_{0}=\inf f,~~~~{\overline {\mathcal {E}}}=\sup \limits _{G_{0}\in \mathbb {P} }\int f\,dG_{0}=\sup f,}
4532:
4482:
78:
1545:
6372:
825:
642:
767:
6536:
prior-dependent. In other words, the IDP test suspends the judgment when the option which minimizes the expected loss changes depending on the
Dirichlet Process base measure we focus on.
6702:
Benavoli, Alessio; Mangili, Francesca; Ruggeri, Fabrizio; Zaffalon, Marco (2014). "Imprecise
Dirichlet Process with application to the hypothesis test on the probability that X< Y".
3650:
4761:
5618:
4616:
2033:
1751:
3060:
205:
3509:
1056:
2117:
2075:
1127:
392:
1376:{\displaystyle P\mid X_{1},\dots ,X_{n}\sim Dp\left(s+n,G_{n}\right),~~~{\text{with}}~~~~~~G_{n}={\frac {s}{s+n}}G_{0}+{\frac {1}{s+n}}\sum \limits _{i=1}^{n}\delta _{X_{i}},}
1418:
5849:
3068:
2159:
714:
321:
295:
261:
4537:
4428:
6573:
1627:
664:
523:
5955:
5888:
4688:
5630:
5981:
5914:
4939:
3609:
6235:
689:
159:
549:
350:
6012:
4571:
3686:
3006:
1683:
1605:
1572:
1445:
576:
377:
232:
105:
6503:
4790:
4645:
3715:
3541:
2276:
2240:
1716:
1656:
6261:
6483:
6281:
4708:
3452:
3432:
3412:
3392:
3369:
3026:
2207:
2187:
1986:
1147:
1076:
1010:
862:
854:
787:
604:
133:
5919:
if only one of the inequality is satisfied (which has necessarily to be the one for the upper), we are in an indeterminate situation, i.e., we cannot decide;
5318:{\displaystyle {\underline {\mathcal {P}}}=\int \limits _{0}^{0.5}\mathrm {Beta} (\theta ;s+n_{<0},n-n_{<0})d\theta =I_{1/2}(s+n_{<0},n-n_{<0}),}
3242:
5568:{\displaystyle {\overline {\mathcal {P}}}=\int \limits _{0}^{0.5}\mathrm {Beta} (\theta ;n_{<0},s+n-n_{<0})d\theta =I_{1/2}(n_{<0},s+n-n_{<0}).}
4798:
6804:
323:
is far from leading to a noninformative prior. Moreover, a-posteriori, it assigns zero probability to any set that does not include the observations.
23:(DP) is one of the most popular Bayesian nonparametric models. It was introduced by Thomas Ferguson as a prior over probability distributions.
6054:
3342:
5621:
6524:
A Bayesian nonparametric near-ignorance model presents several advantages with respect to a traditional approach to hypothesis testing.
267:
by Rubin; in fact it can be proven that the
Bayesian bootstrap is asymptotically equivalent to the frequentist bootstrap introduced by
6765:
6026:
6018:
3512:
161:). According to the Bayesian paradigm these parameters should be chosen based on the available prior information on the domain.
4431:
4947:
5011:
5987:
IDP returns an indeterminate decision when the decision is prior dependent (that is when it would depend on the choice of
6377:
4487:
4437:
32:
3414:
determines how quickly lower and upper posterior expectations converge at the increase of the number of observations,
6682:
Sethuraman, J.; Tiwari, R. C. (1981). "Convergence of
Dirichlet measures and the interpretation of their parameter".
2189:. In other words, by specifying the IDP, we are not giving any prior information on the value of the expectation of
6615:
1450:
525:
is the set of all probability measures. In other words, the IDP is the set of all
Dirichlet processes (with a fixed
6286:
792:
609:
719:
1968:
the lower (upper) bound is obtained by a probability measure that puts all the mass on the infimum (supremum) of
3614:
6596:
6580:
3346:
Lower (red) and Upper (blue) cumulative distribution for the observations {−1.17, 0.44, 1.17, 3.28, 1.44, 1.98}
6237:
encompasses the one-sided frequentist sign test as a test for the median. It can in fact be verified that for
4713:
137:
5581:
4576:
1991:
1721:
6576:
3039:
493:{\displaystyle ~~\mathrm {IDP} :~\left\{\mathrm {DP} \left(s,G_{0}\right):~~G_{0}\in \mathbb {P} \right\}}
297:
has been criticized on diverse grounds. From an a-priori point of view, the main criticism is that taking
167:
6517:
In this respect, the
Imprecise Dirichlet Process has been used for nonparametric hypothesis testing, see
6729:
6591:
6037:
3468:
1015:
3229:{\displaystyle {\underline {\mathcal {E}}}\left,\quad {\overline {\mathcal {E}}}\left\rightarrow S(f),}
2080:
2038:
1081:
6624:
6550:
1389:
326:
The imprecise Dirichlet process has been proposed to overcome these issues. The basic idea is to fix
5822:
2122:
695:
300:
274:
240:
237:
To address this issue, the only prior that has been proposed so far is the limiting DP obtained for
5809:{\displaystyle {\underline {\mathcal {P}}}>1-\gamma ,~~{\overline {\mathcal {P}}}>1-\gamma ,}
4395:
2166:
6556:
2119:). From the above expressions of the lower and upper bounds, it can be observed that the range of
1610:
647:
506:
6703:
5925:
5858:
4658:
3653:
5960:
5893:
4914:
4763:
and the property of the Dirichlet process, it can be shown that the posterior distribution of
6761:
6022:
3549:
27:
20:
6214:
992:
One of the most remarkable properties of the DP priors is that the posterior distribution of
674:
144:
6633:
528:
329:
6647:
5990:
4549:
3659:
2984:
1661:
1583:
1550:
1423:
554:
355:
210:
83:
6742:
6643:
6488:
6030:
4766:
4621:
3691:
3517:
2252:
2216:
1692:
1632:
982:{\displaystyle {\mathcal {E}}={\mathcal {E}}\left=\int f\,d{\mathcal {E}}=\int f\,dG_{0}.}
6240:
6468:
6266:
4693:
3437:
3417:
3397:
3377:
3354:
3332:{\displaystyle S(f)=\lim _{n\rightarrow \infty }{\tfrac {1}{n}}\sum _{i=1}^{n}f(X_{i})}
3011:
2192:
2172:
1971:
1132:
1061:
995:
830:
772:
589:
118:
4901:{\displaystyle F(0)\sim \mathrm {Beta} (\alpha _{0}+n_{<0},\beta _{0}+n-n_{<0})}
6798:
6575:
has a finite number of elements, it is known that the Dirichlet process reduces to a
671:
268:
264:
1574:, we can exploit the previous equations to derive prior and posterior expectations.
6672:
Efron B (1979). Bootstrap methods: Another look at the jackknife. Ann. Stat. 7 1–26
4655:
IDP can also be used for hypothesis testing, for instance to test the hypothesis
2249:
can learn from data. The posterior lower and upper bounds for the expectation of
1629:. This implies that we will get a different prior and posterior expectation of
6638:
6619:
667:
3394:, which is the only parameter left in the prior model. Since the value of
3028:(the concentration parameter). This explains the meaning of the adjective
6201:{\displaystyle F(k;n,p)=\Pr(Z\leq k)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k),}
6784:
6518:
6211:
we can show that the median test derived with th IDP for any choice of
6583:
proposed by Walley as a model for prior (near)-ignorance for chances.
5922:
if both are not satisfied, we can declare that the probability that
3434:
can be chosen so to match a certain convergence rate. The parameter
2981:
It can be observed that the posterior inferences do not depend on
6708:
4536:
3341:
6785:
Open source implementation of hypothesis tests based on the IDP
6579:. In this case, the Imprecise Dirichlet Process reduces to the
1689:
is by computing lower and upper bounds for the expectation of
4434:. Here, to obtain the lower we have exploited the fact that
1420:
is an atomic probability measure (Dirac's delta) centered at
6663:
Rubin D (1981). The Bayesian bootstrap. Ann. Stat. 9 130–134
6384:
6299:
5726:
5637:
5338:
5088:
4582:
4105:
4036:
3804:
3735:
3145:
3075:
2633:
2295:
2128:
1868:
1766:
1456:
939:
896:
868:
809:
701:
626:
5855:
if both the inequalities are satisfied we can declare that
164:
The question is: how should we choose the prior parameters
3350:
1058:
be an independent and identically distributed sample from
16:
Bayesian nonparametric model of probability distributions
5001:{\displaystyle \alpha _{0}=s\int _{-\infty }^{0}dG_{0}}
4941:
is the number of observations that are less than zero,
6789:
5065:{\displaystyle \beta _{0}=s\int _{0}^{\infty }dG_{0}.}
3278:
3008:. To define the IDP, the modeler has only to choose
207:
of the DP, in particular the infinite dimensional one
6559:
6491:
6471:
6380:
6289:
6269:
6243:
6217:
6057:
5993:
5963:
5928:
5896:
5861:
5825:
5633:
5584:
5334:
5084:
5014:
4950:
4917:
4801:
4769:
4716:
4696:
4661:
4624:
4579:
4552:
4490:
4440:
4398:
3726:
3694:
3662:
3617:
3552:
3520:
3471:
3440:
3420:
3400:
3380:
3357:
3245:
3071:
3042:
3014:
2987:
2287:
2255:
2219:
2195:
2175:
2125:
2083:
2041:
1994:
1974:
1762:
1724:
1695:
1664:
1635:
1613:
1586:
1553:
1453:
1426:
1392:
1158:
1135:
1084:
1064:
1018:
998:
865:
833:
795:
775:
722:
698:
677:
650:
612:
592:
557:
531:
509:
395:
358:
332:
303:
277:
243:
213:
170:
147:
121:
86:
35:
4647:
will be included between the lower and upper bound.
141:) is a positive real number (it is often denoted as
6519:
the Imprecise Dirichlet Process statistical package
6458:{\displaystyle {\underline {\mathcal {P}}}>0.95}
4710:is greater than zero. By considering the partition
2213:is therefore a model of prior (near)-ignorance for
6758:Statistical Reasoning with Imprecise Probabilities
6620:"Bayesian analysis of some nonparametric problems"
6567:
6497:
6477:
6457:
6366:
6275:
6255:
6229:
6200:
6006:
5975:
5949:
5908:
5882:
5843:
5808:
5612:
5567:
5317:
5064:
5000:
4933:
4900:
4784:
4755:
4702:
4682:
4639:
4610:
4565:
4526:
4476:
4422:
4381:
3709:
3680:
3644:
3603:
3535:
3503:
3446:
3426:
3406:
3386:
3363:
3331:
3228:
3054:
3020:
3000:
2970:
2270:
2234:
2201:
2181:
2153:
2111:
2069:
2027:
1980:
1957:
1745:
1710:
1677:
1650:
1621:
1599:
1566:
1539:
1439:
1412:
1375:
1141:
1121:
1070:
1050:
1004:
981:
848:
819:
781:
761:
708:
683:
658:
636:
598:
570:
543:
517:
492:
371:
344:
315:
289:
255:
226:
199:
153:
127:
99:
72:
4527:{\displaystyle \sup \mathbb {I} _{(\infty ,x]}=1}
4477:{\displaystyle \inf \mathbb {I} _{(\infty ,x]}=0}
73:{\displaystyle \mathrm {DP} \left(s,G_{0}\right)}
6085:
4491:
4441:
3461:Example: estimate of the cumulative distribution
3262:
2885:
2765:
2547:
2427:
2103:
2061:
1946:
1844:
769:. Then consider a real-valued bounded function
6697:
6695:
6693:
6509:Applications of the Imprecise Dirichlet Process
6505:and, thus, they two tests have the same power.
582:Inferences with the Imprecise Dirichlet Process
1540:{\displaystyle {\mathcal {E}}=\int f\,dG_{n}.}
263:, which has been introduced under the name of
6367:{\displaystyle 1-{\underline {\mathcal {P}}}}
5075:By exploiting this property, it follows that
820:{\displaystyle (\mathbb {X} ,{\mathcal {B}})}
637:{\displaystyle (\mathbb {X} ,{\mathcal {B}})}
578:to span the set of all probability measures.
8:
3656:, we can use IDP to derive inferences about
827:. It is well known that the expectation of
762:{\displaystyle P\sim \mathrm {DP} (s,G_{0})}
6659:
6657:
6017:By exploiting the relationship between the
1685:. A way to characterize inferences for the
352:but do not choose any precise base measure
5624:. We can thus perform the hypothesis test
3645:{\displaystyle \mathbb {I} _{(\infty ,x]}}
856:with respect to the Dirichlet process is
19:In probability theory and statistics, the
6707:
6637:
6561:
6560:
6558:
6490:
6470:
6440:
6421:
6383:
6381:
6379:
6355:
6336:
6298:
6296:
6288:
6268:
6242:
6216:
6162:
6110:
6056:
5998:
5992:
5962:
5957:is lower than the desired probability of
5927:
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5824:
5782:
5763:
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5522:
5505:
5501:
5476:
5448:
5421:
5415:
5410:
5394:
5375:
5337:
5335:
5333:
5300:
5278:
5255:
5251:
5226:
5204:
5171:
5165:
5160:
5144:
5125:
5087:
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5040:
5035:
5019:
5013:
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4922:
4916:
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4400:
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4265:
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3495:
3476:
3470:
3455:
3439:
3419:
3399:
3379:
3356:
3339:. In other words, the IDP is consistent.
3320:
3304:
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2703:
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2529:
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2409:
2400:
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2174:
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2126:
2124:
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2082:
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2012:
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1993:
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1902:
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302:
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120:
91:
85:
80:is completely defined by its parameters:
59:
36:
34:
6790:The imprecise probability group at IDSIA
6040:, where the "probability of success" is
4756:{\displaystyle (-\infty ,0],(0,\infty )}
234:, in case of lack of prior information?
6607:
551:) obtained by letting the base measure
6738:
6727:
4546:Note that, for any precise choice of
3688:The lower and upper posterior mean of
1607:can span the set of all distributions
5613:{\displaystyle I_{x}(\alpha ,\beta )}
4611:{\displaystyle {\mathcal {N}}(x;0,1)}
3511:be i.i.d. real random variables with
2028:{\displaystyle G_{0}=\delta _{X_{0}}}
1746:{\displaystyle G_{0}\in \mathbb {P} }
1129:, then the posterior distribution of
7:
6684:Defense Technical Information Center
6283:-value of the sign test is equal to
5622:regularized incomplete beta function
3055:{\displaystyle n\rightarrow \infty }
200:{\displaystyle \left(s,G_{0}\right)}
4243:
3917:
3374:The IDP is completely specified by
2916:
2808:
2578:
2470:
1333:
115:) is an arbitrary distribution and
5431:
5428:
5425:
5422:
5181:
5178:
5175:
5172:
5041:
4975:
4827:
4824:
4821:
4818:
4747:
4723:
4504:
4454:
4273:
4135:
3947:
3832:
3628:
3584:
3504:{\displaystyle X_{1},\dots ,X_{n}}
3272:
3049:
1051:{\displaystyle X_{1},\dots ,X_{n}}
733:
730:
428:
425:
409:
406:
403:
40:
37:
14:
6805:Nonparametric Bayesian statistics
2112:{\displaystyle X_{0}=\arg \sup f}
2070:{\displaystyle X_{0}=\arg \inf f}
1122:{\displaystyle P\sim Dp(s,G_{0})}
271:. The limiting Dirichlet process
6027:cumulative distribution function
6019:cumulative distribution function
4618:), the posterior expectation of
3513:cumulative distribution function
4432:empirical distribution function
3141:
1413:{\displaystyle \delta _{X_{i}}}
6446:
6405:
6399:
6393:
6361:
6320:
6314:
6308:
6192:
6168:
6146:
6122:
6100:
6088:
6079:
6061:
5938:
5932:
5871:
5865:
5844:{\displaystyle 1-\gamma =0.95}
5788:
5747:
5741:
5735:
5699:
5658:
5652:
5646:
5607:
5595:
5559:
5515:
5485:
5435:
5400:
5359:
5353:
5347:
5309:
5265:
5235:
5185:
5150:
5109:
5103:
5097:
4895:
4831:
4811:
4805:
4779:
4773:
4750:
4738:
4732:
4717:
4671:
4665:
4634:
4628:
4605:
4587:
4513:
4501:
4463:
4451:
4417:
4411:
4405:
4369:
4363:
4357:
4300:
4287:
4282:
4270:
4149:
4144:
4132:
4122:
4059:
4053:
4022:
4016:
4010:
3974:
3961:
3956:
3944:
3881:
3846:
3841:
3829:
3819:
3813:
3758:
3752:
3704:
3698:
3672:
3666:
3637:
3625:
3598:
3593:
3581:
3571:
3562:
3556:
3530:
3524:
3326:
3313:
3269:
3255:
3249:
3220:
3214:
3208:
3168:
3162:
3098:
3092:
3046:
2952:
2939:
2854:
2845:
2786:
2780:
2689:
2654:
2648:
2642:
2614:
2601:
2516:
2507:
2448:
2442:
2351:
2316:
2310:
2304:
2265:
2259:
2229:
2223:
2154:{\displaystyle {\mathcal {E}}}
2148:
2145:
2139:
2133:
1892:
1889:
1883:
1877:
1790:
1787:
1781:
1775:
1705:
1699:
1645:
1639:
1508:
1473:
1467:
1461:
1116:
1097:
950:
944:
888:
885:
879:
873:
843:
837:
814:
796:
756:
737:
709:{\displaystyle {\mathcal {B}}}
631:
613:
606:a probability distribution on
316:{\displaystyle s\rightarrow 0}
307:
290:{\displaystyle s\rightarrow 0}
281:
256:{\displaystyle s\rightarrow 0}
247:
1:
5890:with probability larger than
4423:{\displaystyle {\hat {F}}(x)}
3351:Choice of the prior strength
1753:. A-priori these bounds are:
386:(IDP) is defined as follows:
6760:. London: Chapman and Hall.
6568:{\displaystyle \mathbb {X} }
5730:
5342:
4109:
4040:
3149:
2637:
2165:is the same as the original
1872:
1622:{\displaystyle \mathbb {P} }
659:{\displaystyle \mathbb {X} }
518:{\displaystyle \mathbb {P} }
5950:{\displaystyle F(0)<0.5}
5883:{\displaystyle F(0)<0.5}
4683:{\displaystyle F(0)<0.5}
4573:(e.g., normal distribution
2700:
2362:
1899:
1797:
384:imprecise Dirichlet process
6821:
3036:Finally, observe that for
1447:. Hence, it follows that
1149:given the observations is
6581:Imprecise Dirichlet model
5976:{\displaystyle 1-\gamma }
5909:{\displaystyle 1-\gamma }
4934:{\displaystyle n_{<0}}
1547:Therefore, for any fixed
6597:Robust Bayesian analysis
3604:{\displaystyle F(x)=E}]}
6230:{\displaystyle s\geq 1}
6044:and the sample size is
5851:for instance) and then
4484:and for the upper that
684:{\displaystyle \sigma }
154:{\displaystyle \alpha }
138:concentration parameter
6756:Walley, Peter (1991).
6737:Cite journal requires
6639:10.1214/aos/1176342360
6577:Dirichlet distribution
6569:
6499:
6479:
6459:
6368:
6277:
6257:
6231:
6202:
6008:
5977:
5951:
5910:
5884:
5845:
5810:
5614:
5569:
5420:
5319:
5170:
5066:
5002:
4935:
4902:
4786:
4757:
4704:
4690:, i.e., the median of
4684:
4641:
4612:
4567:
4543:
4528:
4478:
4424:
4383:
4262:
3936:
3711:
3682:
3646:
3605:
3537:
3505:
3448:
3428:
3408:
3388:
3365:
3347:
3333:
3309:
3230:
3056:
3022:
3002:
2972:
2935:
2827:
2597:
2489:
2278:are in fact given by:
2272:
2236:
2203:
2183:
2155:
2113:
2077:(or respectively with
2071:
2029:
1982:
1959:
1747:
1712:
1679:
1652:
1623:
1601:
1568:
1541:
1441:
1414:
1377:
1352:
1143:
1123:
1072:
1052:
1006:
983:
850:
821:
783:
763:
710:
685:
660:
638:
600:
572:
545:
544:{\displaystyle s>0}
519:
494:
373:
346:
345:{\displaystyle s>0}
317:
291:
257:
228:
201:
155:
129:
101:
74:
6592:Imprecise probability
6570:
6551:categorical variables
6545:Categorical variables
6500:
6480:
6460:
6369:
6278:
6258:
6232:
6203:
6038:binomial distribution
6009:
6007:{\displaystyle G_{0}}
5978:
5952:
5911:
5885:
5846:
5811:
5615:
5570:
5406:
5320:
5156:
5067:
5003:
4936:
4903:
4787:
4758:
4705:
4685:
4642:
4613:
4568:
4566:{\displaystyle G_{0}}
4540:
4529:
4479:
4425:
4384:
4242:
3916:
3712:
3683:
3681:{\displaystyle F(x).}
3647:
3606:
3538:
3506:
3449:
3429:
3409:
3389:
3366:
3345:
3334:
3289:
3231:
3057:
3023:
3003:
3001:{\displaystyle G_{0}}
2973:
2915:
2807:
2577:
2469:
2273:
2237:
2204:
2184:
2156:
2114:
2072:
2030:
1983:
1960:
1748:
1713:
1680:
1678:{\displaystyle G_{0}}
1653:
1624:
1602:
1600:{\displaystyle G_{0}}
1569:
1567:{\displaystyle G_{0}}
1542:
1442:
1440:{\displaystyle X_{i}}
1415:
1378:
1332:
1144:
1124:
1073:
1053:
1007:
984:
851:
822:
784:
764:
711:
686:
661:
639:
601:
573:
571:{\displaystyle G_{0}}
546:
520:
495:
374:
372:{\displaystyle G_{0}}
347:
318:
292:
258:
229:
227:{\displaystyle G_{0}}
202:
156:
130:
102:
100:{\displaystyle G_{0}}
75:
6625:Annals of Statistics
6557:
6498:{\displaystyle 0.05}
6489:
6485:-value is less than
6469:
6378:
6287:
6267:
6241:
6215:
6055:
5991:
5961:
5926:
5894:
5859:
5823:
5631:
5582:
5332:
5082:
5012:
4948:
4915:
4799:
4785:{\displaystyle F(0)}
4767:
4714:
4694:
4659:
4651:Example: median test
4640:{\displaystyle F(x)}
4622:
4577:
4550:
4488:
4438:
4396:
3724:
3710:{\displaystyle F(x)}
3692:
3660:
3615:
3550:
3536:{\displaystyle F(x)}
3518:
3469:
3456:Example: median test
3438:
3418:
3398:
3378:
3355:
3243:
3069:
3040:
3012:
2985:
2285:
2271:{\displaystyle E(f)}
2253:
2235:{\displaystyle E(f)}
2217:
2193:
2173:
2123:
2081:
2039:
1992:
1972:
1760:
1722:
1711:{\displaystyle E(f)}
1693:
1662:
1651:{\displaystyle E(f)}
1633:
1611:
1584:
1551:
1451:
1424:
1390:
1156:
1133:
1082:
1062:
1016:
996:
863:
831:
793:
773:
720:
696:
675:
648:
610:
590:
555:
529:
507:
393:
382:More precisely, the
356:
330:
301:
275:
241:
211:
168:
145:
119:
84:
33:
6256:{\displaystyle s=1}
5045:
4984:
1658:for any choice of
1012:is again a DP. Let
6565:
6495:
6475:
6455:
6391:
6364:
6306:
6273:
6253:
6227:
6198:
6004:
5973:
5947:
5906:
5880:
5841:
5806:
5644:
5610:
5565:
5315:
5095:
5062:
5031:
4998:
4967:
4931:
4898:
4782:
4753:
4700:
4680:
4637:
4608:
4563:
4544:
4524:
4474:
4420:
4379:
4377:
3811:
3742:
3707:
3678:
3654:indicator function
3642:
3601:
3533:
3501:
3444:
3424:
3404:
3384:
3361:
3348:
3329:
3287:
3276:
3226:
3082:
3052:
3018:
2998:
2968:
2966:
2723:
2385:
2302:
2268:
2232:
2199:
2179:
2151:
2109:
2067:
2025:
1978:
1955:
1922:
1820:
1773:
1743:
1708:
1675:
1648:
1619:
1597:
1564:
1537:
1437:
1410:
1373:
1139:
1119:
1068:
1048:
1002:
979:
846:
817:
779:
759:
716:) and assume that
706:
681:
656:
634:
596:
568:
541:
515:
490:
369:
342:
313:
287:
265:Bayesian bootstrap
253:
224:
197:
151:
125:
97:
70:
6478:{\displaystyle p}
6382:
6297:
6276:{\displaystyle p}
6023:Beta distribution
5733:
5722:
5719:
5635:
5345:
5086:
4703:{\displaystyle F}
4408:
4360:
4349:
4328:
4307:
4237:
4216:
4112:
4043:
4013:
4002:
3981:
3911:
3802:
3733:
3447:{\displaystyle s}
3427:{\displaystyle s}
3407:{\displaystyle s}
3387:{\displaystyle s}
3364:{\displaystyle s}
3286:
3261:
3152:
3073:
3021:{\displaystyle s}
2959:
2910:
2883:
2805:
2763:
2699:
2640:
2621:
2572:
2545:
2467:
2425:
2361:
2293:
2202:{\displaystyle f}
2182:{\displaystyle f}
1981:{\displaystyle f}
1898:
1875:
1864:
1861:
1858:
1855:
1796:
1764:
1330:
1299:
1269:
1266:
1263:
1260:
1257:
1254:
1250:
1246:
1243:
1240:
1142:{\displaystyle P}
1071:{\displaystyle P}
1005:{\displaystyle P}
849:{\displaystyle E}
782:{\displaystyle f}
599:{\displaystyle P}
466:
463:
418:
401:
398:
128:{\displaystyle s}
109:base distribution
28:Dirichlet process
21:Dirichlet process
6812:
6772:
6771:
6753:
6747:
6746:
6740:
6735:
6733:
6725:
6720:
6714:
6713:
6711:
6699:
6688:
6687:
6679:
6673:
6670:
6664:
6661:
6652:
6651:
6641:
6616:Ferguson, Thomas
6612:
6574:
6572:
6571:
6566:
6564:
6504:
6502:
6501:
6496:
6484:
6482:
6481:
6476:
6464:
6462:
6461:
6456:
6445:
6444:
6426:
6425:
6392:
6387:
6373:
6371:
6370:
6365:
6360:
6359:
6341:
6340:
6307:
6302:
6282:
6280:
6279:
6274:
6262:
6260:
6259:
6254:
6236:
6234:
6233:
6228:
6207:
6205:
6204:
6199:
6167:
6166:
6121:
6120:
6013:
6011:
6010:
6005:
6003:
6002:
5982:
5980:
5979:
5974:
5956:
5954:
5953:
5948:
5915:
5913:
5912:
5907:
5889:
5887:
5886:
5881:
5850:
5848:
5847:
5842:
5815:
5813:
5812:
5807:
5787:
5786:
5768:
5767:
5734:
5729:
5724:
5720:
5717:
5698:
5697:
5679:
5678:
5645:
5640:
5619:
5617:
5616:
5611:
5594:
5593:
5574:
5572:
5571:
5566:
5558:
5557:
5530:
5529:
5514:
5513:
5509:
5484:
5483:
5456:
5455:
5434:
5419:
5414:
5399:
5398:
5380:
5379:
5346:
5341:
5336:
5324:
5322:
5321:
5316:
5308:
5307:
5286:
5285:
5264:
5263:
5259:
5234:
5233:
5212:
5211:
5184:
5169:
5164:
5149:
5148:
5130:
5129:
5096:
5091:
5071:
5069:
5068:
5063:
5058:
5057:
5044:
5039:
5024:
5023:
5007:
5005:
5004:
4999:
4997:
4996:
4983:
4978:
4960:
4959:
4940:
4938:
4937:
4932:
4930:
4929:
4907:
4905:
4904:
4899:
4894:
4893:
4872:
4871:
4859:
4858:
4843:
4842:
4830:
4791:
4789:
4788:
4783:
4762:
4760:
4759:
4754:
4709:
4707:
4706:
4701:
4689:
4687:
4686:
4681:
4646:
4644:
4643:
4638:
4617:
4615:
4614:
4609:
4586:
4585:
4572:
4570:
4569:
4564:
4562:
4561:
4533:
4531:
4530:
4525:
4517:
4516:
4499:
4483:
4481:
4480:
4475:
4467:
4466:
4449:
4429:
4427:
4426:
4421:
4410:
4409:
4401:
4388:
4386:
4385:
4380:
4378:
4362:
4361:
4353:
4350:
4348:
4334:
4329:
4327:
4313:
4308:
4303:
4299:
4298:
4286:
4285:
4268:
4261:
4256:
4240:
4238:
4236:
4222:
4217:
4215:
4201:
4197:
4188:
4184:
4183:
4182:
4164:
4163:
4148:
4147:
4130:
4113:
4108:
4103:
4098:
4094:
4093:
4092:
4074:
4073:
4044:
4039:
4034:
4031:
4015:
4014:
4006:
4003:
4001:
3987:
3982:
3977:
3973:
3972:
3960:
3959:
3942:
3935:
3930:
3914:
3912:
3910:
3896:
3892:
3880:
3879:
3861:
3860:
3845:
3844:
3827:
3812:
3807:
3797:
3793:
3792:
3791:
3773:
3772:
3743:
3738:
3730:
3716:
3714:
3713:
3708:
3687:
3685:
3684:
3679:
3651:
3649:
3648:
3643:
3641:
3640:
3623:
3610:
3608:
3607:
3602:
3597:
3596:
3579:
3542:
3540:
3539:
3534:
3510:
3508:
3507:
3502:
3500:
3499:
3481:
3480:
3453:
3451:
3450:
3445:
3433:
3431:
3430:
3425:
3413:
3411:
3410:
3405:
3393:
3391:
3390:
3385:
3370:
3368:
3367:
3362:
3338:
3336:
3335:
3330:
3325:
3324:
3308:
3303:
3288:
3279:
3275:
3235:
3233:
3232:
3227:
3207:
3203:
3202:
3201:
3183:
3182:
3153:
3148:
3143:
3137:
3133:
3132:
3131:
3113:
3112:
3083:
3078:
3062:, IDP satisfies
3061:
3059:
3058:
3053:
3027:
3025:
3024:
3019:
3007:
3005:
3004:
2999:
2997:
2996:
2977:
2975:
2974:
2969:
2967:
2960:
2955:
2951:
2950:
2934:
2929:
2913:
2911:
2909:
2895:
2884:
2882:
2868:
2860:
2844:
2843:
2842:
2841:
2826:
2821:
2806:
2804:
2790:
2764:
2762:
2748:
2743:
2742:
2722:
2721:
2713:
2712:
2688:
2687:
2669:
2668:
2641:
2636:
2631:
2622:
2617:
2613:
2612:
2596:
2591:
2575:
2573:
2571:
2557:
2546:
2544:
2530:
2522:
2506:
2505:
2504:
2503:
2488:
2483:
2468:
2466:
2452:
2426:
2424:
2410:
2405:
2404:
2384:
2383:
2375:
2374:
2350:
2349:
2331:
2330:
2303:
2298:
2277:
2275:
2274:
2269:
2241:
2239:
2238:
2233:
2208:
2206:
2205:
2200:
2188:
2186:
2185:
2180:
2160:
2158:
2157:
2152:
2132:
2131:
2118:
2116:
2115:
2110:
2093:
2092:
2076:
2074:
2073:
2068:
2051:
2050:
2034:
2032:
2031:
2026:
2024:
2023:
2022:
2021:
2004:
2003:
1987:
1985:
1984:
1979:
1964:
1962:
1961:
1956:
1942:
1941:
1921:
1920:
1912:
1911:
1876:
1871:
1866:
1862:
1859:
1856:
1853:
1840:
1839:
1819:
1818:
1810:
1809:
1774:
1769:
1752:
1750:
1749:
1744:
1742:
1734:
1733:
1717:
1715:
1714:
1709:
1684:
1682:
1681:
1676:
1674:
1673:
1657:
1655:
1654:
1649:
1628:
1626:
1625:
1620:
1618:
1606:
1604:
1603:
1598:
1596:
1595:
1573:
1571:
1570:
1565:
1563:
1562:
1546:
1544:
1543:
1538:
1533:
1532:
1507:
1506:
1488:
1487:
1460:
1459:
1446:
1444:
1443:
1438:
1436:
1435:
1419:
1417:
1416:
1411:
1409:
1408:
1407:
1406:
1382:
1380:
1379:
1374:
1369:
1368:
1367:
1366:
1351:
1346:
1331:
1329:
1315:
1310:
1309:
1300:
1298:
1284:
1279:
1278:
1267:
1264:
1261:
1258:
1255:
1252:
1251:
1248:
1244:
1241:
1238:
1234:
1230:
1229:
1228:
1193:
1192:
1174:
1173:
1148:
1146:
1145:
1140:
1128:
1126:
1125:
1120:
1115:
1114:
1077:
1075:
1074:
1069:
1057:
1055:
1054:
1049:
1047:
1046:
1028:
1027:
1011:
1009:
1008:
1003:
988:
986:
985:
980:
975:
974:
943:
942:
923:
919:
900:
899:
872:
871:
855:
853:
852:
847:
826:
824:
823:
818:
813:
812:
803:
788:
786:
785:
780:
768:
766:
765:
760:
755:
754:
736:
715:
713:
712:
707:
705:
704:
690:
688:
687:
682:
665:
663:
662:
657:
655:
643:
641:
640:
635:
630:
629:
620:
605:
603:
602:
597:
577:
575:
574:
569:
567:
566:
550:
548:
547:
542:
524:
522:
521:
516:
514:
499:
497:
496:
491:
489:
485:
484:
476:
475:
464:
461:
457:
453:
452:
451:
431:
416:
412:
399:
396:
378:
376:
375:
370:
368:
367:
351:
349:
348:
343:
322:
320:
319:
314:
296:
294:
293:
288:
262:
260:
259:
254:
233:
231:
230:
225:
223:
222:
206:
204:
203:
198:
196:
192:
191:
190:
160:
158:
157:
152:
134:
132:
131:
126:
106:
104:
103:
98:
96:
95:
79:
77:
76:
71:
69:
65:
64:
63:
43:
6820:
6819:
6815:
6814:
6813:
6811:
6810:
6809:
6795:
6794:
6781:
6776:
6775:
6768:
6755:
6754:
6750:
6736:
6726:
6722:
6721:
6717:
6701:
6700:
6691:
6681:
6680:
6676:
6671:
6667:
6662:
6655:
6614:
6613:
6609:
6604:
6589:
6555:
6554:
6547:
6511:
6487:
6486:
6467:
6466:
6436:
6417:
6376:
6375:
6351:
6332:
6285:
6284:
6265:
6264:
6239:
6238:
6213:
6212:
6158:
6106:
6053:
6052:
6031:random variable
5994:
5989:
5988:
5959:
5958:
5924:
5923:
5892:
5891:
5857:
5856:
5821:
5820:
5778:
5759:
5689:
5670:
5629:
5628:
5585:
5580:
5579:
5546:
5518:
5497:
5472:
5444:
5390:
5371:
5330:
5329:
5296:
5274:
5247:
5222:
5200:
5140:
5121:
5080:
5079:
5049:
5015:
5010:
5009:
4988:
4951:
4946:
4945:
4918:
4913:
4912:
4882:
4863:
4847:
4834:
4797:
4796:
4765:
4764:
4712:
4711:
4692:
4691:
4657:
4656:
4653:
4620:
4619:
4575:
4574:
4553:
4548:
4547:
4494:
4486:
4485:
4444:
4436:
4435:
4394:
4393:
4376:
4375:
4338:
4317:
4290:
4263:
4241:
4226:
4205:
4198:
4190:
4189:
4174:
4155:
4125:
4118:
4114:
4084:
4065:
4049:
4045:
4029:
4028:
3991:
3964:
3937:
3915:
3900:
3893:
3885:
3884:
3871:
3852:
3822:
3783:
3764:
3748:
3744:
3722:
3721:
3690:
3689:
3658:
3657:
3618:
3613:
3612:
3574:
3548:
3547:
3516:
3515:
3491:
3472:
3467:
3466:
3463:
3436:
3435:
3416:
3415:
3396:
3395:
3376:
3375:
3372:
3353:
3352:
3316:
3241:
3240:
3193:
3174:
3158:
3154:
3123:
3104:
3088:
3084:
3067:
3066:
3038:
3037:
3010:
3009:
2988:
2983:
2982:
2965:
2964:
2942:
2914:
2899:
2872:
2858:
2857:
2833:
2828:
2794:
2752:
2734:
2704:
2692:
2679:
2660:
2627:
2626:
2604:
2576:
2561:
2534:
2520:
2519:
2495:
2490:
2456:
2414:
2396:
2366:
2354:
2341:
2322:
2283:
2282:
2251:
2250:
2215:
2214:
2191:
2190:
2171:
2170:
2121:
2120:
2084:
2079:
2078:
2042:
2037:
2036:
2013:
2008:
1995:
1990:
1989:
1970:
1969:
1933:
1903:
1831:
1801:
1758:
1757:
1725:
1720:
1719:
1691:
1690:
1665:
1660:
1659:
1631:
1630:
1609:
1608:
1587:
1582:
1581:
1554:
1549:
1548:
1524:
1498:
1479:
1449:
1448:
1427:
1422:
1421:
1398:
1393:
1388:
1387:
1358:
1353:
1319:
1301:
1288:
1270:
1220:
1207:
1203:
1184:
1165:
1154:
1153:
1131:
1130:
1106:
1080:
1079:
1060:
1059:
1038:
1019:
1014:
1013:
994:
993:
966:
905:
901:
861:
860:
829:
828:
791:
790:
771:
770:
746:
718:
717:
694:
693:
673:
672:
646:
645:
608:
607:
588:
587:
584:
558:
553:
552:
527:
526:
505:
504:
467:
443:
436:
432:
423:
419:
391:
390:
359:
354:
353:
328:
327:
299:
298:
273:
272:
239:
238:
214:
209:
208:
182:
175:
171:
166:
165:
143:
142:
117:
116:
87:
82:
81:
55:
48:
44:
31:
30:
17:
12:
11:
5:
6818:
6816:
6808:
6807:
6797:
6796:
6793:
6792:
6787:
6780:
6779:External links
6777:
6774:
6773:
6766:
6748:
6739:|journal=
6715:
6689:
6674:
6665:
6653:
6632:(2): 209–230.
6606:
6605:
6603:
6600:
6588:
6585:
6563:
6546:
6543:
6542:
6541:
6537:
6533:
6530:
6510:
6507:
6494:
6474:
6454:
6451:
6448:
6443:
6439:
6435:
6432:
6429:
6424:
6420:
6416:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6390:
6386:
6363:
6358:
6354:
6350:
6347:
6344:
6339:
6335:
6331:
6328:
6325:
6322:
6319:
6316:
6313:
6310:
6305:
6301:
6295:
6292:
6272:
6252:
6249:
6246:
6226:
6223:
6220:
6209:
6208:
6197:
6194:
6191:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6165:
6161:
6157:
6154:
6151:
6148:
6145:
6142:
6139:
6136:
6133:
6130:
6127:
6124:
6119:
6116:
6113:
6109:
6105:
6102:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6078:
6075:
6072:
6069:
6066:
6063:
6060:
6001:
5997:
5985:
5984:
5972:
5969:
5966:
5946:
5943:
5940:
5937:
5934:
5931:
5920:
5917:
5905:
5902:
5899:
5879:
5876:
5873:
5870:
5867:
5864:
5840:
5837:
5834:
5831:
5828:
5817:
5816:
5805:
5802:
5799:
5796:
5793:
5790:
5785:
5781:
5777:
5774:
5771:
5766:
5762:
5758:
5755:
5752:
5749:
5746:
5743:
5740:
5737:
5732:
5728:
5716:
5713:
5710:
5707:
5704:
5701:
5696:
5692:
5688:
5685:
5682:
5677:
5673:
5669:
5666:
5663:
5660:
5657:
5654:
5651:
5648:
5643:
5639:
5609:
5606:
5603:
5600:
5597:
5592:
5588:
5576:
5575:
5564:
5561:
5556:
5553:
5549:
5545:
5542:
5539:
5536:
5533:
5528:
5525:
5521:
5517:
5512:
5508:
5504:
5500:
5496:
5493:
5490:
5487:
5482:
5479:
5475:
5471:
5468:
5465:
5462:
5459:
5454:
5451:
5447:
5443:
5440:
5437:
5433:
5430:
5427:
5424:
5418:
5413:
5409:
5405:
5402:
5397:
5393:
5389:
5386:
5383:
5378:
5374:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5344:
5340:
5326:
5325:
5314:
5311:
5306:
5303:
5299:
5295:
5292:
5289:
5284:
5281:
5277:
5273:
5270:
5267:
5262:
5258:
5254:
5250:
5246:
5243:
5240:
5237:
5232:
5229:
5225:
5221:
5218:
5215:
5210:
5207:
5203:
5199:
5196:
5193:
5190:
5187:
5183:
5180:
5177:
5174:
5168:
5163:
5159:
5155:
5152:
5147:
5143:
5139:
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5133:
5128:
5124:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
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5090:
5073:
5072:
5061:
5056:
5052:
5048:
5043:
5038:
5034:
5030:
5027:
5022:
5018:
4995:
4991:
4987:
4982:
4977:
4974:
4970:
4966:
4963:
4958:
4954:
4928:
4925:
4921:
4909:
4908:
4897:
4892:
4889:
4885:
4881:
4878:
4875:
4870:
4866:
4862:
4857:
4854:
4850:
4846:
4841:
4837:
4833:
4829:
4826:
4823:
4820:
4816:
4813:
4810:
4807:
4804:
4781:
4778:
4775:
4772:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4699:
4679:
4676:
4673:
4670:
4667:
4664:
4652:
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4636:
4633:
4630:
4627:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4584:
4560:
4556:
4523:
4520:
4515:
4512:
4509:
4506:
4503:
4498:
4493:
4473:
4470:
4465:
4462:
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4456:
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4448:
4443:
4419:
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4404:
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4371:
4368:
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4356:
4347:
4344:
4341:
4337:
4332:
4326:
4323:
4320:
4316:
4311:
4306:
4302:
4297:
4293:
4289:
4284:
4281:
4278:
4275:
4272:
4267:
4260:
4255:
4252:
4249:
4245:
4235:
4232:
4229:
4225:
4220:
4214:
4211:
4208:
4204:
4199:
4195:
4192:
4191:
4187:
4181:
4177:
4173:
4170:
4167:
4162:
4158:
4154:
4151:
4146:
4143:
4140:
4137:
4134:
4129:
4124:
4121:
4117:
4111:
4107:
4101:
4097:
4091:
4087:
4083:
4080:
4077:
4072:
4068:
4064:
4061:
4058:
4055:
4052:
4048:
4042:
4038:
4032:
4030:
4027:
4024:
4021:
4018:
4012:
4009:
4000:
3997:
3994:
3990:
3985:
3980:
3976:
3971:
3967:
3963:
3958:
3955:
3952:
3949:
3946:
3941:
3934:
3929:
3926:
3923:
3919:
3909:
3906:
3903:
3899:
3894:
3890:
3887:
3886:
3883:
3878:
3874:
3870:
3867:
3864:
3859:
3855:
3851:
3848:
3843:
3840:
3837:
3834:
3831:
3826:
3821:
3818:
3815:
3810:
3806:
3800:
3796:
3790:
3786:
3782:
3779:
3776:
3771:
3767:
3763:
3760:
3757:
3754:
3751:
3747:
3741:
3737:
3731:
3729:
3706:
3703:
3700:
3697:
3677:
3674:
3671:
3668:
3665:
3639:
3636:
3633:
3630:
3627:
3622:
3600:
3595:
3592:
3589:
3586:
3583:
3578:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3532:
3529:
3526:
3523:
3498:
3494:
3490:
3487:
3484:
3479:
3475:
3462:
3459:
3443:
3423:
3403:
3383:
3371:
3360:
3349:
3328:
3323:
3319:
3315:
3312:
3307:
3302:
3299:
3296:
3292:
3285:
3282:
3274:
3271:
3268:
3264:
3260:
3257:
3254:
3251:
3248:
3237:
3236:
3225:
3222:
3219:
3216:
3213:
3210:
3206:
3200:
3196:
3192:
3189:
3186:
3181:
3177:
3173:
3170:
3167:
3164:
3161:
3157:
3151:
3147:
3140:
3136:
3130:
3126:
3122:
3119:
3116:
3111:
3107:
3103:
3100:
3097:
3094:
3091:
3087:
3081:
3077:
3051:
3048:
3045:
3017:
2995:
2991:
2979:
2978:
2963:
2958:
2954:
2949:
2945:
2941:
2938:
2933:
2928:
2925:
2922:
2918:
2908:
2905:
2902:
2898:
2893:
2890:
2887:
2881:
2878:
2875:
2871:
2866:
2863:
2861:
2859:
2856:
2853:
2850:
2847:
2840:
2836:
2831:
2825:
2820:
2817:
2814:
2810:
2803:
2800:
2797:
2793:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2761:
2758:
2755:
2751:
2746:
2741:
2737:
2733:
2729:
2726:
2720:
2716:
2711:
2707:
2702:
2698:
2695:
2693:
2691:
2686:
2682:
2678:
2675:
2672:
2667:
2663:
2659:
2656:
2653:
2650:
2647:
2644:
2639:
2635:
2629:
2628:
2625:
2620:
2616:
2611:
2607:
2603:
2600:
2595:
2590:
2587:
2584:
2580:
2570:
2567:
2564:
2560:
2555:
2552:
2549:
2543:
2540:
2537:
2533:
2528:
2525:
2523:
2521:
2518:
2515:
2512:
2509:
2502:
2498:
2493:
2487:
2482:
2479:
2476:
2472:
2465:
2462:
2459:
2455:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2423:
2420:
2417:
2413:
2408:
2403:
2399:
2395:
2391:
2388:
2382:
2378:
2373:
2369:
2364:
2360:
2357:
2355:
2353:
2348:
2344:
2340:
2337:
2334:
2329:
2325:
2321:
2318:
2315:
2312:
2309:
2306:
2301:
2297:
2291:
2290:
2267:
2264:
2261:
2258:
2245:A-posteriori,
2231:
2228:
2225:
2222:
2198:
2178:
2150:
2147:
2144:
2141:
2138:
2135:
2130:
2108:
2105:
2102:
2099:
2096:
2091:
2087:
2066:
2063:
2060:
2057:
2054:
2049:
2045:
2020:
2016:
2011:
2007:
2002:
1998:
1977:
1966:
1965:
1954:
1951:
1948:
1945:
1940:
1936:
1932:
1928:
1925:
1919:
1915:
1910:
1906:
1901:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1874:
1870:
1852:
1849:
1846:
1843:
1838:
1834:
1830:
1826:
1823:
1817:
1813:
1808:
1804:
1799:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1772:
1768:
1741:
1737:
1732:
1728:
1707:
1704:
1701:
1698:
1672:
1668:
1647:
1644:
1641:
1638:
1617:
1594:
1590:
1561:
1557:
1536:
1531:
1527:
1523:
1519:
1516:
1513:
1510:
1505:
1501:
1497:
1494:
1491:
1486:
1482:
1478:
1475:
1472:
1469:
1466:
1463:
1458:
1434:
1430:
1405:
1401:
1396:
1384:
1383:
1372:
1365:
1361:
1356:
1350:
1345:
1342:
1339:
1335:
1328:
1325:
1322:
1318:
1313:
1308:
1304:
1297:
1294:
1291:
1287:
1282:
1277:
1273:
1237:
1233:
1227:
1223:
1219:
1216:
1213:
1210:
1206:
1202:
1199:
1196:
1191:
1187:
1183:
1180:
1177:
1172:
1168:
1164:
1161:
1138:
1118:
1113:
1109:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1067:
1045:
1041:
1037:
1034:
1031:
1026:
1022:
1001:
990:
989:
978:
973:
969:
965:
961:
958:
955:
952:
949:
946:
941:
936:
932:
929:
926:
922:
918:
915:
911:
908:
904:
898:
893:
890:
887:
884:
881:
878:
875:
870:
845:
842:
839:
836:
816:
811:
806:
802:
798:
778:
758:
753:
749:
745:
742:
739:
735:
732:
728:
725:
703:
680:
666:is a standard
654:
633:
628:
623:
619:
615:
595:
583:
580:
565:
561:
540:
537:
534:
513:
501:
500:
488:
483:
479:
474:
470:
460:
456:
450:
446:
442:
439:
435:
430:
427:
422:
415:
411:
408:
405:
366:
362:
341:
338:
335:
312:
309:
306:
286:
283:
280:
252:
249:
246:
221:
217:
195:
189:
185:
181:
178:
174:
150:
124:
94:
90:
68:
62:
58:
54:
51:
47:
42:
39:
15:
13:
10:
9:
6:
4:
3:
2:
6817:
6806:
6803:
6802:
6800:
6791:
6788:
6786:
6783:
6782:
6778:
6769:
6767:0-412-28660-2
6763:
6759:
6752:
6749:
6744:
6731:
6719:
6716:
6710:
6705:
6698:
6696:
6694:
6690:
6685:
6678:
6675:
6669:
6666:
6660:
6658:
6654:
6649:
6645:
6640:
6635:
6631:
6627:
6626:
6621:
6617:
6611:
6608:
6601:
6599:
6598:
6594:
6593:
6586:
6584:
6582:
6578:
6553:, i.e., when
6552:
6544:
6538:
6534:
6531:
6527:
6526:
6525:
6522:
6520:
6515:
6508:
6506:
6492:
6472:
6452:
6449:
6441:
6437:
6433:
6430:
6427:
6422:
6418:
6414:
6411:
6408:
6402:
6396:
6388:
6356:
6352:
6348:
6345:
6342:
6337:
6333:
6329:
6326:
6323:
6317:
6311:
6303:
6293:
6290:
6270:
6250:
6247:
6244:
6224:
6221:
6218:
6195:
6189:
6186:
6183:
6180:
6177:
6174:
6171:
6163:
6159:
6155:
6152:
6149:
6143:
6140:
6137:
6134:
6131:
6128:
6125:
6117:
6114:
6111:
6107:
6103:
6097:
6094:
6091:
6082:
6076:
6073:
6070:
6067:
6064:
6058:
6051:
6050:
6049:
6047:
6043:
6039:
6035:
6032:
6028:
6024:
6020:
6015:
5999:
5995:
5970:
5967:
5964:
5944:
5941:
5935:
5929:
5921:
5918:
5903:
5900:
5897:
5877:
5874:
5868:
5862:
5854:
5853:
5852:
5838:
5835:
5832:
5829:
5826:
5803:
5800:
5797:
5794:
5791:
5783:
5779:
5775:
5772:
5769:
5764:
5760:
5756:
5753:
5750:
5744:
5738:
5714:
5711:
5708:
5705:
5702:
5694:
5690:
5686:
5683:
5680:
5675:
5671:
5667:
5664:
5661:
5655:
5649:
5641:
5627:
5626:
5625:
5623:
5604:
5601:
5598:
5590:
5586:
5562:
5554:
5551:
5547:
5543:
5540:
5537:
5534:
5531:
5526:
5523:
5519:
5510:
5506:
5502:
5498:
5494:
5491:
5488:
5480:
5477:
5473:
5469:
5466:
5463:
5460:
5457:
5452:
5449:
5445:
5441:
5438:
5416:
5411:
5407:
5403:
5395:
5391:
5387:
5384:
5381:
5376:
5372:
5368:
5365:
5362:
5356:
5350:
5328:
5327:
5312:
5304:
5301:
5297:
5293:
5290:
5287:
5282:
5279:
5275:
5271:
5268:
5260:
5256:
5252:
5248:
5244:
5241:
5238:
5230:
5227:
5223:
5219:
5216:
5213:
5208:
5205:
5201:
5197:
5194:
5191:
5188:
5166:
5161:
5157:
5153:
5145:
5141:
5137:
5134:
5131:
5126:
5122:
5118:
5115:
5112:
5106:
5100:
5092:
5078:
5077:
5076:
5059:
5054:
5050:
5046:
5036:
5032:
5028:
5025:
5020:
5016:
4993:
4989:
4985:
4980:
4972:
4968:
4964:
4961:
4956:
4952:
4944:
4943:
4942:
4926:
4923:
4919:
4890:
4887:
4883:
4879:
4876:
4873:
4868:
4864:
4860:
4855:
4852:
4848:
4844:
4839:
4835:
4814:
4808:
4802:
4795:
4794:
4793:
4776:
4770:
4744:
4741:
4735:
4729:
4726:
4720:
4697:
4677:
4674:
4668:
4662:
4650:
4648:
4631:
4625:
4602:
4599:
4596:
4593:
4590:
4558:
4554:
4539:
4535:
4521:
4518:
4510:
4507:
4471:
4468:
4460:
4457:
4433:
4414:
4402:
4372:
4366:
4354:
4345:
4342:
4339:
4335:
4330:
4324:
4321:
4318:
4314:
4309:
4304:
4295:
4291:
4279:
4276:
4258:
4253:
4250:
4247:
4233:
4230:
4227:
4223:
4218:
4212:
4209:
4206:
4202:
4193:
4185:
4179:
4175:
4171:
4168:
4165:
4160:
4156:
4152:
4141:
4138:
4119:
4115:
4099:
4095:
4089:
4085:
4081:
4078:
4075:
4070:
4066:
4062:
4056:
4050:
4046:
4025:
4019:
4007:
3998:
3995:
3992:
3988:
3983:
3978:
3969:
3965:
3953:
3950:
3932:
3927:
3924:
3921:
3907:
3904:
3901:
3897:
3888:
3876:
3872:
3868:
3865:
3862:
3857:
3853:
3849:
3838:
3835:
3816:
3808:
3798:
3794:
3788:
3784:
3780:
3777:
3774:
3769:
3765:
3761:
3755:
3749:
3745:
3739:
3720:
3719:
3718:
3701:
3695:
3675:
3669:
3663:
3655:
3634:
3631:
3590:
3587:
3568:
3565:
3559:
3553:
3544:
3527:
3521:
3514:
3496:
3492:
3488:
3485:
3482:
3477:
3473:
3460:
3458:
3457:
3441:
3421:
3401:
3381:
3358:
3344:
3340:
3321:
3317:
3310:
3305:
3300:
3297:
3294:
3290:
3283:
3280:
3266:
3258:
3252:
3246:
3223:
3217:
3211:
3204:
3198:
3194:
3190:
3187:
3184:
3179:
3175:
3171:
3165:
3159:
3155:
3138:
3134:
3128:
3124:
3120:
3117:
3114:
3109:
3105:
3101:
3095:
3089:
3085:
3079:
3065:
3064:
3063:
3043:
3034:
3031:
3015:
2993:
2989:
2961:
2956:
2947:
2943:
2936:
2931:
2926:
2923:
2920:
2906:
2903:
2900:
2896:
2891:
2888:
2879:
2876:
2873:
2869:
2864:
2862:
2851:
2848:
2838:
2834:
2829:
2823:
2818:
2815:
2812:
2801:
2798:
2795:
2791:
2783:
2777:
2774:
2771:
2768:
2759:
2756:
2753:
2749:
2744:
2739:
2735:
2731:
2727:
2724:
2714:
2709:
2705:
2696:
2694:
2684:
2680:
2676:
2673:
2670:
2665:
2661:
2657:
2651:
2645:
2623:
2618:
2609:
2605:
2598:
2593:
2588:
2585:
2582:
2568:
2565:
2562:
2558:
2553:
2550:
2541:
2538:
2535:
2531:
2526:
2524:
2513:
2510:
2500:
2496:
2491:
2485:
2480:
2477:
2474:
2463:
2460:
2457:
2453:
2445:
2439:
2436:
2433:
2430:
2421:
2418:
2415:
2411:
2406:
2401:
2397:
2393:
2389:
2386:
2376:
2371:
2367:
2358:
2356:
2346:
2342:
2338:
2335:
2332:
2327:
2323:
2319:
2313:
2307:
2299:
2281:
2280:
2279:
2262:
2256:
2248:
2243:
2226:
2220:
2212:
2196:
2176:
2168:
2164:
2142:
2136:
2106:
2100:
2097:
2094:
2089:
2085:
2064:
2058:
2055:
2052:
2047:
2043:
2018:
2014:
2009:
2005:
2000:
1996:
1975:
1952:
1949:
1943:
1938:
1934:
1930:
1926:
1923:
1913:
1908:
1904:
1895:
1886:
1880:
1850:
1847:
1841:
1836:
1832:
1828:
1824:
1821:
1811:
1806:
1802:
1793:
1784:
1778:
1770:
1756:
1755:
1754:
1735:
1730:
1726:
1702:
1696:
1688:
1670:
1666:
1642:
1636:
1592:
1588:
1580:
1575:
1559:
1555:
1534:
1529:
1525:
1521:
1517:
1514:
1511:
1503:
1499:
1495:
1492:
1489:
1484:
1480:
1476:
1470:
1464:
1432:
1428:
1403:
1399:
1394:
1370:
1363:
1359:
1354:
1348:
1343:
1340:
1337:
1326:
1323:
1320:
1316:
1311:
1306:
1302:
1295:
1292:
1289:
1285:
1280:
1275:
1271:
1235:
1231:
1225:
1221:
1217:
1214:
1211:
1208:
1204:
1200:
1197:
1194:
1189:
1185:
1181:
1178:
1175:
1170:
1166:
1162:
1159:
1152:
1151:
1150:
1136:
1111:
1107:
1103:
1100:
1094:
1091:
1088:
1085:
1065:
1043:
1039:
1035:
1032:
1029:
1024:
1020:
999:
976:
971:
967:
963:
959:
956:
953:
947:
934:
930:
927:
924:
920:
916:
913:
909:
906:
902:
891:
882:
876:
859:
858:
857:
840:
834:
804:
776:
751:
747:
743:
740:
726:
723:
692:
678:
669:
621:
593:
581:
579:
563:
559:
538:
535:
532:
486:
477:
472:
468:
458:
454:
448:
444:
440:
437:
433:
420:
413:
389:
388:
387:
385:
380:
364:
360:
339:
336:
333:
324:
310:
304:
284:
278:
270:
269:Bradley Efron
266:
250:
244:
235:
219:
215:
193:
187:
183:
179:
176:
172:
162:
148:
140:
139:
122:
114:
110:
92:
88:
66:
60:
56:
52:
49:
45:
29:
24:
22:
6757:
6751:
6730:cite journal
6718:
6683:
6677:
6668:
6629:
6623:
6610:
6595:
6590:
6548:
6523:
6516:
6512:
6210:
6045:
6041:
6033:
6016:
5986:
5818:
5577:
5074:
4910:
4654:
4545:
4391:
3545:
3464:
3373:
3238:
3035:
3029:
2980:
2246:
2244:
2210:
2209:. A-priori,
2162:
1967:
1686:
1578:
1576:
1385:
991:
585:
502:
383:
381:
325:
236:
163:
136:
113:base measure
112:
108:
25:
18:
6374:. Thus, if
6025:, and the
789:defined on
670:with Borel
668:Borel space
6602:References
6465:then the
2161:under the
6709:1402.2755
6431:…
6415:∣
6389:_
6346:…
6330:∣
6304:_
6294:−
6222:≥
6187:−
6156:−
6129:−
6115:−
6095:≤
5971:γ
5968:−
5904:γ
5901:−
5833:γ
5830:−
5801:γ
5798:−
5773:…
5757:∣
5731:¯
5712:γ
5709:−
5684:…
5668:∣
5642:_
5605:β
5599:α
5544:−
5492:θ
5470:−
5439:θ
5408:∫
5385:…
5369:∣
5343:¯
5294:−
5242:θ
5220:−
5189:θ
5158:∫
5135:…
5119:∣
5093:_
5042:∞
5033:∫
5017:β
4976:∞
4973:−
4969:∫
4953:α
4880:−
4865:β
4836:α
4815:∼
4748:∞
4724:∞
4721:−
4505:∞
4455:∞
4406:^
4358:^
4274:∞
4244:∑
4169:…
4153:∣
4136:∞
4110:¯
4079:…
4063:∣
4041:¯
4011:^
3948:∞
3918:∑
3866:…
3850:∣
3833:∞
3809:_
3778:…
3762:∣
3740:_
3629:∞
3585:∞
3486:…
3291:∑
3273:∞
3270:→
3209:→
3188:…
3172:∣
3150:¯
3118:…
3102:∣
3080:_
3050:∞
3047:→
2917:∑
2830:δ
2809:∑
2775:∫
2725:∫
2715:∈
2674:…
2658:∣
2638:¯
2579:∑
2492:δ
2471:∑
2437:∫
2387:∫
2377:∈
2336:…
2320:∣
2300:_
2101:
2059:
2010:δ
1924:∫
1914:∈
1873:¯
1822:∫
1812:∈
1771:_
1736:∈
1515:∫
1493:…
1477:∣
1395:δ
1355:δ
1334:∑
1195:∼
1179:…
1163:∣
1089:∼
1033:…
957:∫
928:∫
907:∫
727:∼
679:σ
478:∈
308:→
282:→
248:→
149:α
6799:Category
6618:(1973).
6587:See also
3611:, where
1988:, i.e.,
6648:0350949
6036:from a
6021:of the
5620:is the
4430:is the
3652:is the
1718:w.r.t.
1577:In the
6764:
6724:2014).
6646:
5819:(with
5721:
5718:
5578:where
4911:where
4542:zero".
4392:where
3546:Since
3239:where
1863:
1860:
1857:
1854:
1386:where
1268:
1265:
1262:
1259:
1256:
1253:
1245:
1242:
1239:
691:-field
644:(here
503:where
465:
462:
417:
400:
397:
6704:arXiv
6529:loss.
6029:of a
2167:range
2035:with
135:(the
107:(the
6762:ISBN
6743:help
6549:For
6493:0.05
6453:0.95
6450:>
6409:<
6324:<
6263:the
5942:<
5875:<
5839:0.95
5792:>
5751:<
5703:>
5662:<
5552:<
5524:<
5478:<
5450:<
5363:<
5302:<
5280:<
5228:<
5206:<
5113:<
5008:and
4924:<
4888:<
4853:<
4675:<
3717:are
3465:Let
3030:near
1249:with
1078:and
586:Let
536:>
337:>
6634:doi
6412:0.5
6327:0.5
6014:).
5945:0.5
5878:0.5
5754:0.5
5665:0.5
5417:0.5
5366:0.5
5167:0.5
5116:0.5
4792:is
4678:0.5
4492:sup
4442:inf
3263:lim
2886:sup
2766:sup
2701:sup
2548:inf
2428:inf
2363:inf
2247:IDP
2211:IDP
2169:of
2163:IDP
2104:sup
2098:arg
2062:inf
2056:arg
1947:sup
1900:sup
1845:inf
1798:inf
1687:IDP
1579:IDP
111:or
6801::
6734::
6732:}}
6728:{{
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2602:(
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2539:+
2536:s
2532:s
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2443:(
2440:f
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2308:E
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2146:)
2143:f
2140:(
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2134:[
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2107:f
2095:=
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2086:X
2065:f
2053:=
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2044:X
2019:0
2015:X
2006:=
2001:0
1997:G
1976:f
1953:,
1950:f
1944:=
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1935:G
1931:d
1927:f
1918:P
1909:0
1905:G
1896:=
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1887:f
1884:(
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1878:[
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1851:,
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1807:0
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1794:=
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1535:.
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