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)');