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Presentation of Question 3 Sun Jie U093351B
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Prior1: Be(.5,.5) pv=seq(0.00001,0.05,0.00001) xv=51; nv=8197 logf1= (xv-.5)*log(pv)+(nv-xv-.5)*log(1-pv) f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dbeta(pv,.5,.5) plot(pv, prior,col="red",ylim=range(0:500)) points(pv,post)
![Prior1: Be(.5,.5) pv=seq( ,0.05, ) xv=51; nv=8197 logf1= (xv-.5)*log(pv)+(nv-xv-.5)*log(1-pv) f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dbeta(pv,.5,.5) plot(pv, prior,col= red ,ylim=range(0:500)) points(pv,post)](http://images.slideplayer.com./12/3414877/slides/slide_2.jpg "Prior1: Be(.5,.5) pv=seq( ,0.05, ) xv=51; nv=8197 logf1= (xv-.5)*log(pv)+(nv-xv-.5)*log(1-pv) f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dbeta(pv,.5,.5) plot(pv, prior,col= red ,ylim=range(0:500)) points(pv,post)")
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Prior1: Be(.5,.5)
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Point Estimate pmean=sum(pv*post)/sum(post) pcdf=cumsum(post)/sum(post) pmedian=.5*(max(pv[pcdf.5])) pmode=pv[which.max(post)]
![Point Estimate pmean=sum(pv*post)/sum(post) pcdf=cumsum(post)/sum(post) pmedian=.5*(max(pv[pcdf.5])) pmode=pv[which.max(post)]](http://images.slideplayer.com./12/3414877/slides/slide_4.jpg "Point Estimate pmean=sum(pv*post)/sum(post) pcdf=cumsum(post)/sum(post) pmedian=.5*(max(pv[pcdf.5])) pmode=pv[which.max(post)]")
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Interval Estimate CI1=c(max(pv[pcdf.975])) threshold=max(post) coverage=0 for(i in seq(.999,.001,-.001)) { threshold=i*max(post) within=which(post>=threshold) coverage=pcdf[max(within)]-pcdf[min(within)] if(coverage>=.95)break() } CI2=pv[range(within)]
![Interval Estimate CI1=c(max(pv[pcdf.975])) threshold=max(post) coverage=0 for(i in seq(.999,.001,-.001)) { threshold=i*max(post) within=which(post>=threshold) coverage=pcdf[max(within)]-pcdf[min(within)] if(coverage>=.95)break() } CI2=pv[range(within)]](http://images.slideplayer.com./12/3414877/slides/slide_5.jpg "Interval Estimate CI1=c(max(pv[pcdf.975])) threshold=max(post) coverage=0 for(i in seq(.999,.001,-.001)) { threshold=i*max(post) within=which(post>=threshold) coverage=pcdf[max(within)]-pcdf[min(within)] if(coverage>=.95)break() } CI2=pv[range(within)]")
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> pmean [1] 0.00628202 > pmedian [1] 0.006235 > pmode [1] 0.00616 > CI1 [1] 0.00468 0.00810 > CI2 [1] 0.00461 0.00802
![> pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1]](http://images.slideplayer.com./12/3414877/slides/slide_6.jpg "> pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1]")
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Prior2: N(1.5%,.25%) pv=seq(0.00001,0.05,0.00001) xv=51; nv=8197 logf1= xv*log(pv)+(nv-xv)*log(1-pv)-(pv-.015)^2/(2*.0025^2) f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dnorm(pv,mean=.015,sd=.0025) plot(pv, prior,col="red",ylim=range(0:500)) points(pv,post)
![Prior2: N(1.5%,.25%) pv=seq( ,0.05, ) xv=51; nv=8197 logf1= xv*log(pv)+(nv-xv)*log(1-pv)-(pv-.015)^2/(2*.0025^2) f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dnorm(pv,mean=.015,sd=.0025) plot(pv, prior,col= red ,ylim=range(0:500)) points(pv,post)](http://images.slideplayer.com./12/3414877/slides/slide_7.jpg "Prior2: N(1.5%,.25%) pv=seq( ,0.05, ) xv=51; nv=8197 logf1= xv*log(pv)+(nv-xv)*log(1-pv)-(pv-.015)^2/(2*.0025^2) f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dnorm(pv,mean=.015,sd=.0025) plot(pv, prior,col= red ,ylim=range(0:500)) points(pv,post)")
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Prior2: N(1.5%,.25%)
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> pmean[1] 0.007414297 > pmedian[1] 0.007375 > pmode[1] 0.00731 > CI1 [1] 0.00565 0.00936 > CI2 [1] 0.00559 0.00930
![> pmean[1] > pmedian[1] > pmode[1] > CI1 [1] > CI2 [1]](http://images.slideplayer.com./12/3414877/slides/slide_9.jpg "> pmean[1] > pmedian[1] > pmode[1] > CI1 [1] > CI2 [1]")
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Prior3: Exp(100) pv=seq(0.00001,0.05,0.00001) xv=51; nv=8197 logf1= xv*log(pv)+(nv-xv)*log(1-pv)-100*pv f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dexp(pv,rate=100) plot(pv, prior,col="red",ylim=range(0:500)) points(pv,post)
![Prior3: Exp(100) pv=seq( ,0.05, ) xv=51; nv=8197 logf1= xv*log(pv)+(nv-xv)*log(1-pv)-100*pv f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dexp(pv,rate=100) plot(pv, prior,col= red ,ylim=range(0:500)) points(pv,post)](http://images.slideplayer.com./12/3414877/slides/slide_10.jpg "Prior3: Exp(100) pv=seq( ,0.05, ) xv=51; nv=8197 logf1= xv*log(pv)+(nv-xv)*log(1-pv)-100*pv f2=exp(logf1-max(logf1)) intf2=sum(f2)*(pv[2]-pv[1]) post=f2/intf2 prior=dexp(pv,rate=100) plot(pv, prior,col= red ,ylim=range(0:500)) points(pv,post)")
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Prior3: Exp(100) > pmean [1] 0.006266297 > pmedian [1] 0.006225 > pmode [1] 0.00615 > CI1 [1] 0.00467 0.00807 > CI2 [1] 0.00461 0.00799
![Prior3: Exp(100) > pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1]](http://images.slideplayer.com./12/3414877/slides/slide_11.jpg "Prior3: Exp(100) > pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1]")
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Comparison > pmean [1] 0.00628202 > pmedian [1] 0.006235 > pmode [1] 0.00616 > CI1 [1] 0.00468 0.00810 > CI2 [1] 0.00461 0.00802 Prior1: Be(.5,.5) > pmean[1] 0.007414297 > pmedian[1] 0.007375 > pmode[1] 0.00731 > CI1 [1] 0.00565 0.00936 > CI2 [1] 0.00559 0.00930 Prior2: N(1.5%,.25%) Prior3: Exp(100) > pmean [1] 0.006266297 > pmedian [1] 0.006225 > pmode [1] 0.00615 > CI1 [1] 0.00467 0.00807 > CI2 [1] 0.00461 0.00799
![Comparison > pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1] Prior1: Be(.5,.5) > pmean[1] > pmedian[1] > pmode[1] > CI1 [1] > CI2 [1] Prior2: N(1.5%,.25%) Prior3: Exp(100) > pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1]](http://images.slideplayer.com./12/3414877/slides/slide_12.jpg "Comparison > pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1] Prior1: Be(.5,.5) > pmean[1] > pmedian[1] > pmode[1] > CI1 [1] > CI2 [1] Prior2: N(1.5%,.25%) Prior3: Exp(100) > pmean [1] > pmedian [1] > pmode [1] > CI1 [1] > CI2 [1]")
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