{"id":235,"date":"2013-12-13T00:26:33","date_gmt":"2013-12-13T06:26:33","guid":{"rendered":"http:\/\/rodrigosotomoreno.com\/revistanew\/?p=235"},"modified":"2017-11-14T12:04:46","modified_gmt":"2017-11-14T18:04:46","slug":"tema-34-el-teorema-de-bayes","status":"publish","type":"post","link":"https:\/\/cienciauanl.uanl.mx\/?p=235","title":{"rendered":"Tema 34: El Teorema de Bayes"},"content":{"rendered":"

Peter B. Mandeville*<\/p>\n

[W]hen people thought the earth was flat, they were wrong.
\nWhen people thought the earth was spherical, they were wrong.
\nBut if you think that thinking the earth
\nis spherical is just as wrong as thinking the earth is flat,
\nthen your view is wronger than both of them put together.
\nIsaac Asimov(1)<\/p>\n

CIENCIA UANL \/ A\u00d1O 16, No. 64, OCTUBRE-DICIEMBRE 2013<\/a><\/p>\n

Las probabilidades simples, conjuntas\u00a0y condicionales<\/p>\n

Sean A y B dos eventos asociados con un experimento,\u00a0\u03b5. Sea n el tama\u00f1o de la muestra, n(A) el n\u00famero\u00a0de puntos con caracter\u00edstica A, y n(B) el n\u00famero de\u00a0puntos con caracter\u00edstica B, entonces las probabilidades\u00a0simples de Ay B son: (2-13)<\/p>\n

\"panan\"<\/a><\/p>\n

\"pbnbn\"<\/a><\/p>\n

La probabilidad que ambos eventos sucedan es la\u00a0probabilidad conjunta que es: (2-13)<\/p>\n

\"pabnaybn\"<\/a>\u00a0\"nabn\"<\/a><\/p>\n

La probabilidad de condicional del evento B, dado\u00a0que el evento A ha sucedido es<\/p>\n

\"pbapbapa\"<\/a><\/p>\n

que se relacionen las probabilidades simples, conjuntas\u00a0y condicionales. \u00a0\u00a0\"PAB\"<\/a>representa la proporci\u00f3n\u00a0de los n puntos con caracter\u00edstica A, que tambi\u00e9n tienen\u00a0caracter\u00edstica B. Cada vez que se calcula\"PAB\"<\/a>se calcula P(B) con respecto al espacio reducido de A en lugar del espacio original. (2-13)<\/p>\n

El Teorema de Bayes<\/p>\n

El Teorema de Bayes fue desarrollado por el reverendo\u00a0Thomas Bayes en Essay towards solving a problem\u00a0in the doctrine of chances, que fue publicado por la\u00a0Royal Society of London en 1763, dos a\u00f1os despu\u00e9s de\u00a0su fallecimiento.(14)<\/p>\n

Hubo poco inter\u00e9s en las ideas de Bayes hasta que\u00a0fueron redescubiertas independientemente por Pierre-Simon, marqu\u00e9s de Laplace, quien public\u00f3 la formulaci\u00f3n\u00a0moderna en su Th\u00e9orie analytique des\u00a0probabilities (1812). (14)<\/p>\n

Se supone que la causa A sucede antes del resultado\u00a0B. De la definici\u00f3n de la probabilidad condicional se obtiene<\/p>\n

\"pbapappba\"<\/a><\/p>\n

lo que equivale a<\/p>\n

\"PABPBPAB\"<\/a><\/p>\n

y dado que<\/p>\n

\"pbapab\"<\/a><\/p>\n

entonces<\/p>\n

\"PBPABPAPBA\"<\/a><\/p>\n

 <\/p>\n

Las ecuaciones est\u00e1n relacionadas por el Teorema de Bayes<\/p>\n

\"pabpapbapb\"<\/a><\/p>\n

Entonces, dado\u00a0un resultado, B puede calcular la\u00a0probabilidad de la causa, A: (2-13)<\/p>\n

El problema es que hay pocas veces que se conoce\u00a0P(B).<\/p>\n

 <\/p>\n

El Teorema de la Probabilidad Total<\/strong><\/p>\n

Sea B alg\u00fan evento con respecto a S y sea de A1<\/sub>, A2<\/sub>,\u2026,Ak\u00a0<\/sub>de una partici\u00f3n de S.\u00a0Se supone que\u00a0los eventos \"bnabnabna\"<\/a>\u00a0\u00a0 se excluyen mutuamente, por lo tanto,\u00a0se puede aplicar la Regla\u00a0de Adici\u00f3n y escribir<\/p>\n

\"bbababa\"<\/a><\/p>\n

\"k1ba\"<\/a><\/p>\n

y<\/p>\n

\"pbpbapbapba\"<\/a><\/p>\n

\"k1pba\"<\/a><\/p>\n

Dada la definici\u00f3n de la probabilidad condicional,\u00a0se puede expresar cada t\u00e9rmino<\/p>\n

\"pbak\"<\/a><\/p>\n

como<\/p>\n

\"pakpbak\"<\/a><\/p>\n

 <\/p>\n

y por tanto se obtiene el Teorema de la probabilidad\u00a0total: (2-13)<\/p>\n

\"pbpapbapapbapapba\"<\/a><\/p>\n

 <\/p>\n

\"k1papba\"<\/a><\/p>\n

 <\/p>\n

Por el Teorema de la probabilidad total se puede escribir\u00a0el Teorema de Bayes<\/p>\n

\"pabpapbapapba\"<\/a><\/p>\n

 <\/p>\n

Aplicaci\u00f3n<\/strong><\/p>\n

Se puede utilizar la siguiente funci\u00f3n en R15 para solucionar\u00a0problemas que requiere el Teorema de Bayes:<\/p>\n

Bayes <- function(e,p,pse){
\ncat(\u201c\\n Bayes Theorem\\n\\n\u201d)
\njoint <- p*pse
\nps <- sum(joint)
\nbayes <-
\nround(cbind(p,pse,joint,joint\/ps),4)
\ncolnames(bayes) <-
\nc(\u201cP(E)\u201d,\u201cP(S|E)\u201d,\u201cP(E)P(S|E)\u201d,\u201cP(E|S)\u201d)
\nrownames(bayes) <- c(e)
\nprint(bayes)
\ncat(\u201c\\nP(S):\u201d,round(ps,4),\u201c\\n\u201d)
\n}<\/p>\n

Los historiales m\u00e9dicos indican que muchas condiciones\u00a0distintas pueden producir s\u00edntomas id\u00e9nticos.\u00a0Suponga que un conjunto particular de s\u00edntomas,\u00a0al cual se denota como el suceso H, ocurre solamente\u00a0con base en una de las tres condiciones A, B o C. Para\u00a0simplificar, suponga que las condiciones A, B y C son\u00a0mutuamente excluyentes. Ciertos estudios muestran que\u00a0las probabilidades de contraer las tres condiciones son:
\nP(A) = 0.01
\nP(B) = 0.005
\nP(C) = 0.02<\/p>\n

Las probabilidades de desarrollar los s\u00edntomas H,\u00a0dada una condici\u00f3n espec\u00edfica, son:<\/p>\n

P(H|A) = 0.90
\nP(H|B) = 0.95
\nP(H|C) = 0.75<\/p>\n

Suponiendo que una paciente tiene los s\u00edntomas H,<\/p>\n

> evento <- c(\u201cA\u201d,\u201cB\u201d,\u201cC\u201d)
\n> probabilidad <- c(0.01,0.005,0.02)
\n> condicional <- c(0.9,0.95,0.75)<\/p>\n

><\/p>\n

Bayes(evento,probabilidad,condicional)<\/p>\n

Bayes Theorem<\/p>\n

– \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 P(E) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0P(S|E) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 P(E) P(S|E) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0P(E|S)
\nA. \u00a0 \u00a0 \u00a00.010 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.90 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a00.0090 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a00.3130
\nB. \u00a0 \u00a0 \u00a00.005 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a00.95 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a00.0048 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.1652
\nC. \u00a0 \u00a0 \u00a00.020 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a00.75 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a00.0150 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 0.5217<\/p>\n

P(S): 0.0288<\/p>\n

\u00bfCu\u00e1l es la probabilidad de que tenga la condici\u00f3n\u00a0A?<\/p>\n

P(B|H) = 0.3130<\/p>\n

\u00bfCu\u00e1l es la probabilidad de que tenga la condici\u00f3n\u00a0B?<\/p>\n

P(B|H) = 0.1652<\/p>\n

\u00bfCu\u00e1l es la probabilidad de que tenga la condici\u00f3n\u00a0C?<\/p>\n

P(C|H) = 0.5217<\/p>\n

\u00bfCu\u00e1l es la probabilidad de los s\u00edntomas H?<\/p>\n

P(H) = 0.0288<\/p>\n

Se recomienda el texto de Bertsch McGrayne (14) para\u00a0una fascinante revisi\u00f3n de la importancia del Teorema\u00a0de Bayes.<\/p>\n

 <\/p>\n

*Consultor Asociado, BioEstad\u00edstica, S.C.
\n(www.bioestadistica.com)
\nContacto: peter.mandeville@bioestadistica.com<\/p>\n

Referencias<\/strong><\/p>\n

1. Isaac Asimov. The Relativity of Wrong. The Skeptical\u00a0Inquirer, 14(1), 35-44. Fall 1989.<\/p>\n

2. Kenneth Baclawski. Introduction to Probability with\u00a0R. Texts in Statistical Science. Chapman & Hall\/CRC,\u00a0Boca Raton, FL, USA. 2008.<\/p>\n

3. Jay L. Devore. Probability and Statistics for\u00a0Engineering and the Sciences. Fifth edition. Duxbury,\u00a0Pacific Groove, CA, USA. 2000.<\/p>\n

4. William Feller. Introducci\u00f3n a la teor\u00eda de probabilidades\u00a0sus aplicaciones, volumen I. Editorial Limusa,\u00a0S.A., M\u00e9xico 1, D.F., M\u00e9xico, 1983.<\/p>\n

5. Brian S. Everitt. Chance Rules: An Informal Guide to\u00a0Probability, Risk, and Statistics. Springer-Verlag New\u00a0York, Inc., New York, NY, USA. 1999.<\/p>\n

6. John E. Freund. Modern Elementary Statistics.\u00a0Pentice-Hall, Inc., Englewood Cliffs, NJ, USA, 1973.<\/p>\n

7. J. K. Lindsey. Introductory Statistics: A Modelling\u00a0Approach. Oxford Science Publications. Oxford University\u00a0Press, Oxford, UK, 1995.<\/p>\n

8. Paul L. Meyer. Probabilidad y aplicaciones estad\u00edsticas.\u00a0Fondo Educativo Interamericano, S. A., M\u00e9xico,\u00a0DF, MEX. 1973. Versi\u00f3n espa\u00f1ola de la segunda edici\u00f3n\u00a0de la obra Introductory Probability and Statistical\u00a0Applications. 1970.<\/p>\n

9. Irwin Miller y John E. Freund. Probabilidad y estad\u00edstica\u00a0para ingenieros. Tercera Edici\u00f3n. Prentice-Hall\u00a0Hispanoamericana, S. A., Naucalpan de Ju\u00e1rez, Estado\u00a0de M\u00e9xico. 1985.<\/p>\n

10. David S. Moore and George P. McCabe. Introduction\u00a0to the Practice of Statistics. Fourth Edition. W. H.\u00a0Freeman and Company, New York, NY, USA. 2003.<\/p>\n

11. Emanuel Parzen. Teor\u00eda moderna de probabilidades y\u00a0sus aplicaciones. Editorial Limusa, S.A. de C.V. M\u00e9xico,\u00a0DF, MEX. 1991. Versi\u00f3n autorizada en espa\u00f1ol de\u00a0la edici\u00f3n publicada en ingl\u00e9s por John Wiley & Sons,\u00a0bajo el t\u00edtulo de Modern Probability Theory and Its\u00a0Aplications.<\/p>\n

12. Leonard J. Savage. The Foundations of Statistics.\u00a0Second edition. Dover Publications, Inc., New York,\u00a0NY, USA. 1972.<\/p>\n

13. Taro Yamane. Estad\u00edstica. Nueva Edici\u00f3n. HARLA\u00a0S.A. de C.V., M\u00e9xico 4, DF, M\u00e9xico. 1979.<\/p>\n

14. Sharon Bertsch McGrayne. The theory that would not\u00a0die: How Bayes\u2019 rule cracked the enigma code, hunted\u00a0down Russian submarines & emerged triumphant from\u00a0two centuries of controversy. Yale University Press, New\u00a0Haven, CN, USA. 2011.<\/p>\n

15. R Core Team. R: A language and environment for\u00a0statistical computing. R Foundation for Statistical\u00a0Computing, Vienna, Austria. URL http:\/\/www.Rproject.\u00a0org\/. 2013.<\/p>\n","protected":false},"excerpt":{"rendered":"

Peter B. Mandeville* [W]hen people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together. Isaac Asimov(1) CIENCIA UANL \/ A\u00d1O […]<\/p>\n","protected":false},"author":3,"featured_media":716,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[19],"tags":[],"class_list":["post-235","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tips-bioestadisticos"],"_links":{"self":[{"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/posts\/235","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=235"}],"version-history":[{"count":31,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/posts\/235\/revisions"}],"predecessor-version":[{"id":3458,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/posts\/235\/revisions\/3458"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=\/wp\/v2\/media\/716"}],"wp:attachment":[{"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cienciauanl.uanl.mx\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}