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## Bayesian Statistics

by: Vance Bode Sr.

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# Bayesian Statistics 22S 138

Vance Bode Sr.
UI
GPA 3.72

Mary Cowles

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Mary Cowles
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## Popular in Natural Sciences and Mathematics

This 3 page Class Notes was uploaded by Vance Bode Sr. on Friday October 23, 2015. The Class Notes belongs to 22S 138 at University of Iowa taught by Mary Cowles in Fall. Since its upload, it has received 43 views. For similar materials see /class/228077/22s-138-university-of-iowa in Natural Sciences and Mathematics at University of Iowa.

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Date Created: 10/23/15
Introduction to Hierarchical Models 228 138 Bayesian Statistics Lecture 12 Oct 87 2008 Kate Cowles7 PhD Example Pump failure data 0 A hierarchical model is t to data on failr ure rates of the pump at each of 10 power plants The number of failures for the 1 th pump is assumed to follow a Poisson distribution P01550nlt0iti 1110 where 01 is the failure rate for pump 1 and t1 is the length of operation time of the pump in 1000s of hours 0 Important point we do not assume that all the pumps have the same failure rate In fact7 one of the questions of interest is to estimate the rates for the individual pumps oWe do not consider the zbti pairs ew changeable Hierarchical models 0 Bayesian models with more than two leve els or stages 0 may arise for several reasons i we have insuf cient knowledge to spec ify the parameters of priors we wish to model data or parameters that cannot be considered exchanger able but that are related 0 Write the likelihood of the data 0 Recall that the de nition of exchangeable observations is their likelihood is invarir ant to permutations of the indices o If we exchanged the subscripts on two xi t1 pairs7 and did not change the indices of the corresponding 02s the evaluation of the likelihood would change oThe rst stage of a hierarchical model is the joint distribution of the observed data given certain parameters7 or the like lihood The second stage 0 The second stage gives priors on the pa rameters that appeared in the rst stage 0 In the pump failures example7 a conjugate gamma prior distribution is adopted for the failure rates 01 Gammaa i 1 10 o This says that7 although the failure rates for the individual pumps are not the same7 they are related They are all drawn from a common distribution oWe do not know enough about failure rates of pumps in nuclear power plants to be able to specify xed numbers for the prior parameters a and ln fact7 we want the data to inform us about these values 0 Consequently7 we will make a and addir tional unknown paramters in the model 5 WinBUGS program to t Pump model model for i in 1N thetai quot dgammalta1phabeta 1ambdai lt7 thetaiti xi 39 dpois1ambdai alpha 39 dexp10 beta 39 dgamma0110 hyperparameters 0 At the third stage of the hierarchical model for pump failures7 the following priors are speci ed for the hyperparameters a and a N Exponential N Gamma01010 Data and initial values listCt C943157629126052431410510521 105 x clt51514319114221110 listCalpha 10 beta 10 theta Co101o1o1o101o1o1o101 Results node mean Sd MC error alpha 0 70010 0 26990 0047060 0 beta 0 92900 0 53250 0097800 0 theta1 0 05980 0 02542 0002680 0 theta2 0 10080 0 07855 0008177 0 theta3 0 08927 0 03759 0003702 0 theta4 0 11600 0 03048 0003170 0 theta5 0 60560 0 31500 0030870 0 theta6 0 61050 0 13930 0014000 0 theta7 0 90250 0 72520 0079370 0 theta8 0 89640 0 72500 0082620 0 theta9 1 59000 0 77670 0090040 0 theta10 1 99300 0 42510 0049150 1 25 28510 26400 Hwoooooooooo than thetas for other observations othetas far from the common mean are shrunk more than those near it MWMMOHOOOOMH start 1001 1001 Sam 10m oooooooooo Q Q Q Q Q Q Q Q Q Q 10m Compare to maximum likelihood es timates for individual pumps hours failures mle theta 94 30 5 0530 0598 1570 1 0637 1008 6290 5 0795 0893 12600 14 1111 1160 5 24 3 5725 6056 3140 19 6051 6105 105 1 9528 9025 105 1 9524 8964 210 4 19048 15900 1050 22 20952 19309 oindividual estimates are shrunk away from mle toward a common mean 0 individual estimates borrow strength from the rest of the data 0 thetas for observations with large 7 sam7 ple size time observed are shrunk less 10

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