1172:) concerning racial discrimination, the petitioner, a black man named McCleskey was charged with the murder of a white police officer during a robbery. Expert testimony for McClesky introduced a statistical proof showing that "defendants charged with killing white victims were 4.3 times as likely to receive a death sentence as charged with killing blacks.". Nonetheless, the statistics was insufficient "to prove that the decisionmakers in his case acted with discriminatory purpose." It was further argued that there were "inherent limitations of the statistical proof", because it did not refer to the specifics of the individual. Despite the statistical demonstration of an increased probability of discrimination, the legal burden of proof (it was argued) had to be examined on a case-by-case basis.
1107:(Pr ) to produce a posterior probability distribution of the parameter (Pr ). The posterior probability is the likelihood that the parameter is correct given the observed data or samples statistics. Hypotheses can be compared using Bayesian inference by means of the Bayes factor, which is the ratio of the posterior odds to the prior odds. It provides a measure of the data and if it has increased or decreased the likelihood of one hypothesis relative to another.
842:. True values of a population, which are unknowable in practice, are called parameters of the population. Researchers sample from populations, which provide estimates of the parameters, to calculate the mean or standard deviation. If the entire population is sampled, then the sample statistic mean and distribution will converge with the parametric distribution.
1115:. Some researches have suggested that Popper's quest to define corroboration on the premise of probability put his philosophy in line with the Bayesian approach. In this context, the likelihood of one hypothesis relative to another may be an index of corroboration, not confirmation, and thus statistically proven through rigorous objective standing.
849:
that the sample statistic is sufficiently different from the null-model than can be explained by chance alone is given prior to the test. Most statisticians set the prior probability value at 0.05 or 0.1, which means if the sample statistics diverge from the parametric model more than 5 (or 10) times
66:
rests on the demonstrable application of the statistical method, the disclosure of the assumptions, and the relevance that the test has with respect to a genuine understanding of the data relative to the external world. There are adherents to several different statistical philosophies of inference,
1164:
proof" of discrimination, resulting in a shift of the burden of proof from plaintiff to defendant. Since that ruling, statistical proof has been used in many other cases on inequality, discrimination, and DNA evidence. However, there is not a one-to-one correspondence between statistical proof and
760:
of randomness, or objective chance from where modern statistical theory has advanced. Experimental data, however, can never prove that the hypotheses (h) is true, but relies on an inductive inference by measuring the probability of the hypotheses relative to the empirical data. The proof is in the
110:
claims and explain theory. Science cannot achieve absolute certainty nor is it a continuous march toward an objective truth as the vernacular as opposed to the scientific meaning of the term "proof" might imply. Statistical proof offers a kind of proof of a theory's falsity and the means to learn
1110:
The statistical proof is the
Bayesian demonstration that one hypothesis has a higher (weak, strong, positive) likelihood. There is considerable debate if the Bayesian method aligns with Karl Poppers method of proof of falsification, where some have suggested that "...there is no such thing as
1973:
cited in Meier (1986) Ibid. who states "Thus, in the space of less than half a year, the
Supreme Court had moved from the traditional legal disdain for statistical proof to a strong endorsement of it as being capable, on its own, of establishing a prima facie case against a
367:
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out of 100, then the discrepancy is unlikely to be explained by chance alone and the null-hypothesis is rejected. Statistical models provide exact outcomes of the parametric and estimates of the sample statistics. Hence, the
131:, 1) conventions that are taken as true that should be avoided because they cannot be tested, and 2) hypotheses. Proof in the theory of probability was built on four axioms developed in the late 17th century:
1111:"accepting" hypotheses at all. All that one does in science is assign degrees of belief..." According to Popper, hypotheses that have withstood testing and have yet to be falsified are not verified but
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1103:) “given” the data (or empirical observation), where the horizontal bar refers to "given". The right hand side of the formula calculates the prior probability of a statistical model (Pr ) with the
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2008:
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the legal burden of proof. "The
Supreme Court has stated that the degrees of rigor required in the fact finding processes of law and science do not necessarily correspond."
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Statistical proof was not regularly applied in decisions concerning United States legal proceedings until the mid 1970s following a landmark jury discrimination case in
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through repeated statistical trials and experimental error. Statistical proof also has applications in legal matters with implications for the
43:
that can be drawn from the test scores. Statistical methods are used to increase the understanding of the facts and the proof demonstrates the
1214:
1687:
1346:
Wiley, E. O. (1975). "Karl R. Popper, Systematics, and
Classification: A Reply to Walter Bock and Other Evolutionary Taxonomists".
1919:
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is applied to samples of a population, the test determines if the sample statistics are significantly different from the assumed
83:. These methods of reason have direct bearing on statistical proof and its interpretations in the broader philosophy of science.
1833:
810:. Hence, proof is a form of inference by means of a statistical test. Statistical tests are formulated on models that generate
1248:
Thomas Kuhn's "Linguistic Turn" and the Legacy of
Logical Empiricism: Incommensurability, Rationality and the Search for Truth
447:
59:
has two essential aims: the first is to convince and the second is to explain the proposition through peer and public review.
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The
Harvard Law Review Association (1995). "Developments in the Law: Confronting the New Challenges of Scientific Evidence".
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98:, which is a well-established practice in the tradition of statistics. Other modes of inference, however, may include the
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138:
518:
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190:
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Fienberg, S. E.; Kadane, J. B. (1983). "The presentation of
Bayesian statistical analyses in legal proceedings".
1834:"Legal Standards and Statistical Proof in Title VII Litigation: In Search of a Coherent Disparate Impact Model"
770:
362:{\displaystyle {\bigg \{}\Pr \left(h_{1}\right)+\Pr \left(h_{2}\right)=\Pr \left(h_{1}orh_{2}\right){\bigg \}}}
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1971:
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rests in the sample statistics that provide estimates of a statistical model. Statistical models contain the
1920:"Normative Constitutional Fact-Finding": Exploring the Empirical Component of Constitutional Interpretation"
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are mutually exclusive, then the sum of their probabilities is equal to the probability of their
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689:{\displaystyle {\bigg \{}\Pr(h_{1}|h_{2})={\frac {\Pr(h_{1}\And h_{2})}{\Pr(h_{2})}}{\bigg \}}}
106:
modes of proof. Scientists do not use statistical proof as a means to attain certainty, but to
1272:
Pedemont, B. (2007). "How can the relationship between argumentation and proof be analysed?".
1131:"Where gross statistical disparities can be shown, they alone may in a proper case constitute
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Statistical proof in a legal proceeding can be sorted into three categories of evidence:
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A History of
Parametric Statistical Inference from Bernoulli to Fisher, 1713-1935
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1158:. The US Supreme Court ruled that gross statistical disparities constitutes "
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1449:"Proof-Theoretical Semantics and Fregean Identity Criteria for Propositions"
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1757:"Falsifications and corroborations: Karl Popper's influence on systematics"
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1627:"Bayesian Inference of Phylogeny and Its Impact on Evolutionary Biology"
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The formula is read as the probability of the parameter (or hypothesis
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Howson, Colin; Urbach, Peter (1991). "Bayesian reasoning in science".
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The preceding axioms provide the statistical proof and basis for the
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and logic of inference with explicit reference to a hypothesis, the
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that give exact descriptions of variables that behave according to
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An introduction to experimental design and statistics for biology
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2017:
777:
135:
The probability of a hypothesis is a non-negative real number:
128:
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of the parametric values and their probability distributions.
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are based on a different philosophical approach for proof of
1731:
Reconstructing the Past: Parsimony, Evolution, and
Inference
1624:
52:
19:
is the rational demonstration of degree of certainty for a
1625:
Huelsenbeck, J. P.; Ronquist, F.; Bollback, J. P. (2001).
814:. Examples of probability distributions might include the
500:{\displaystyle {\bigg \{}\Pr(h_{1}\And h_{2}){\bigg \}}}
1543:(3rd ed.). W.H. Freeman & Company. pp.
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Proof and other dilemmas: Mathematics and philosophy
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The occurrence of an event, act, or type of conduct,
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descended from its Latin roots (provable, probable,
437:{\displaystyle {\Bigg \{}\Pr(h_{1}|h_{2}){\Bigg \}}}
1135:proof of a pattern or practice of discrimination."
885:. The mathematical formula for Bayes's theorem is:
745:{\displaystyle {\bigg \{}\Pr(h_{2})>0{\bigg \}}}
1799:Journal of the Royal Statistical Society, Series D
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845:Using the scientific method of falsification, the
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177:{\displaystyle {\bigg \{}\Pr(h)\geqq 0{\bigg \}}}
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31:that is used to convince others subsequent to a
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187:The probability of necessary truth equals one:
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558:{\displaystyle {\bigg \{}\Pr(h_{2}){\bigg \}}}
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1147:The identity of the individual(s) responsible
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1086:{\displaystyle Pr={\frac {Pr\times Pr}{Pr}}}
229:{\displaystyle {\bigg \{}\Pr(t)=1{\bigg \}}}
515:, divided by the unconditional probability
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1688:"Bayesian methods in conservation biology"
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1219:. Mathematics Association of America Inc.
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2007:) is being considered for deletion. See
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1755:Helfenbein, K. G.; DeSalle, R. (2005).
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1168:In an example of a death row sentence (
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1924:University of Pennsylvania Law Review
1838:University of Pennsylvania Law Review
1764:Molecular Phylogenetics and Evolution
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761:rational demonstration of using the
94:proof of falsification developed by
1966:Supreme Court of the United States
1535:Sokal, R. R.; Rohlf, F. J. (1995).
1314:"Damned Liars and Expert Witnesses"
569:where that probability is positive
13:
1522:10.1111/j.1467-9639.1996.tb00300.x
1493:"Statisticians have a Word for it"
1274:Educational Studies in Mathematics
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2011:to help reach a consensus. ›
1990:
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1734:. A Bradford Book. p. 284.
1707:10.1046/j.1523-1739.2000.99415.x
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372:The conditional probability of h
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1251:. Ashgate Pub Co. p. 277.
51:, the facts, the test, and the
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1604:. Springer. p. 260.
1286:10.1007/s10649-006-9057-x
812:probability distributions
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2009:templates for discussion
1953:
1213:; Simons, R. A. (2008).
1918:Faigman, D. L. (1991).
1654:10.1126/science.1065889
1468:10.5840/monist199477315
127:There are two kinds of
1832:Garaud, M. C. (1990).
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1598:Hald, Anders (2006).
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117:legal burden of proof
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2033:Logic and statistics
1983:481 U.S. 279 (1987).
1968:Castaneda v. Partida
1695:Conservation Biology
1686:Wade, P. R. (2000).
1491:Bissell, D. (1996).
1156:Castaneda v. Partida
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1640:(5550): 2310–2314.
1500:Teaching Statistics
1410:1991Natur.350..371H
1245:Gattei, S. (2008).
879:Bayesian statistics
239:If two hypotheses h
73:likelihood function
1876:Harvard Law Review
1728:Sober, E. (1991).
1349:Systematic Zoology
1312:Meier, P. (1986).
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1921:
1914:
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1910:
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1897:
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1889:
1885:
1881:
1877:
1870:
1868:
1859:
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1835:
1828:
1820:
1816:
1812:
1808:
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1800:
1793:
1785:
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1777:
1773:
1769:
1765:
1758:
1751:
1743:
1737:
1733:
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1724:
1716:
1712:
1708:
1704:
1700:
1696:
1689:
1682:
1680:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1628:
1621:
1613:
1607:
1603:
1602:
1594:
1586:
1580:
1577:. CRC Press.
1576:
1575:
1567:
1560:
1556:
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1514:
1509:
1505:
1501:
1494:
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1473:
1469:
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1457:
1450:
1443:
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1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1398:
1390:
1388:
1379:
1375:
1371:
1367:
1363:
1359:
1356:(2): 233–43.
1355:
1351:
1350:
1342:
1334:
1330:
1326:
1322:
1315:
1308:
1306:
1304:
1295:
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1197:
1188:
1187:Data analysis
1185:
1183:
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1173:
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1166:
1163:
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1157:
1149:
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1126:
1116:
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1098:
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1074:
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1065:
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1024:
1018:
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1000:
997:
994:
991:
988:
985:
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974:
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968:
965:
959:
956:
950:
944:
941:
938:
935:
927:
924:
921:
918:
915:
912:
909:
906:
903:
897:
894:
886:
884:
880:
875:
869:
868:Bayes theorem
862:Bayes theorem
859:
857:
853:
848:
843:
841:
837:
833:
832:random chance
829:
825:
821:
817:
813:
809:
805:
801:
795:
785:
783:
779:
776:
772:
768:
764:
759:
732:
729:
721:
717:
668:
664:
647:
643:
634:
630:
617:
609:
605:
594:
590:
540:
536:
482:
478:
469:
465:
419:
415:
404:
400:
371:
348:
342:
338:
334:
331:
326:
322:
317:
310:
306:
301:
297:
293:
286:
282:
277:
273:
269:
250:
238:
216:
213:
207:
186:
164:
161:
155:
134:
133:
132:
130:
120:
118:
114:
113:heuristically
109:
105:
101:
97:
93:
89:
84:
82:
78:
74:
70:
69:Bayes theorem
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
18:
2002:
1979:
1967:
1962:
1927:
1923:
1879:
1875:
1841:
1837:
1827:
1802:
1798:
1792:
1767:
1763:
1750:
1730:
1723:
1698:
1694:
1637:
1633:
1620:
1600:
1593:
1573:
1566:
1558:
1538:
1530:
1506:(3): 87–89.
1503:
1499:
1486:
1459:
1455:
1442:
1401:
1395:
1353:
1347:
1341:
1324:
1320:
1280:(1): 23–41.
1277:
1273:
1267:
1247:
1215:
1169:
1167:
1159:
1155:
1153:
1139:
1132:
1130:
1113:corroborated
1109:
1096:
1094:
887:
877:
844:
828:natural laws
807:
806:L.) meaning
803:
799:
797:
782:significance
755:
126:
85:
61:
16:
15:
1997:‹ The
1974:defendant."
1161:prima facie
1133:prima facie
249:disjunction
96:Karl Popper
88:non-science
71:versus the
21:proposition
2014:Statistics
1477:1887/11990
1456:The Monist
1193:References
1105:likelihood
872:See also:
840:null-model
834:. When a
77:positivism
41:inferences
25:hypothesis
1559:biometry.
1508:CiteSeerX
1426:1476-4687
1370:0039-7989
1294:121547580
1013:×
883:inference
798:The term
778:reasoning
775:deductive
640:&
475:&
162:≧
104:abductive
100:inductive
2027:Category
1999:template
1784:15737596
1715:55853118
1662:11743192
1539:Biometry
1211:Gold, B.
1176:See also
696:, where
67:such as
45:validity
37:evidence
2001:below (
1970:, 1977
1944:3312337
1896:1341808
1858:3312286
1819:2987595
1670:2138288
1642:Bibcode
1634:Science
1434:5419177
1406:Bibcode
1378:2412764
808:to test
804:probare
771:testing
376:given h
108:falsify
90:is the
79:versus
2018:Curlie
2004:Curlie
1942:
1894:
1856:
1817:
1782:
1738:
1713:
1668:
1660:
1608:
1581:
1551:
1510:
1432:
1424:
1397:Nature
1376:
1368:
1292:
1255:
1223:
1101:axioms
820:normal
816:binary
773:, and
129:axioms
123:Axioms
29:theory
1954:Notes
1940:JSTOR
1892:JSTOR
1854:JSTOR
1815:JSTOR
1760:(PDF)
1711:S2CID
1691:(PDF)
1666:S2CID
1630:(PDF)
1496:(PDF)
1452:(PDF)
1430:S2CID
1374:JSTOR
1317:(PDF)
1290:S2CID
822:, or
800:proof
511:and h
243:and h
75:, or
57:Proof
1780:PMID
1736:ISBN
1658:PMID
1606:ISBN
1579:ISBN
1549:ISBN
1422:ISSN
1366:ISSN
1253:ISBN
1221:ISBN
767:math
758:laws
730:>
565:of h
102:and
62:The
53:odds
2016:at
1932:doi
1928:139
1884:doi
1880:108
1846:doi
1842:139
1807:doi
1772:doi
1703:doi
1650:doi
1638:294
1545:887
1518:doi
1472:hdl
1464:doi
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