Bayesian Inference in a Normal Population

Transcription

1 Bayesian Inference in a Normal Population September 20, 2007 Casella & Berger Chapter 7, Gelman, Carlin, Stern, Rubin Sec 2.6, 2.8, Chapter 3. Bayesian Inference in a Normal Population p. 1/16

2 Normal Model IID observations Y = (Y 1, Y 2,... Y n ) Y i N(µ, σ 2 ) unknown parameters µ and σ 2. From a Bayesian perspective, it is easier to work with the precision, φ, where φ = 1/σ 2. Likelihood L(µ, φ Y ) = n i=1 1 2π φ 1/2 exp{ 1 2 φ(y i µ) 2 } φ n/2 exp{ 1 2 φ i (Y i µ) 2 } Bayesian Inference in a Normal Population p. 2/16

3 Likelihood L(µ, φ Y ) φ n/2 exp{ 1 2 φ i (Y i µ) 2 } φ n/2 exp{ 1 2 φ i [ (Yi Ȳ ) (µ Ȳ )] 2 } φ n/2 exp{ 1 2 φ [ i (Y i Ȳ )2 + n(µ Ȳ )2 ]} φ n/2 exp{ 1 2 φs2 (n 1)} exp{ 1 2 φn(µ Ȳ )2 } where s 2 = i (Y i Ȳ )2 /(n 1) is the usual sample variance. Bayesian Inference in a Normal Population p. 3/16

4 Prior Distributions Conjugate prior distribution for (µ, φ) is Normal-Gamma. µ φ N(m 0, 1/(n 0 φ)) φ Gamma(v 0 /2, (v 0 s 2 0)/2) p(φ) φ v 0/2 1 exp{ φv 0 s 2 0/2} Non-informative prior distribution (improper) Reference Distribution p(µ, φ) = 1/φ µ is uniform on the real line, log(φ) is uniform on real line: invariance to scale and location changes of the data. Bayesian Inference in a Normal Population p. 4/16

5 Reference Posterior Distribution p(µ, φ Y ) L(µ, φ)p(µ, φ) = φ n/2 exp{ 1 2 φs2 (n 1)} exp{ 1 2 φn(µ Ȳ )2 }φ 1 = {φ } { n e { 1 2 φs2 (n 1)} φ 1/2 e 1 φn(µ Ȳ )2} 2 ( ) ( ) n 1 Gamma, (n 1) s2 1 N Ȳ, 2 2 φn = p(φ Y )p(µ φ, Y ) Bayesian Inference in a Normal Population p. 5/16

6 Marginal Distribution for µ Y Obtain the marginal distribution for µ by integrating out φ from the joint posterior distribution, and recognizing the kernel of the distribution! p(µ Y ) p(µ, φ Y )dφ = φ n/2 1 exp[ 1 2 {φs2 (n 1) + φn(µ Ȳ )2 }]dφ Bayesian Inference in a Normal Population p. 6/16

7 Continued p(µ Y ) φ n/2 1 exp{ φ 1 2 [s2 (n 1) + n(µ Ȳ )2 ]}dφ This has the form of a Gamma integral with α = n/2 and β equal to the mess multiplying φ, so that the result is β α (at least that is all that matters) p(µ Y ) (s 2 (n 1) + n(µ Ȳ )2 ) n/2 ( (µ Ȳ )2 n 1 s 2 /n ) (n 1+1)/2 Student-t n 1 (Ȳ, s2 /n) location Ȳ, df = n-1, scale s2 /n) Bayesian Inference in a Normal Population p. 7/16

8 Proper Conjugate Priors Under the Normal-Gamma Prior (proper prior distributions): Find the (conditional) posterior distribution µ φ Find the (marginal) posterior distribution of φ Find the (marginal) posterior distribution of µ Hint: Expand quadratics in µ to read off the posterior precision p n and mean m n then complete the square and factor.5(p n µ 2 2p n m n µ + p n m 2 n) =.5p n (µ m n ) 2 Bayesian Inference in a Normal Population p. 8/16

9 Example: SPF A Sunlight Protection Factor (SPF) of 5 means an individual that can tolerate X minutes of sunlight without any sunscreen can tolerate 5X minutes with sunscreen. Tolerance to Sunlight (min) PreTreatment During Treatment Bayesian Inference in a Normal Population p. 9/16

10 Pairing A paired design may be more powerful than two sample design because of patient to patient variability. Analysis should take into account pairing which induces dependence between observations use differences use ratios or log(ratios) difference in logs Ratios make more sense given the goals: how much longer can a person be exposed to the sun relative to their baseline. Bayesian Inference in a Normal Population p. 10/16

11 Data Differences Normal Q Q Plot Theoretical Quantiles Post PreTreatment Time Sample Quantiles log(ratio) Normal Q Q Plot Theoretical Quantiles Bayesian Inference in a Normal Population p. 11/16 log(post/pretreatment Time) Sample Quantiles

12 Model for SPF Model Y = log(trt) - log(control) as N(µ, 1/φ) E(log(TRT/CONTROL)) = µ = log(spf) Want distribution of exp µ SPF Summary statistics ybar = s2 = n = 13 Bayesian Inference in a Normal Population p. 12/16

13 Samples from the Posterior To draw samples of SPF from the posterior distribution: Draw φ Y phi = rgamma(10000, (n-1)/2, rate=(n-1)*s2/2) Draw µ φ, Y mu = rnorm(10000, ybar, 1/sqrt(phi*n)) or Draw µ Y directly mu = ybar + rt(10000, n-1)*sqrt(s2/n) summarize, transform (exp(mu)), etc quantile(exp(mu), c(.025,.5,.975)) Bayesian Inference in a Normal Population p. 13/16

14 Distributions Posterior Distribution of µ Density µ Posterior Distribution of σ Density σ Bayesian Inference in a Normal Population p. 14/16

15 Distribution for SPF Posterior Distribution of SPF Density exp(µ) 95% Probability Interval 5 to 12 (equal tail area) Bayesian Inference in a Normal Population p. 15/16

16 HPD Region p(4.42 < exp(µ) < y) = 0.95) posterior Density exp(µ) Bayesian Inference in a Normal Population p. 16/16