December 2010T. A. Louis: Basic Bayes 1 Basic Bayes December 2010T. A. Louis: Basic Bayes 2 December 2010T. A. Louis: Basic Bayes 3 December 2010T. A. Louis: Basic Bayes 4 December 2010T. A. Louis: Basic Bayes 5 December 2010T. A. Louis: Basic Bayes 6 December 2010T. A. Louis: Basic Bayes 7 December 2010T. A. Louis: Basic Bayes 8 December 2010T. A. Louis: Basic Bayes 9 December 2010T. A. Louis: Basic Bayes 10 To find a method for: the probability that an event has to happen, in given circumstances Bayes Rule: Pr( |Y) Pr(Y| ) Pr( ) Reverend Thomas Bayes Reverent Thomas Bayes December 2010T. A. Louis: Basic Bayes 11 December 2010T. A. Louis: Basic Bayes 12 December 2010T. A. Louis: Basic Bayes 13 December 2010T. A. Louis: Basic Bayes 14 December 2010T. A. Louis: Basic Bayes 15 Example: N=50 December 2010T. A. Louis: Basic Bayes 16 Example: N=100 (50 more) December 2010T. A. Louis: Basic Bayes 17 Example: N=150 (50 more) December 2010T. A. Louis: Basic Bayes 18 December 2010T. A. Louis: Basic Bayes 19 December 2010T. A. Louis: Basic Bayes 20 December 2010T. A. Louis: Basic Bayes 21 CABAG DEATH RATE December 2010T. A. Louis: Basic Bayes 22 TOXOPLASMOSIS RATES (centered) December 2010T. A. Louis: Basic Bayes 23 December 2010T. A. Louis: Basic Bayes 24 December 2010T. A. Louis: Basic Bayes 25 V(b) December 2010T. A. Louis: Basic Bayes 26 December 2010T. A. Louis: Basic Bayes 27 regression line Pop line 45 o line December 2010T. A. Louis: Basic Bayes 28 December 2010T. A. Louis: Basic Bayes 29 December 2010T. A. Louis: Basic Bayes 30 December 2010T. A. Louis: Basic Bayes 31 December 2010T. A. Louis: Basic Bayes 32 December 2010T. A. Louis: Basic Bayes 33 Improving regression estimates Improving regression estimates Similar to the BUGS rat data Dependent variable (Y ij ) is weight for rat i at age X ij i = 1,..., I (=10); j = 1,..., J (=5) X ij = X j = (-14, -7, 0, 7, 14) = (8-22, 15-22, 22-22, ) Y ij = b i0 + b i1 X j + ij As usual, the intercept depends on the centering Analyses Each rat has its own line All rats follow the same line: b i0 = 0, b i1 = 1 A compromise between these two December 2010T. A. Louis: Basic Bayes 34 Each rat has its own (LSE, MLE) line Each rat has its own (LSE, MLE) line (with the population line) Pop line December 2010T. A. Louis: Basic Bayes 35 A multi-level model: A multi-level model: Each rat has its own line, but the lines come from the same distribution The b i0 are independent Normal( 0, 0 2 ) The b i1 are independent N( 1, 1 2 ) Overdispersion Sample variance of the OLS estimated intercepts: 345 = SE int 2 + 0 2 = 0 2 0 2 = 25, 0 = 5 Sample variance of the OLS estimated slopes 4.25 = SE slope 2 + 1 2 = 1 2 1 2 = 1.00, 1 = 1.00 December 2010T. A. Louis: Basic Bayes 36 A compromise: each rat has its own line, but the lines come from the same distribution Pop line December 2010T. A. Louis: Basic Bayes 37 Observed & Predicted Deviations of Annual Charges (in dollars) for Specialist Services vs. Primary Care Services Deviation, Specialists Charges Square (blue) = Posterior Mean of Predicted Deviation Dot (red) = Posterior Mean of Observed Deviation December 2010T. A. Louis: Basic Bayes 38 Observed and Predicted Deviations for Specialist Services: Log(Charges>$0) and Probability of Any Use of Service Mean Deviation of Log(Charges >$0) Dot (red) = Posterior Mean of Observed Deviation Square (blue) = Posterior Mean of Predicted Deviation December 2010T. A. Louis: Basic Bayes 39 December 2010T. A. Louis: Basic Bayes 40 December 2010T. A. Louis: Basic Bayes 41 December 2010T. A. Louis: Basic Bayes 42 December 2010T. A. Louis: Basic Bayes 43 BACK TO HISTORICAL CONTROLS December 2010T. A. Louis: Basic Bayes 44 December 2010T. A. Louis: Basic Bayes 45 December 2010T. A. Louis: Basic Bayes 46 December 2010T. A. Louis: Basic Bayes 47 December 2010T. A. Louis: Basic Bayes 48 December 2010T. A. Louis: Basic Bayes 49 December 2010T. A. Louis: Basic Bayes 50 Bayes for Frequentist Goals December 2010T. A. Louis: Basic Bayes 51 December 2010T. A. Louis: Basic Bayes 52Summary December 2010T. A. Louis: Basic Bayes 53 SURGICAL SURGICAL Hospital # of ops# of deaths A [1] 47 0 B C119 8 D E211 8 F G148 9 H I J 97 8 K L360 24 December 2010T. A. Louis: Basic Bayes 54 model { for( i in 1 : N ) { p[i] ~ dbeta(1.0, 1.0) #need to specify the prior r[i] ~ dbin(p[i], n[i]) } righttail