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or Yes ... it is possible to consider the frequency properties (coverage probabilities) of credibility intervals, in which case the ends of a credibility interval are treated as being random when considering the coverage probabilities (ie. via stochastic simulation studies). Of course the ends of the
2039:
In discussing frequentist confidence intervals with colleagues, it seems people want to interpret them as the range of values (read, effect sizes) consistent with the data. It is my vague sense that this is what a
Bayesian credible interval actual does describe. If this is true, I think it would be
2008:
I am not a math or stats person. Now, I use
Knowledge quite often. Today, reading an article on Reuters I came across the phrase "credibility interval" for the first time. So I go to Knowledge for enlightenment. The article is incomprehensible to me. My impression is that you have to know your
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2012:
Perhaps this is as it should be. On the other hand, it could be that the author(s) of the article are all math/stats folks who aren't in the habit of thinking about audience. If the audience for this article is intended to reach the non-specialist, general reader, it doesn't.
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1981:
The article mentions "It is possible to frame the choice of a credible interval within decision theory and, in that context, an optimal interval will always be a highest probability density set". I didn't think this was right, so I asked the guys over at cross validated
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245:
I wasn't sure if the article would benefit from an example, to show the different thought-patterns behind credible intervals and confidence intervals. In the end, per NOTATEXTBOOK, I thought not, but perhaps it's useful to place here instead.
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That argument is quite different to the credible-interval calculation, which proceeds by finding the posterior probability, which is equal to the prior times the likelihood,
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For an example that has quite simple arithmetic, consider for instance the following question. Suppose a real-valued random variable is drawn from a uniform distribution,
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good to state that clearly and towards the top (i.e. in terms of how I would "read" a credible interval, and not simply in terms of a definition of one).
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A frequentist confidence interval is a RANDOM interval that contains a FIXED POINT a specified percentage of the time in repeated sampling.
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1174:{\displaystyle \mathrm {P} (n|x_{\rm {obs}})\,{\rm {d}}n\propto {\frac {1}{n^{2}}}\,{\rm {d}}n,\qquad \mathrm {if} \,n\geq x_{\rm {obs}}}
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correspond to the credible interval (even using the appropriate
Jeffreys' prior), see eg interval estimation for a binomial proportion,
1347:{\displaystyle \int _{x_{\rm {obs}}}^{\infty }{\frac {1}{n^{2}}}\;{\rm {d}}n=\left_{x_{\rm {obs}}}^{\infty }={\frac {1}{x_{\rm {obs}}}}}
2009:
way around stats to such an extent that if you could understand the article, you wouldn't have come to the article in the first place.
2020:
817:{\displaystyle \mathrm {P} (n|x_{\rm {obs}})\,{\rm {d}}n\propto \mathrm {P} (n|\mathrm {I} )\mathrm {P} (x_{\rm {obs}}|n)\,{\rm {d}}n}
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A credible interval is a FIXED INTERVAL such that the values of a certain RANDOM VARIABLE fall within it with a specified probability.
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1497:{\displaystyle \mathrm {P} (n|x_{\rm {obs}})\,{\rm {d}}n={\frac {x_{\rm {obs}}}{n^{2}}}\;{\rm {d}}n,\;n\geq x_{\rm {obs}}}
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as credibility intervals, whereas the ends of a confidence interval are treated as random when confidence intervals are
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http://stats.stackexchange.com/questions/83153/is-this-statement-about-credible-intervals-from-wikipedia-correct
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In this case thereofore the confidence interval and the credible interval coincide; but this is only because
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1986:), and most seem to agree with me. Maybe it's just unclearly worded? Either way it need attention...
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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The confidence-interval analysis for a 95% interval with two equal tails runs as follows:
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1036:{\displaystyle \mathrm {P} (n|x_{\rm {obs}})=0,\,\mathrm {if} \,n<x_{\rm {obs}}}
1774:{\displaystyle \int _{0}^{n_{2}}\mathrm {P} (n|x_{\rm {obs}})\,{\rm {d}}n=0.975,}
1620:{\displaystyle \int _{0}^{n_{1}}\mathrm {P} (n|x_{\rm {obs}})\,{\rm {d}}n=0.025,}
1963:
97:
600:{\displaystyle \int _{0}^{x_{\rm {obs}}}{\rm {P}}(x|n_{1})\,{\rm {d}}x=0.975}
407:{\displaystyle \int _{0}^{x_{\rm {obs}}}{\rm {P}}(x|n_{2})\,{\rm {d}}x=0.025}
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1945:{\displaystyle \scriptstyle \mathrm {Pr} (n|\mathrm {I} )\,\propto \,1/n}
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credibility region are treated as fixed when credibility intervals are
490:; this will lead to an interval with end-points that are greater than
1952:; and it is apparent that the logic in each case is very different.
297:; this will lead to an interval with end-points that are less than
929:{\displaystyle \scriptstyle \mathrm {Pr} (n|I)\,\propto \,1/n}
15:
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Integrating this to establish the normalisation, we have that
253:; and we seek an interval estimate for the unknown parameter
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This leaves 95% of the time that the interval will cover
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264:We place the left hand end of the interval at
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463:{\displaystyle n_{2}=x_{\rm {obs}}/0.025}
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2005:What is the audience for this article?
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478:2.5% of the time an unknown parameter
285:2.5% of the time an unknown parameter
7:
90:This article is within the scope of
2099:High-importance Statistics articles
2035:Interpretation of Credible Interval
49:It is of interest to the following
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1881:{\displaystyle \mathrm {Pr} (n|I)}
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867:{\displaystyle \mathrm {Pr} (n|I)}
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271:/0.975; and the right hand end at
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110:Knowledge:WikiProject Statistics
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2104:WikiProject Statistics articles
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1888:was the Jeffreys' distribution
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113:Template:WikiProject Statistics
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153:Is a credible interval random?
1:
2050:21:51, 13 November 2019 (UTC)
1135:
486:that is greater than 0.975 *
233:16:42, 25 February 2008 (UTC)
157:is credible interval random?
104:and see a list of open tasks.
1996:17:51, 24 January 2014 (UTC)
1972:00:27, 8 December 2010 (UTC)
206:17:51, 8 December 2007 (UTC)
1977:"within decision theory..."
1846:was a scale parameter, and
874:the Jeffreys' distribution
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2064:Start-Class vital articles
293:that is less than 0.025 *
176:) 01:56, 20 September 2007
2029:12:20, 30 July 2016 (UTC)
482:will give an observation
289:will give an observation
223:as confidence intervals.
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257:, which is real-valued.
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278:/0.025, because:
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136:importance scale
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132:High-importance
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43:on Knowledge's
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166:128.100.74.204
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102:the discussion
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2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2061:
2059:
2052:
2051:
2047:
2043:
2034:
2032:
2030:
2026:
2022:
2018:
2010:
2006:
2000:
1998:
1997:
1993:
1989:
1988:66.29.243.106
1985:
1976:
1974:
1973:
1969:
1965:
1961:
1958:
1953:
1938:
1934:
1930:
1925:
1907:
1872:
1864:
1845:
1824:
1820:
1802:
1798:
1793:
1789:
1768:
1765:
1762:
1759:
1730:
1721:
1706:
1702:
1696:
1692:
1684:
1670:
1666:
1648:
1644:
1639:
1635:
1614:
1611:
1608:
1605:
1576:
1567:
1552:
1548:
1542:
1538:
1530:
1529:
1522:
1515:
1511:
1510:
1509:
1508:
1477:
1473:
1470:
1465:
1462:
1446:
1442:
1424:
1418:
1415:
1386:
1377:
1362:
1361:
1357:
1356:
1325:
1321:
1316:
1292:
1286:
1280:
1277:
1272:
1268:
1263:
1260:
1244:
1240:
1236:
1210:
1205:
1197:
1196:
1195:
1194:
1188:
1187:
1186:
1185:
1154:
1150:
1147:
1132:
1129:
1113:
1109:
1105:
1100:
1097:
1068:
1059:
1044:
1016:
1012:
1009:
994:
991:
988:
968:
959:
944:
943:
942:
941:
922:
918:
914:
909:
901:
893:
858:
850:
831:
830:
829:
828:
811:
796:
774:
749:
738:
735:
706:
697:
682:
681:
680:
679:
678:
672:
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667:
650:
646:
628:
624:
619:
615:
594:
591:
588:
571:
567:
558:
529:
523:
519:
510:
509:
505:
501:
497:
493:
489:
485:
481:
477:
476:
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474:
457:
453:
435:
431:
426:
422:
401:
398:
395:
378:
374:
365:
336:
330:
326:
317:
316:
312:
308:
304:
300:
296:
292:
288:
284:
283:
282:
281:
274:
267:
263:
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256:
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247:
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234:
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199:
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137:
133:
127:
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103:
99:
95:
94:
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86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2038:
2021:24.65.41.120
2015:— Preceding
2011:
2007:
2004:
1980:
1956:
1954:
1843:
1841:
1520:
1513:
676:
670:
503:
499:
495:
491:
487:
483:
479:
310:
306:
302:
298:
294:
290:
286:
272:
265:
259:
254:
250:
248:
244:
220:
216:
160:— Preceding
156:
131:
91:
51:WikiProjects
34:
221:interpreted
217:interpreted
41:Start-class
2058:Categories
251:x ~ U(0,n)
107:Statistics
98:statistics
70:Statistics
1781:given by
1627:given by
1526:such that
39:is rated
2017:unsigned
2001:Audience
301:/39 ...
225:Melcombe
174:contribs
162:unsigned
2042:Muniche
241:Example
134:on the
1964:Jheald
494:... 39
198:Blaise
47:scale.
1825:0.025
1766:0.975
1671:0.975
1612:0.025
936:gives
651:0.975
607:when
595:0.975
511:i.e.
458:0.025
414:when
402:0.025
318:i.e.
28:This
2046:talk
2025:talk
1992:talk
1968:talk
1519:and
1013:<
229:talk
202:talk
170:talk
126:High
1957:not
276:obs
269:obs
182:No.
2060::
2048:)
2027:)
1994:)
1970:)
1962:.
1926:∝
1693:∫
1539:∫
1474:≥
1358:so
1312:∞
1273:−
1230:∞
1206:∫
1151:≥
1101:∝
910:∝
739:∝
520:∫
327:∫
231:)
204:)
172:•
2044:(
2023:(
1990:(
1982:(
1966:(
1939:n
1935:/
1931:1
1921:)
1917:I
1912:|
1908:n
1905:(
1901:r
1898:P
1876:)
1873:I
1869:|
1865:n
1862:(
1858:r
1855:P
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1794:2
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1763:=
1760:n
1755:d
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1742:s
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1398:s
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1375:(
1371:P
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980:s
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969:x
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960:n
957:(
953:P
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902:I
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887:r
884:P
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859:I
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780:o
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771:(
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712:o
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698:n
695:(
691:P
671:n
647:/
640:s
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572:1
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563:|
559:x
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551:P
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492:n
488:n
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480:n
454:/
447:s
444:b
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432:=
427:2
423:n
399:=
396:x
391:d
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379:2
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370:|
366:x
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358:P
348:s
345:b
342:o
337:x
331:0
313:.
311:n
307:n
303:n
299:n
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291:x
287:n
273:x
266:x
255:n
227:(
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168:(
138:.
53::
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