540:
478:
889:
827:
1291:
375:
149:
Suppose we wish to estimate the effect of attending an elite college on future earnings. Simply regressing earnings on college rating will not give an unbiased estimate of the target effect because elite colleges are highly selective, and students attending them are likely to have qualifications for
124:
are held constant. In most cases, error terms are excluded from the graph. However, if the graph author suspects that the error terms of any two variables are dependent (e.g. the two variables have an unobserved or latent common cause) then a bidirected arc is drawn between them. Thus, the presence
68:
under the rubric "path diagrams". They were later adopted by social scientists and, to a lesser extent, by economists. These models were initially confined to linear equations with fixed parameters. Modern developments have extended graphical models to non-parametric analysis, and thus achieved a
547:
Figure 1 is a causal graph that represents this model specification. Each variable in the model has a corresponding node or vertex in the graph. Additionally, for each equation, arrows are drawn from the independent variables to the dependent variables. These arrows reflect the direction of
57:. As communication devices, the graphs provide formal and transparent representation of the causal assumptions that researchers may wish to convey and defend. As inference tools, the graphs enable researchers to estimate effect sizes from non-experimental data, derive
1441:
1045:
165:
137:, which allows researchers to determine, by inspection, whether the causal structure implies that two sets of variables are independent given a third set. In recursive models without correlated error terms (sometimes called
729:
1312:
1050:
661:
170:
1307:
1286:{\displaystyle {\begin{aligned}Q_{1}&=U_{1}\\A&=a\cdot Q_{1}+U_{2}\\C&=b\cdot A+U_{3}\\Q_{2}&=e\cdot Q_{1}+d\cdot C+U_{4}\\S&=c\cdot C+f\cdot Q_{2}+U_{5},\end{aligned}}}
370:{\displaystyle {\begin{aligned}Q_{1}&=U_{1}\\C&=a\cdot Q_{1}+U_{2}\\Q_{2}&=c\cdot C+d\cdot Q_{1}+U_{3}\\S&=b\cdot C+e\cdot Q_{2}+U_{4},\end{aligned}}}
1605:
1521:
1012:
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921:
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813:
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high-earning jobs prior to attending the school. Assuming that the causal relationships are linear, this background knowledge can be expressed in the following
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1561:
1541:
1032:
988:
968:
752:
643:
623:
472:
452:
77:
The causal graph can be drawn in the following way. Each variable in the model has a corresponding vertex or node and an arrow is drawn from a variable
656:
2255:
2222:
1806:
1763:
1674:
53:
Causal graphs can be used for communication and for inference. They are complementary to other forms of causal reasoning, for instance using
125:
of latent variables is taken into account through the correlations they induce between the error terms, as represented by bidirected arcs.
1645:
116:
Causal models often include "error terms" or "omitted factors" which represent all unmeasured factors that influence a variable
1949:
1436:{\displaystyle {\begin{aligned}A&=a\cdot Q_{1}+U_{A}\\C&=b\cdot A+U_{C}\\S&=\beta \cdot C+U_{S},\end{aligned}}}
1903:
Bareinmboim, Elias; Pearl, Judea (2014). "External
Validity: From do-calculus to Transportability across Populations".
69:
generality and flexibility that has transformed causal analysis in computer science, epidemiology, and social science.
35:
2336:
151:
61:
implications of the assumptions encoded, test for external validity, and manage missing data and selection bias.
548:
causation. In some cases, we may label the arrow with its corresponding structural coefficient as in Figure 1.
54:
1587:, a necessary and sufficient graphical condition for the identification of a structural coefficients, like
2280:
1014:
is not identified in Model 2. However, if we include the strength of an individual's college application,
43:
1736:
Bareinboim, Elias; Pearl, Judea (2012). "Causal
Inference by Surrogate Experiments: z-Identifiability".
645:
can be attributed to their error terms. By removing them, we obtain the following model specification:
1779:
Tian, Jin; Pearl, Judea (2002). "On the
Testable Implications of Causal Models with Hidden Variables".
1794:
1751:
2285:
2271:
Geiger, Dan; Pearl, Judea (1993). "Logical and
Algorithmic Properties of Conditional Independence".
1823:
2169:
2134:
2117:
Jöreskog, K. G. (1969). "A general approach to confirmatory maximum likelihood factor analysis".
2072:
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953:
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141:), these conditional independences represent all of the model's testable implications.
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2138:
1738:
Proceedings of the Twenty-Eighth
Conference on Uncertainty in Artificial Intelligence
65:
2076:
1934:
1889:
2040:
2023:
1781:
Proceedings of the
Eighteenth Conference on Uncertainty in Artificial Intelligence
539:
2240:
Counterfactuals and causal inference: Methods and principles for social research
1629:
724:{\displaystyle {\begin{aligned}C&=U_{C}\\S&=\beta C+U_{S}\end{aligned}}}
2309:
2152:
Goldberger, A. S. (1972). "Structural equation models in the social sciences".
991:
2294:
2247:
1842:
734:
The background information specified by Model 1 imply that the error term of
30:
In statistics, econometrics, epidemiology, genetics and related disciplines,
17:
2189:"Linking granger causality and the pearl causal model with settable systems"
1988:
1880:
1667:
Proceedings of the
Eighteenth National Conference on Artificial Intelligence
1662:
1719:
Identifiability in Causal
Bayesian Networks: A Sound and Complete Algorithm
477:
888:
454:
contains attributes representing the quality of the college attended, and
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By removing the latent variables from the model specification we obtain:
93:
when all other variables are being held constant. Variables connected to
58:
2173:
2130:
1926:
2165:
2090:
Duncan, O. D. (1976). "Introduction to structural equation models".
1972:"Recovering from Selection Bias in Causal and Statistical Inference"
2103:
2068:
1917:
826:
1789:
1746:
887:
825:
538:
476:
27:
Directed graph that models causal relationships between variables
2213:
Rothman, Kenneth J.; Greenland, Sander; Lash, Timothy (2008).
2055:
Duncan, O. D. (1966). "Path analysis: Sociological examples".
50:
used to encode assumptions about the data-generating process.
1976:
Proceedings of the AAAI Conference on
Artificial Intelligence
1868:
Proceedings of the AAAI Conference on
Artificial Intelligence
543:
Figure 2: Unidentified model with latent variables summarized
407:
represents the individual's qualifications prior to college,
1864:"Testable Implications of Linear Structural Equation Models"
1523:
is identified and can be estimated using the regression of
892:
Figure 4: Identified model with latent variables summarized
1824:"Complete Identification Methods for the Causal Hierarchy"
1692:"Complete Identification Methods for the Causal Hierarchy"
2310:"Graphical Tools for Linear Structural Equation Modeling"
2196:
Causality in Time Series Challenges in Machine Learning
1663:"A general identification condition for causal effects"
1034:, as shown in Figure 3, we obtain the following model:
605:
are unobserved or latent variables their influence on
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Figure 1: Unidentified model with latent variables (
1970:Bareinboim, Elias; Tian, Jin; Pearl, Judea (2014).
1950:"Graphical Models for Inference with Missing Data"
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1599:
1575:
1555:
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1515:
1492:
1465:
1435:
1285:
1026:
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982:
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830:Figure 3: Identified model with latent variables (
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527:
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466:
446:
426:
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369:
1957:Advances in Neural Information Processing Systems
1948:Mohan, Karthika; Pearl, Judea; Tian, Jin (2013).
2003:Wright, S. (1921). "Correlation and causation".
815:. As a result, we add a bidirected arc between
64:Causal graphs were first used by the geneticist
2187:White, Halbert; Chalak, Karim; Lu, Xun (2011).
2024:"Correlational analysis and causal inferences"
8:
133:A fundamental tool in graphical analysis is
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1987:
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434:represents qualifications after college,
418:
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299:
286:
251:
237:
224:
194:
177:
169:
167:
2242:. New York: Cambridge University Press.
1616:
2217:. Lippincott Williams & Wilkins.
1822:Shpitser, Ilya; Pearl, Judea (2008).
1690:Shpitser, Ilya; Pearl, Judea (2008).
7:
1831:Journal of Machine Learning Research
1699:Journal of Machine Learning Research
2238:Morgan, S. L.; Winship, C. (2007).
1862:Chen, Bryant; Pearl, Judea (2014).
89:is judged to respond to changes in
25:
1583:. This can be verified using the
97:through direct arrows are called
2005:Journal of Agricultural Research
1716:Huang, Y.; Valtorta, M. (2006).
1661:Tian, Jin; Pearl, Judea (2002).
48:probabilistic graphical models
1:
2092:American Journal of Sociology
2057:American Journal of Sociology
2041:10.1525/aa.1960.62.4.02a00060
1640:. Cambridge, MA: MIT Press.
73:Construction and terminology
2308:Chen, B.; Pearl, J (2014).
2353:
474:the individual's salary.
152:structural equation model
2248:10.1017/cbo9781107587991
55:causal equality notation
2028:American Anthropologist
2022:Blalock, H. M. (1960).
1989:10.1609/aaai.v28i1.9074
1881:10.1609/aaai.v28i1.9065
105:, or "direct causes of
2295:10.1214/aos/1176349407
1601:
1600:{\displaystyle \beta }
1577:
1557:
1537:
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1516:{\displaystyle \beta }
1494:
1467:
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1007:{\displaystyle \beta }
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109:," and are denoted by
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1585:single-door criterion
1578:
1558:
1538:
1518:
1495:
1493:{\displaystyle U_{S}}
1468:
1466:{\displaystyle U_{A}}
1438:
1288:
1029:
1009:
985:
965:
945:
943:{\displaystyle U_{C}}
918:
916:{\displaystyle U_{S}}
891:
879:
877:{\displaystyle Q_{2}}
852:
850:{\displaystyle Q_{1}}
829:
810:
808:{\displaystyle U_{C}}
781:, is correlated with
776:
774:{\displaystyle U_{S}}
749:
726:
640:
620:
600:
598:{\displaystyle Q_{2}}
573:
571:{\displaystyle Q_{1}}
542:
530:
528:{\displaystyle Q_{2}}
503:
501:{\displaystyle Q_{1}}
480:
469:
449:
429:
427:{\displaystyle Q_{2}}
402:
400:{\displaystyle Q_{1}}
372:
154:(SEM) specification.
2273:Annals of Statistics
1607:, using regression.
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2215:Modern epidemiology
1905:Statistical Science
1799:2013arXiv1301.0608T
1783:. pp. 519–27.
1756:2012arXiv1210.4842B
923:is correlated with
2131:10.1007/bf02289343
1597:
1573:
1553:
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1513:
1490:
1463:
1433:
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1283:
1281:
1024:
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884:) shown explicitly
874:
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823:, as in Figure 2.
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771:
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719:
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615:
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568:
545:
537:
535:) shown explicitly
525:
498:
464:
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397:
367:
365:
2257:978-1-107-06507-9
2224:978-0-7817-5564-1
1927:10.1214/14-sts486
1837:(64): 1941–1979.
1808:978-1-55860-897-9
1765:978-0-9749039-8-9
1676:978-0-262-51129-2
1576:{\displaystyle A}
1556:{\displaystyle C}
1536:{\displaystyle S}
1027:{\displaystyle A}
983:{\displaystyle C}
963:{\displaystyle C}
747:{\displaystyle S}
638:{\displaystyle S}
618:{\displaystyle C}
467:{\displaystyle S}
447:{\displaystyle C}
129:Fundamental tools
40:Bayesian networks
16:(Redirected from
2344:
2337:Graphical models
2321:
2320:
2317:Technical Report
2314:
2305:
2299:
2298:
2288:
2279:(4): 2001–2021.
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1473:correlated with
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950:and, therefore,
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2166:10.2307/1913851
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34:(also known as
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2125:(2): 183–202.
2109:
2104:10.1086/226377
2098:(3): 731–733.
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2069:10.1086/224256
2047:
2034:(4): 624–631.
2014:
1995:
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1940:
1911:(4): 579–595.
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1854:
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1162:
1161:
1156:
1152:
1148:
1145:
1142:
1139:
1136:
1133:
1131:
1129:
1126:
1125:
1120:
1116:
1112:
1107:
1103:
1099:
1096:
1093:
1090:
1088:
1086:
1083:
1082:
1077:
1073:
1069:
1066:
1064:
1060:
1056:
1052:
1051:
1023:
1003:
979:
959:
937:
933:
910:
906:
871:
867:
844:
840:
802:
798:
768:
764:
743:
732:
731:
714:
710:
706:
703:
700:
697:
694:
692:
690:
687:
686:
681:
677:
673:
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634:
614:
592:
588:
565:
561:
522:
518:
495:
491:
463:
443:
421:
417:
394:
390:
378:
377:
362:
357:
353:
349:
344:
340:
336:
333:
330:
327:
324:
321:
318:
315:
313:
311:
308:
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298:
294:
289:
285:
281:
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275:
272:
269:
266:
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250:
246:
245:
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227:
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216:
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202:
197:
193:
189:
186:
184:
180:
176:
172:
171:
146:
143:
130:
127:
81:to a variable
74:
71:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2349:
2338:
2335:
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2163:
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2119:Psychometrika
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2110:
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2025:
2018:
2015:
2010:
2006:
1999:
1996:
1990:
1985:
1981:
1977:
1973:
1966:
1963:
1958:
1951:
1944:
1941:
1936:
1932:
1928:
1924:
1919:
1914:
1910:
1906:
1899:
1896:
1891:
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1882:
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1873:
1869:
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1832:
1825:
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1815:
1810:
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1800:
1796:
1791:
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1301:
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1177:
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1021:
1001:
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632:
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392:
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316:
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276:
273:
270:
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234:
230:
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211:
209:
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195:
191:
187:
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178:
174:
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160:
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119:
114:
112:
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100:
96:
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84:
80:
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70:
67:
66:Sewall Wright
62:
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41:
37:
36:path diagrams
33:
32:causal graphs
19:
18:Causal graphs
2316:
2303:
2276:
2272:
2266:
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2208:
2199:
2195:
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2154:Econometrica
2153:
2147:
2122:
2118:
2112:
2095:
2091:
2085:
2060:
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2031:
2027:
2017:
2008:
2004:
1998:
1979:
1975:
1965:
1956:
1943:
1908:
1904:
1898:
1871:
1867:
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1846:. Retrieved
1834:
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1705:: 1941–1979.
1702:
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1635:
1630:Pearl, Judea
1584:
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1299:
1298:
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1037:
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895:
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816:
788:error term,
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135:d-separation
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2011:: 557–585.
1918:1503.01603
1848:2024-08-11
1611:References
992:endogenous
2281:CiteSeerX
2139:186236320
1843:1533-7928
1790:1301.0608
1747:1210.4842
1636:Causality
1595:β
1511:β
1408:⋅
1405:β
1372:⋅
1329:⋅
1251:⋅
1239:⋅
1203:⋅
1184:⋅
1141:⋅
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1002:β
699:β
335:⋅
323:⋅
280:⋅
268:⋅
218:⋅
139:Markovian
85:whenever
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2331:Category
2077:59428866
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59:testable
2174:1913851
1935:5586184
1890:1612893
1795:Bibcode
1752:Bibcode
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1038:Model 3
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158:Model 1
145:Example
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1785:arXiv
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