Kari Lock. Department of Statistics, Harvard University Joint Work with Andrew Gelman (Columbia University)
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1 Bayesian Combinaion of Sae Polls and Elecion Forecass Kari Lock Deparmen of Saisics, Harvard Universiy Join Work wih Andrew Gelman (Columbia Universiy Harvard Insiue for Quaniaive Social Science Feb 4 h, 9
2 Forecasing an Elecion Polling Daa Pas Elecions Model-based Esimaes Our Goal: Uilize Bayesian echniques o combine hese sources Empirically esimae he uncerainy associaed wih predicion from each of hese sources
3 AL AK Using Pas Elecions o Predic he Fuure AZ AR CA CO CT DE FL GA '76 '9 '4 HI '76 '9 '4 ID '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 IL IN IA KS KY LA '76 '9 '4 ME '76 '9 '4 MD '76 '9 '4 MA '76 '9 '4 MI '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 MN MS MO MT NE NV '76 '9 '4 NH '76 '9 '4 NJ '76 '9 '4 NM '76 '9 '4 NY '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 NC ND OH OK OR PA '76 '9 '4 RI '76 '9 '4 SC '76 '9 '4 SD '76 '9 '4 TN '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 TX UT VT VA WA WV '76 '9 '4 WI '76 '9 '4 WY '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4 '76 '9 '4
4 How well does one presidenial elecion predic he nex? Kerr ry '4 Gore ' Clino on '96 Clino on ' Gore ' Clinon ' Clinon ' Dukakis '88 Dukakis ' Mondale '84 Mondale ' Carer '8.7 => > Separae he naional voe and he sae posiions relaive o he naional voe Carer ' Carer '76
5 Esimaing Variance from Elecion o Elecion WANT: va r( d d sy, sy, 4 psy, = Two-pary Democraic voe share in sae s, year y d = Posiion of sae s in year y, relaive o naion = p p sy, s, y Naion, y OPTION 1: Separae Esimae for Each Sae ( d sy d, sy, OPTION 3: 4 y Elecion years '8 - '4 var( dsy, + 4 ds, y = OPTION : Common Esimae for All Saes 5 7 ( d d s, y s, y 4 s= 1 y Elecion years '8 - '4 var( dsy, + 4 dsy, = 7*5 PARTIAL POOLING (Shrinkage Esimaion
6 Esimaing Variance from Elecion o Elecion Esimaes for SD( dsy, dsy, 4 :
7 Forecasing an Elecion Polling Pas Model-based Daa Elecions Esimaes Our Goal: Uilize Bayesian echniques o combine hese sources Empirically esimae he uncerainy associaed wih predicion from each of hese sources
8 Model-Based Esimae Douglas Hibbs Bread and Peace Model: Predics he success of he incumben pary ybased on The US Economy weighed-average growh of per capia real personal disposable income over he previous erm Miliary Faaliies cumulaive US miliary faaliies owing o unprovoked hosile deploymens of American armed forces in foreign conflics
9 Variance of Hibbs Bread and Peace Esimae Voe Share for In ncumben Pa ary Hibbs' Esimae p i var ( p p 13 i i, Hibbs ( p i, Hibbs pi i= 1 = 13 =.1 : Naional Two-Pary Democraic Voe Share i Elecions ago p ihibbs, : Esimae for pi based on he Bread and Peace Model
10 Forecasing an Elecion Polling Daa Pas Elecions Model-based Esimaes Our Goal: Uilize Bayesian echniques o combine hese sources Empirically esimae he uncerainy associaed wih predicion from each of hese sources
11 Esimaing Polling Uncerainy How accurae are polls??? We esimae his empirically using hisorical polling daa. For he naional popular voe: Gallup Polls, For he sae relaive posiions: Annenberg Sae Polls, and 4
12 Polls monhs before he Elecion p : Naional wo-pary Democraic voe share monhs before he elecion p : Esimae for p based on a poll monhs before he elecion p : Naional wo-pary Democraic voe share in he elecion 1 p (1 p p p ~ Bin ( n, p N, p n n ( var( p p N p, var( p p WANT : p p ( N p,???
13 Polls monhs before he Elecion ( ( var( p p = E var( p p, p p + var E( p p, p p p(1 p = E p + var ( p p n ( ( E p E p p p var p n n = + ( p ( ( p + var p p p = + var n n p(1 p ( n 1 = + var ( p p n n p(1 p + var ( p p n ( p p
14 Esimaing Polling Variance p (1 p var ( p p var p p n ( = + p :Naional Two-Pary opary Democraic Voe Share Monhs Before he Elecion p : Esimae for p basedonapo a poll monhs before he elecion p :Naional Two-Pary Democraic Voe Share in he Elecion
15 Esimaing Polling Variance p (1 p var ( p p var p p ( var p p = + var p p n p (1 p var ( p p = var( p p n p (1 p var ( p p = var ( p p n
16 Esimaing Polling Variance p : Naional Two-Pary Democraic Voe Share Monhs Before he Elecion var p p ( p : Naional Two-Pary Democraic Voe Share in he Elecion p : Esimae for based on Poll i, p i ni, :Sample size of poll i N :# of polls we have daa for monhs before an elecion p (1 p var( p p = = n (1 p N ( p i, p i= 1 ni, Calculaed using Gallup Polls from each monh Calculaed using Gallup Polls from each monh, for elecions N p
17 Esimaing Polling Variance Esimaes for SD( p p : var ( p p p (1 p n = + (.11
18 Esimaing Polling Variance Esimaes for SD( d d : s, s, var( d d = p s, s, (1 p n + (.49 Calculaed using Annenberg sae polls from each monh, from he and 4 elecions
19 Forecasing an Elecion Polling Daa Pas Elecions Model-based Esimaes Our Goal: Uilize Bayesian echniques o combine hese sources Empirically esimae he uncerainy associaed Empirically esimae he uncerainy associaed wih predicion from each of hese sources
20 Creaing Poseriors ( prio r PRIOR: μ μ ~ N μ, σ rue prior r prio ( daa rue N rue a LIKELIHOOD: μ μ ~ μ, σ da 1 Info = Informaion = Variance POSTERIOR: μ rue μdaa, μprior ~ Info Info daa prior 1 N daa prior, Info daa Info μ + μ prior Info prior Info + + prior Info daa + Info prior MEAN VARIANCE
21 Creaing Poseriors POSTERIOR: μ rue μdaa, μprior ~ Info Info d 1 daa prior N daa prio, Info r daa Info μ + μ prior Info prior Info + + prior Info daa + Info prior MEAN VARIANCE 1 Infodaa σ daa Wdaa = = Info daa + Info prior σ σ daa prior μ rue μdaa, μprior ~ N Wdaa μdaa + ( 1 Wd aa μprior, Info daa 1 + Info prior
22 Creaing Poseriors Polling Daa Pas Elecions Model-based Esimaes Daa Pi Prior for Pi Prior for (Likelihood Sae Naional Relaive popular voe Posiions
23 Creaing Poseriors 1. Combine naional polling daa and he model-based esimae o ge a poserior disribuion for he naional popular voe p = p, Hibbs p Acual Elecion Resul, Naionwide Bread and Peace Model Esimae Esimae based on a naionwide poll monhs before he elecion. Combine sae-wide polling daa and pas elecion resuls o ge 5 poserior disribuions; one for each sae relaive posiion d s, = d sp, rev, d s, Acual relaive posiion of sae s in Elecion Relaive posiion of sae s in previous elecion Esimaed relaive posiion of sae s based on a poll monhs before he elecion 3. Add he sae relaive poseriors o he naional poserior o ge a poserior disribuion for he way each sae will voe
24 Creaing Naionwide Poserior ( Hibbs Prior: p phibbs ~ N p,.1 Likelihood p p ( 1 p, + 1 : p ~ N p (. 1 n => Can ge Poserior: p p, p p =, p, Hibbs p Acual Elecion Resul, Naionwide Bread and Peace Model Esimae Hibbs Esimae based on a naionwide poll monhs before he elecion
25 Creaing Naionwide Poserior p 1 1 p, ~ (1 Hibbs p N Wpoll p + W poll phibbs, + var( p p var( p phibbs 1 W poll = 1 var( p p var( p var p ( p phibbs = p 1 p(1 p + (.11 n (1 p + (.11.1 n p =, p, Hibbs p Acual Elecion Resul, Naionwide Bread and Peace Model Esimae Esimae based on a naionwide poll monhs before he elecion
26 Creaing Sae Relaive Poseriors d s, = ds, prev N ds, prev ( Prior : ~,.37 p (1 p hood: d d, 49 s, s, Likeli s, ds, ~ N s, + (. n s, => Can ge Poserior: d d, d s, = s, s, prev d s, =, dsp, rev, ds, Acual relaive posiion of sae s in Elecion Relaive posiion of sae s in previous elecion Esimaed relaive posiion of sae s based on a poll monhs before he elecion
27 Creaing Sae Relaive Posiion Poseriors 1 1 ps,(1 ps, var( + (.49 d d ns, var( d var(,,,,,(1, s s prev.37 s d d d p s s ps + (.49 n s, s, Wpoll = = s, 1 1, d d d ~ N W d + (1 W d, + + (.49 ns, s, s, prev s, poll s, poll s, prv e ps, (1 ps,.37 1 d s, =, dsp, rev, ds, Acual relaive posiion of sae s in Elecion Relaive posiion of sae s in previous elecion Esimaed relaive posiion of sae s based on a poll monhs before he elecion
28 Forecasing he 8 Elecion Polling Daa: Survey USA Polls 6 people in each sae Conduced in February p p p N p p Feb, Hibbs ~ (.4 Feb +.96 Hibbs,. ( 537 ~ N.537,. For a ypical sae: d d, d ~ N(.35 d + 5 d,.3 s, s, feb s, prev s, feb s, prev p =, p, Hibbs p Acual Elecion Resul, Naionwide Bread and Peace Model Esimae Esimae based on a naionwide poll monhs before he elecion d s, =, dsp, rev, ds, Acual relaive posiion of sae s in Elecion Relaive posiion of sae s in previous elecion Esimaed relaive posiion of sae s based on a poll monhs before he elecion
29 Forecasing he 8 Elecion If hose same polls had been conduced in Ocober p p p N p p p p p N p p Oc, Hibbs ~ (.93 Oc +.7 Hibbs,.5 Feb, Hibbs ~ (.4 Feb +.96 Hibbs,. For a ypical sae: ds, ds, oc, ds, prev ~ N(7 (.7 ds, oc +.3 ds, prev,. d d, d ~ N(.35 d + 5 d,.3 s, s, feb s, prev s, feb s, prev p =, p, Hibbs p Acual Elecion Resul, Naionwide Bread and Peace Model Esimae Esimae based on a naionwide poll monhs before he elecion d s, =, dsp, rev, ds, Acual relaive posiion of sae s in Elecion Relaive posiion of sae s in previous elecion Esimaed relaive posiion of sae s based on a poll monhs before he elecion
30 <- Relaive Posiion, ds
31 A Simulaed Elecion STATE 4 Poll Difference Simulaed Differences Differences Poseriors 8 Difference ( p pfeb, phibbs ~ N.537,. Simulaed 8 Popular Voe Simulaed Sae Oucomes Winner? AK N(-.11, DEM AL N(-.9, REP AZ N(-.1, REP AR N(-.8, DEM CA.5 N(, DEM WA.5 N(.5, DEM WV N(-.8, REP WI.1.5 N(., DEM WY N(-.16, REP
32 Los of Simulaed Elecions Of 1, simulaed elecions, Obama won 99,886
33 How Did We Do?
34 How Did We Do? Roo Mean Square Error (RMSE for Our Resuls:.3 (RMSE for fivehiryeigh.com: i h.5 RMSE for Relaive Posiions:.31 Average Poserior SD for Relaive Posiions:.9 48 f 5 d i h i 95% P i I l 48 ou of 5 saes capured in heir 95% Poserior Inervals => 96% Coverage!
35 Key Conribuions Separaely modeling he naion and he sae relaive posiions Quanifying addiional variance of a poll due o ime before he elecion Ineresed? Google he ile of his alk and you will find our paper online. lock@sa.harvard.edu Thanks!
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