Standing Between a Bayesian and a Frequentist: An Emperical Bayes Exploration of Movies, Baseball, and Williams College.
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1 Standing Between a Bayesian and a Frequentist: An Emperical Bayes Exploration of Movies, Baseball, and Williams College Arthur Berg Pennsylvania State University
2 Bayesian and Frequentist Representatives Rev. Thomas Bayes FRS ( ) English Mathematician Presbyterian Minister Sir Ronald Fisher FRS ( ) English Statistician Evolutionary Biologist, Geneticist P (H E) = P (E H)P (H) P (E) Let the data speak for itself. Arthur Berg Standing Between a Bayesian and a Frequentist 2 / 27
3 Bayes Estimator as a Convex Combination 1 st Goal: List the top 250 movies of all time. Movies are rated on a scale of 1 to 10. Some movies are rated by many people, and some by only a few. Movies with fewer than 3000 votes are not considered. All movies have an average rating of C = 6.9. µ i represents the mean rating by everyone who has seen movie i. The real goal is to construct the best estimate of µ i, then pick the top 250. The frequentist approach uses only X i, the average rating for movie i. = X i ˆµ (Fisher) i The Bayesian approach shrinks X i towards C with more shrinking applied when the number of votes for movie i is small. ˆµ (Bayes) i = α i Xi + (1 α i )C where α i (0, 1) Arthur Berg Standing Between a Bayesian and a Frequentist 3 / 27
4 Internet Movie Database Top 250 Rank WR R Title Votes The Shawshank Redemption (1994) 546, The Godfather (1972) 427, The Godfather: Part II (1974) 257, The Good, the Bad and the Ugly (1966) 170, Pulp Fiction (1994) 436, Inception (2010) 265, Schindler s List (1993) 289, Angry Men (1957) 126, One Flew Over the Cuckoo s Nest (1975) 225, The Dark Knight (2008) 487, Black Swan (2010) 20, Avatar (2009) 285, True Grit (2010) 6,444 Arthur Berg Standing Between a Bayesian and a Frequentist 4 / 27
5 IMDb Weighted Ranking a true Bayesian estimate WR i = v ir i + mc v i + m = v i v i + m α i R i = average rating of the movie i ( X i ) R i + Xi v i = total number of votes from regular voters m v i + m 1 α i m = minimum # of votes to make the list = 3000 C = grand mean across all movies in the database = 6.9 C Arthur Berg Standing Between a Bayesian and a Frequentist 5 / 27
6 A Bayesian Calculation X i = (X i,1,..., X i,vi ) represents the v i ratings of movie i. prior: µ i N (µ 0, σ 2 0 ) conditional: X i,j µ i iid N (µi, σ 2 ) (j = 1,..., v i ) ˆµ (Bayes) i = E[µ i X i ] v i = ( v i + σ 2 /σ0 2 ) X i + ( σ2 /σ0 2 v i + σ 2 /σ0 2 ) µ 0 v i = v i + m R m i + v i + m C µ 0 = C, m = σ 2 /σ0 2 Arthur Berg Standing Between a Bayesian and a Frequentist 6 / 27
7 1 Does 2 How shrinking really help? much to shrink by? Prediction Error = n i=1 (µ i ˆµ i ) 2
8 Standing Between a Bayesian and a Frequentist In 1956, Charles Stein proved the existence of an estimator better than the sample mean under certain assumptions. In 1961, Willard James and Charles Stein explicitly constructed such an estimator. Arthur Berg Standing Between a Bayesian and a Frequentist 8 / 27
9 The James-Stein Estimator (n 4) µ i N (µ 0, σ 2 0) X i µ i iid N (µi, σ 2 ) (i = 1,... n) σ 2 ˆµ (Bayes) i = E [µ i X i ] = ( σ0 2 + σ2 α )µ 0 + ( σ0 2 σ0 2 + )X i σ2 1 α (n ˆµ (JS) 3)σ2 i = ( (X i X) 2 α ) X + ( 1 In practice, if σ 2 is unknown, an estimate is used. (n 3)σ2 (X i X) 2 )X i 1 α Arthur Berg Standing Between a Bayesian and a Frequentist 9 / 27
10 Predicting Batting Averages 2 nd Goal: Predict final batting averages from pre-season performances. Pre-season batting averages for 18 major league players are provided. Season final batting averages for the same players are also recorded. Data is from the 1970 season and is published in JASA (1975) and Scientific American (1977) by Efron and Morris. The frequentist approach uses only X i, the pre-season batting average for player i. = X i ˆp (Fisher) i The Emperical Bayes approach shrinks X i towards X by some empirically determined amount. ˆp (Stein) i = ˆαX i + (1 ˆα) X where ˆα (0, 1) Arthur Berg Standing Between a Bayesian and a Frequentist 10 / 27
11 Name hits/ab pre-season (ˆµ (ML) ) season final (µ) 1 Clemente 18/ Robinson 17/ Howard 16/ Johnstone 15/ Berry 14/ Spencer 14/ Kessinger 13/ Alvarado 12/ Santo 11/ Swoboda 11/ Unser 10/ Williams 10/ Scott 10/ Petrocelli 10/ Rodriguez 10/ Campaneris 9/ Munson 8/ Alvis 7/ Arthur Berg Standing Between a Bayesian and a Frequentist 11 / 27
12 Batting Average Dataset 1977 Batting Averages Dataset (Efron) Batting Average pre season season final Arthur Berg Standing Between a Bayesian and a Frequentist 12 / 27
13 James-Stein Estimation of Batting Averages 1977 Batting Averages Dataset (Efron) Batting Average pre season season final Arthur Berg Standing Between a Bayesian and a Frequentist 13 / 27
14 Ranking Bias Emperical Bayes + Order Statistics 1977 Batting Averages Dataset (Efron) Genome-wide association studies SNPS: AA/Aa/aa or 0/1/2 ( 10 7 ) Batting Average pre season season final ranking bias estimator part frequentist, part Bayesian with robust properties Estimated effects of the top SNPs are biased up. (winner s curse) Applied to 2 GWAS studies with 2,000 cases and 3,000 controls Crohn s Disease Type 1 Diabetes Arthur Berg Standing Between a Bayesian and a Frequentist 14 / 27
15 Williams College Book Survey In the summer of 2009, Williams faculty members were asked to list three books they felt that students should read. 150 faculty members responded. 25 departments are represented. 394 different books were recommended. The original publication dates were added (wikipedia/openlibrary.org). Books with unknown publication dates (13 in total) were approximated. Arthur Berg Standing Between a Bayesian and a Frequentist 15 / 27
16 The Top Picks Most Picked Authors (4+ hits) Fyodor Dostoyevsky (6) The Brothers Karamazov (4) Crime and Punishment (1) Notes from the Underground (1) Gabriel García Márquez (5) One Hundred Years of Solitude (5) Leo Tolstoy (5) Anna Karenina (4) War and Peace (1) Bill Bryson (4) A Short History of Nearly Everything (3) In a Sunburned Country (1) George Eliot (4) Middlemarch (4) Henry David Thoreau (4) Walden (4) Vladimir Nabokov (4) Speak, Memory (3) Lolita (1) Most Picked Titles (3+ hits) One Hundred Years of Solitude (5) Anna Karenina (4) Middlemarch (4) The Brothers Karamazov (4) Walden (4) Independent People (3) Speak, Memory (3) The Death and Life of Great American Cities (3) The Things They Carried (3) Arthur Berg Standing Between a Bayesian and a Frequentist 16 / 27
17 Average Publication Year Predictions Let µ i represent average publication year for department i. Let X i be the average publication year for department i based on only the first book selected. 3 rd Goal: Estimate µ i with only X i. Arthur Berg Standing Between a Bayesian and a Frequentist 17 / 27
18 Observed Data: First Book (Red), Truth : All Books (Gray) Classics Asian Stud Anth & Soc Religion Humanities Political Sci Philosophy Geosciences Music Math & Stat English Art Astronomy Comp Sci Psychology History Theater Ger & Rus Biology Economics Amer Stud Physics Comp Lit Chemistry Rom. Lang Arthur Berg Standing Between a Bayesian and a Frequentist 18 / 27
19 Results µ i N (µ 0, σ 2 0) X i µ i iid N (µi, σ 2 i ) (i = 1,... 25) Set 1 σ i 2 = n (X i X) 2 n i where n i = the number of observed books in department i. 1 ˆµ (1) i 2 ˆµ (2) i 3 ˆµ (3) i = X i = ˆα i X i + (1 ˆα i ) X = ˆα i X i + (1 ˆα i ) X where X denotes the median of X s. Prediction Error = 25 (ˆµ (j) i=1 i µ i ) 2 pe 2 pe 1 =.583 pe 3 pe 1 =.543 Arthur Berg Standing Between a Bayesian and a Frequentist 19 / 27
20 James-Stein Shrinkage Toward the Median Unequal Variances Case Classics Asian Stud Anth & Soc Religion Humanities Political Sci Philosophy Geosciences Music Math & Stat English Art Astronomy Comp Sci Psychology History Theater Ger & Rus Biology Economics Amer Stud Physics Comp Lit Chemistry Rom. Lang Arthur Berg Standing Between a Bayesian and a Frequentist 20 / 27
21 4 th Goal: Investigate how the departments cluster based on the book survey. Departments are classified in the following groups Natural Sciences: Astronomy, Biology, Chemistry, Geosciences, Physics Social Sciences: American Studies, Anthropology & Sociology, Asian Studies, Economics, History, Political Science, Psychology Formal Sciences: Computer Science, Mathematics & Statistics Humanities: Art, Classics, Comparative Literature, English, German & Russian, Humanities, Music, Philosophy, Religion, Romance Languages, Theater Arthur Berg Standing Between a Bayesian and a Frequentist 21 / 27
22 Departments Ranked by Publication Year Philosophy Anth & Soc Classics Asian Stud Political Sci Religion Math & Stat Humanities Astronomy Geosciences English Economics Comp Sci Ger & Rus Music Art Amer Stud History Psychology Comp Lit Theater Rom. Lang Physics Biology Chemistry Arthur Berg Standing Between a Bayesian and a Frequentist 22 / 27
23 Distance Measures Author/Title Data: Jaccard distance=1 A B A B = A B A B A B Year data: absolute value of the two sample t-statistic (non-metric distance measure) Homework Prove the Jaccard distance is a proper metric. Arthur Berg Standing Between a Bayesian and a Frequentist 23 / 27
24 Dendrogram of the Year Distances (Philosophy Removed) Philosophy Anth & Soc Classics Asian Stud Political Sci Religion Math & Stat Humanities Astronomy Geosciences English Economics Comp Sci Ger & Rus Music Art Amer Stud History Psychology Comp Lit Theater Rom. Lang Physics Biology Chemistry Chemistry Anth & Soc Math & Stat Religion Political Sci Asian Stud Classics Economics English Humanities Astronomy Geosciences Art Music Comp Sci Ger & Rus Psychology Amer Stud History Biology Physics Comp Lit Rom. Lang Theater Arthur Berg Standing Between a Bayesian and a Frequentist 24 / 27
25 Multidimensional Scaling of Author Distances Anth & Soc Chemistry Biology Political Sci Classics Theater Rom. Lang AstronomyMath & Stat Asian Stud Religion Comp Lit Ger & Rus Art Amer Stud Comp Sci Humanities History Economics English Physics PsychologyMusic Geosciences Philosophy Arthur Berg Standing Between a Bayesian and a Frequentist 25 / 27
26 Summary There are often multiple statistical approaches to a single problem. The complete statistician makes use of all available tools. When reporting the mean values of several related quantities, think about shrinkage! Arthur Berg Standing Between a Bayesian and a Frequentist 26 / 27
27 Thank You!! Williams.ArthurBerg.com Arthur Berg Standing Between a Bayesian and a Frequentist 27 / 27
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