PROB & STATISTICS
PROB & STATISTICS PSTAT 120C
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This 3 page Class Notes was uploaded by Douglas Simonis on Thursday October 22, 2015. The Class Notes belongs to PSTAT 120C at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 66 views. For similar materials see /class/226915/pstat-120c-university-of-california-santa-barbara in Statistics and Probability at University of California Santa Barbara.
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Date Created: 10/22/15
PSTAT 1200 Bayesian estimation of proportions May 28 2009 ll Bayesian estimation 2 Prior distribution 3 Posterior distribution 4 Bayesian estimation of a Binomial Some Probability Review 0 Law of total probability7 for a partition B17 i i i Bk k WA EMA l BjlP Bj j1 0 Ba es Rule y MW W i BjgtPltBjgt 1 PM Ei wl ww o For a continuous random variables With density fz7 y7 nw and then t 7 17y fIly 7 Jew o Bayes rule becomes y I f1 l yfy ffr l yfy dy Bayesian Statistics We want a Bayesian estimate of p in a Binomialn7 p model 0 Statistical Model n 119 H ri l 9 i1 the likelihood depends on the random variables and the parameter 0 A conditional probability mass function PXIlpfrlp 0 Prior information about the parameter is given by a density on 07 l 7rpi Prior distribution 0 The information about the parameter from the data is therefore fp l X Posterior distri bution o Bayesian estimator Posterior expected value US eta distribution 0 We need to x a prior distribution on p Which sits on 01 0 Take a number of polynomial functions 10 C10 1 7 Pb o lntegrate to get 1 7 7 lquotalquot a 1 1 7 7 0 z 1 1 dz 7 Ba Flta for a and B gt 01 0 Therefore a beta distribution With parameters a and B has density WU mpa l 70 3 1 0 To get expectations7 we integrate 1 1 p g 1E WP dp 7100quot1 1 ldp 0 0 3045 Tim0 10011 7 10V dp 7 Blta 16 7 30175 W 1 MUMNa 6 W 130710 a 7 WWW 10 vamp a 2a 1 7 16 Beta prior 0 The Beta prior is convenient o It is exible we can choose from a range of means and variances o The posterior calculation 21010 7 0 0 quot11 7 PW lB 01 9pm 7 pgtn7rp71lt1e pgt 713 m p quot11 7 P 1 fPlXI dp 1 xa71 7 7 1 n 06H 1 Bzan7z p p o Conveniently the posterior distribution is also a beta distribution 0 Therefore the expected value za 15EltP XIgtm gives us our Bayes estimator 0 In our example on Thursday we used a uniform distribution which is a Beta with a 1 Thus we got o If a B which gives a symmetric prior distribution A7Xa7 17 2a l 2a p7n2a7n n2a 2 n2a 0 Another version of the prior 1 17 7r 101 7 0 which is inversely proportional to the standard deviation X g n 1 5 n25X12then 0i4808 n7Xlthen 50l1875 n3X3then 0l875 n10X8then 37150l7727 1 n 1000X 800 then 1a 07997