bayesian statistics and r - 统计之都 statistics and r peng ding, school of mathematical...
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Bayesian Statistics and R
Peng Ding,School of Mathematical Sciences,
Peking Univ.
December 16, 2008
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Introduction
• What is Probability?• Frequentist: LLN,CLT• Bayesian: Prior and Posterior• What is Statistics?
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Statistical Methodology
• Moment Estimation( Karl Pearson)• Maximum Likelihood Estimation( Gauss, R. A. Fisher)• Bayesian Method( Bayes)• Empirical Bayesian( Robbins)• ...?
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Frequentist vs Bayesian
• Frequentist:· parameters are constant to be estimate· point estimation and interval estimation• Bayesian:· parameters are random variables· Prior + Model→ Posterior· all information are contained in posterior distribution
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Bayes’ Formula and Bayesian Statistics
• Bayes’ Formula:If θ ∈ Θ has prior distribution π(θ), and the observed data ycomes from conditional distribution p(y |θ). Then the posteriordistribution of θ given y is
π(θ|y) =p(y |θ)π(θ)∫
Θ p(y |θ)π(θ)dθ
• Bayesian Statictics:· choose prior· model observed data· inference based on posterior distribution
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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How to choose Prior?
• Congugate Prior Distribution:· Prior and Posterior have the same form· Congugate Prior for Exponential Family
p(yi |θ) = f (yi )g(θ) exp{
φ(θ)Tu(yi )}
p(y |θ) =
(n∏
i=1
f (yi )
)g(θ)n exp
{φ(θ)T
n∑
i=1
u(yi )
}
Choose Prior as:
p(θ) ∝ g(θ)η exp{φ(θ)Tν}The Posterior is:
p(θ|y) ∝ g(θ)n+η exp{φ(θ)T (ν +n∑
i=1
u(yi ))}
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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How to choose Prior?
• Non-informative Prior Distribution:· Baysian Assumption
p(θ) ∝ constant, θ ∈ Θ
· Jefferys Prior
π(θ) ∝ |I (θ)|1/2
where I (θ) is the Fisher Information Matrix.
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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How to explore the Posterior?
• Direct Caculation:· The posterior has explicit and simple form!• Simulation the Posterior:· Sampling from the posterior distribution· Markov Chain Monte Carlo( MCMC):
Gibbs Sampler and Metropolis- Hasting Algorithm
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Gibbs Sampling
• X ∼ π(x), x = (x1, ..., xn)
• Initial value x (0) = (x(0)1 , ..., x
(0)n )
• The t-th iteration:· Sample x
(t)1 ∼ π(x1|x (t−1)
2 , ..., x(t−1)n );
· ...· Sample x
(t)i ∼ π(xi |x (t)
1 , ..., x(t)i−1, x
(t−1)i+1 , ..., x
(t−1)n );
· ...· Sample x
(t)n ∼ π(xn|x (t)
1 , ..., x(t)n−1).
• Under some regular conditions, the distribution of x converges tothe stationary distribution of the Markov Chain: π(x).
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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M-H Algorithm
• Given an irreducible transition probability q(·, ·),• Given a function
α(·, ·) = min{1,π(x ′)q(x ′, x)
π(x)q(x , x ′)}, 0 < α ≤ 1.
• At time t, X (t) = x ,· Generate a potential transition x → x ′ by q(x , ·);· With probability α(x , x ′), accept x ′; with probability 1− α(x , x ′)stay at x .• Under some regular conditions, π(x) is the stationary distributionof this Markov Chain.
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Bayesian Statistics in R
• MCMCpack
library(lattice)library(coda)library(MASS)library(MCMCpack)
• Famous software WinBUGS: Bayesian inference Using GibbsSampling.
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 1: Binomial Distribution
• Modelp(y |θ) ∝ θy (1− θ)n−y
• Priorp(θ) ∝ θα−1(1− θ)β−1
• Posteriorp(θ|y) ∝ θα+y−1(1− θ)β+n−y−1
• R code
posterior <-MCbinomialbeta(y=3,n=12,alpha=1,beta=1,mc=5000)summary(posterior)plot(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Result for Binomial Distribution
1. Empirical mean and standard deviation for each variable,plus standard error of the mean:
Mean SD Naive SE Time-series SE0.286442 0.116600 0.001649 0.001924
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%0.08972 0.19926 0.27805 0.36167 0.53906
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Binomial Parameter
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 2: Poisson Distribution
• Model
p(y |λ) ∝n∏
i=1
λyi e−λ
• Priorp(λ) ∝ eβλλα−1
• Posteriorλ|y ∼ Gamma(α + ny , β + n)
• R code
y<-rpois(1000,lambda=2)posterior <- MCpoissongamma(y, 15, 1, 5000)summary(posterior)plot(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Poisson Parameter
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 3: Normal Distribution with Variance known
• Model
p(y |µ) ∝ exp{− 1
2σ2
n∑
i=1
(yi − µ)2}
• Prior
p(µ) ∝ exp{− 1
2τ20
(µ− µ0)2}
• Posteriorµ|y ∼ N(µ1, τ
21 )
where
µ1 =
µ0
τ20
+ nyσ2
1τ20
+ 1σ2
, y =n∑
i=1
yi/n,
1
τ21
=1
τ20
+1
σ2.
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 3: Normal Distribution with Variance known
• R code
y<-rnorm(1000,5,1)posterior <-MCnormalnormal(y, sigma2=1, mu0=0,tau20=100, mc=5000)summary(posterior)plot(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Normal Parameter: µ
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 4: Normal Distribution with unknown Variance
• Model
p(y |µ) ∝ exp{− 1
2σ2
n∑
i=1
(yi − µ)2}
• Prior( Semi-Congugate)
µ ∼ N(b0,B20 )
σ2 ∼ Inverse− χ2(2c0, 2d0)
• R code
y<-rnorm(1000,2,1)posterior<-MCMCregress(y~1, b0 = 0, B0 =0, c0= 0.001, d0 = 0.001)summary(posterior)plot(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Normal Parameter: µ and σ
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 5: Multinomial Distribution
• Model
p(y |θ) ∝n∏
i=1
θyii
• Prior
p(θ|α) ∝n∏
i=1
θαi−1i
• R code
posterior <-MCmultinomdirichlet(c(727,583,137), c(1,1,1), mc=10000)summary(posterior)plot(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Multinomial Parameter
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Generalized Linear Model
• E (y |x) = g−1(βT x)• Different g(·)–Link Function, different models· Linear Regression: g(t) = t: E (y |x) = βT x
· Logistic Regression: g(t) = log( t1−t ): log( P(y=1|x)
1−P(y=1|x)) = βT x
· Probit Regression: g(t) = Φ−1(t): Φ−1(P(y = 1|x)) = βT x· Poisson Regression: g(t) = log(t): log(E (y |x)) = βT x
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 6: Linear Regression
X<-rnorm(100,2,1)Y<-1+2*X+rnorm(100,0,1)posterior <- MCMCregress(Y~X,b0 = 0, B0 = 0,
c0 = 0.001, d0 = 0.001,verbose=1000)plot(posterior)summary(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Linear Regression
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 7: Logistic Regression
x<-rnorm(1000)y<-rbinom(1000,1,exp(1-x)/(1+exp(1-x)))posterior <-MCMClogit(y~x, b0=0, B0=.001)plot(posterior)summary(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Logistic Regression
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 8: Probit Regression
y<-rbinom(1000,1,pnorm(1-x))posterior <- MCMCprobit(y~x, b0=0,B0=.001)plot(posterior)summary(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Probit Regression
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Exmaple 9: Poisson Regression
x<-rnorm(100)y<-rpois(100,exp(1+x))posterior <- MCMCpoisson(y ~x)plot(posterior)summary(posterior)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Posterior for Probit Regression
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Other Models
• Gaussian Mixture Model• Latent Class Analysis• Hierachical Models• Perhaps any parametric models: Examples in WinBUGS• Bayesians believes that all inference and more is Bayesianterritory.—Bayesian Nonparametrics, J. K. Ghosh and R. V.Ramamoorthi, Springer(2003)
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R
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Reference and recommendatory books
• Andrew Gelman, John B Carlin, Hal S Stern and Donald BRubin( 2004), Bayesian Data Analysis, Chapman&Hall/CRC• Martin A. Tanner( 1996), Tools for Statistical Inference:Methods for Exploration of Posterior Distribution and LikelihoodFunctions, Springer• Bradley P. Carlin and Thomas A. Lious(2000), Bayes andEmpirical Bayes Methods for Data Analysis, Chapman&Hall/CRC• Mao Shi-song, Wang Jing-long and Pu Xiao-long( 2006),Advanced Mathematical Statistics, Higher Education Press(inChinese).
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Acknowledgement
• I would like to thank the organizers of the first R conference inChina.• Also I am grateful to my mentor Professor Zhi Geng for hisintroduction of GLM and Bayesian Methods.
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The End
Thank you!
Peng Ding, School of Mathematical Sciences, Peking Univ. Bayesian Statistics and R