Chapter 3: priors for the binomial likelihood
Chapter 3: priors for the binomial likelihood
beta1=.5; beta2=2;
stepsize=.001; Rtheta=[0 1];
theta=Rtheta(1)+stepsize/2:stepsize:Rtheta(2)-stepsize/2;
fpriorJ=@(t) (t.*(1-t)).^(-.5);
fpriorC=@(t,beta) (t.^(beta-1)).*((1-t).^(beta-1));
fpriorU=@(t) ones(length(t),1)./length(t);
flike=@(t,N,S) (t.^S).*(1-t).^(N-S);
fprior3015=flike(theta,30,15);
Ns=[1:5 10:5:20 50 100 500];
figure; subplot(4,3,1); hold on
fpJ=fpriorJ(theta); fpC1=fpriorC(theta,beta1); fpC2=fpriorC(theta,beta2); fpU=fpriorU(theta);
piJ=fpJ/sum(fpJ); piC=fpC2/sum(fpC2); piU=fpU/sum(fpU); pi3015=fprior3015/sum(fprior3015);
pis=[piJ(:) piC(:) piU(:) pi3015(:)];
plot(theta,pis(:,1),'k-','LineWidth',1.05)
plot(theta,pis(:,2),'k-.','LineWidth',1.1)
plot(theta,pis(:,3),'k:','LineWidth',2.1)
plot(theta,pis(:,4),'k--','LineWidth',1.1)
axis([0 1 0 max(pi3015)])
Bdraw=brand(1,.63,[500 1]);
for n=1:length(Ns),
N=Ns(n); S=sum(Bdraw(1:N));
fnow=flike(theta,N,S); fnow=fnow(:)*ones(1,size(pis,2));
post=fnow.*pis; pnorm=max(post); post=post./(ones(size(post,1),1)*pnorm);
subplot(4,3,n+1), hold on
plot(theta,post(:,1),'k-','LineWidth',1.05)
plot(theta,post(:,2),'k-.','LineWidth',1.1)
plot(theta,post(:,3),'k:','LineWidth',2.1)
plot(theta,post(:,4),'k--','LineWidth',1.1)
meandef(post,theta)
sqrt(vardef(post,theta))
axis([0 1 0 1]); end
xlabel('theta'); ylabel('p(theta|data)');