Bayesian linear regression Dr Merlise Clyde body fat Source https commonimedia orgwikiFile Obesity waist circumference svg body fat data 80 100 120 140 0 10 20 30 40 abdomen circumference (cm.Bayesian linear regression Dr Merlise Clyde body fat Source https commonimedia orgwikiFile Obesity waist circumference svg body fat data 80 100 120 140 0 10 20 30 40 abdomen circumference (cm.
Bayesian linear regression Dr Merlise Clyde body fat Source: https://commons.wikimedia.org/wiki/File:Obesity-waist_circumference.svg body fat data ˆ ˆ + xi fitted yˆi = ↵ σ \at = Bodyf MSE = 39.28 + 0.63 Abdomen 10 Bodyfat 20 30 40 values residuals "ˆi = yi 80 100 120 140 abdomen circumference (cm) yˆi Pn "ˆi i=1 n model and prior ‣ model Y i = ↵ + xi + "i iid ‣ "i ⇠ N(0, ) conjugate bivariate normal-gamma distribution ↵| | 1/ ⇠ N(a0 , ⇠ N(bbo0 , ⇠ S↵ ) cov(↵, S ) G(⌫0 /2, ⌫0 /2) | )= S↵, reference prior and posterior distributions reference prior p(↵, , reference posterior ) / 1/ | y , , y n ⇠ tn ↵ | y1 , yn ⇠ tn ↵ + xi | y1 , yn ⇠ tn s yi = 2 ⇣ ⇣ sY |X ( n ˆ, sd( )2 ↵ ˆ , sd(↵) ⌘ 2 ˆ ↵ ˆ + xi , syi + (xi x ¯) Sxx ) ⌘ estimates post mean post sd 2.5% 97.5% (intercept) -39.28 2.66 -44.52 -34.04 abdomen 0.63 0.03 0.58 0.69 posterior mean ± t1 ↵/2,n posterior standard deviation predicting body fat ‣ posterior predictive distribution for a new case yn+1 = ↵ + xn+1 + "n+1 ‣ is also a Student t distribution with n − df yn+1 |y1 , yn ⇠ tn yˆn+1 syn+1 ⇣ yˆn+1 , syn+1 ˆ =↵ ˆ + xn+1 ⇣ = ˆ 1+ n + ⌘ ⌘ (x x ¯ ) n+1 P (xi x ¯ )2 predicting body fat (continued) syn+1 =ˆ ⇣ 1+ n + ⌘ (x x ¯ ) n+1 P (xi x ¯ )2 posterior uncertainty about α + βxn+1 ‣ depends on xn+1 spread ‣ is higher for xn+1 far from x additional variability +sY |X due to εn+1 30 20 10 bodyfat 40 prediction intervals 80 100 120 abdomen 140 summary ‣ under reference prior, point estimates and Bayesian credible intervals are equivalent to frequentist estimates and confidence intervals ‣ use standard software to obtain ‣ change in interpretation ‣ reference analysis ... prediction intervals 80 100 120 abdomen 140 summary ‣ under reference prior, point estimates and Bayesian credible intervals are equivalent to frequentist estimates and confidence intervals ‣ use