Chapter 3: conjugate and jeffreys priors for the poisson parameter

Jeffreys prior

d=[15, 12, 5, 13]; n=length(d); pmax=nan(n,1);

SS=linspace(0,30,301); inds=find(rem(SS,1)==0);

f=@(nvar) exp(-nvar.*SS).*SS.^(sum(d(1:nvar))-.5);

for nnow=1:n, figure(20+nnow); clf, hold on; box off; axis([0 max(SS) 0 1.1]);

pSS=f(nnow); pmax(nnow)=median(SS(pSS==max(pSS)));

plot(SS,pSS/max(pSS),'k--','LineWidth',1.5);

plot(SS(inds),pSS(inds)/max(pSS),'ko','MarkerFaceColor','k','MarkerSize',12)

plot(d(1:nnow),1.05,'ko','MarkerSize',8,'LineWidth',1.7); end

Conjugate prior #1

d2=[12, 5, 13]; n2=length(d2);

f2=@(nvar) exp(-(nvar+1).*SS).*SS.^(14.5+sum(d2(1:nvar)));

for nnow=0:n2, figure(20+n+nnow); clf, hold on; box off; axis([0 max(SS) 0 1.1]);

pSS=f2(nnow); pmax(nnow+1)=median(SS(pSS==max(pSS)));

plot(SS,pSS/max(pSS),'k--','LineWidth',1.5);

plot(SS(inds),pSS(inds)/max(pSS),'ko','MarkerFaceColor','k','MarkerSize',12)

if ~isempty(d(1:nnow)), plot(d(1:nnow),1.05,'ko','MarkerSize',8,'LineWidth',1.7); end, end

Conjugate prior #2

d3=[13]; n3=length(d3);

f3=@(nvar) exp(-(nvar+diff([n3 n])).*SS).*SS.^(diff([n3 3])*10.5+sum(d3(1:nvar)));

for nnow=0:n3, figure(20+n+n2+nnow); clf, hold on; box off; axis([0 max(SS) 0 1.1]);

pSS=f3(nnow); pmax(nnow+1)=median(SS(pSS==max(pSS)));

plot(SS,pSS/max(pSS),'k--','LineWidth',1.5);

plot(SS(inds),pSS(inds)/max(pSS),'ko','MarkerFaceColor','k','MarkerSize',12)

if ~isempty(d(1:nnow)), plot(d(1:nnow),1.05,'ko','MarkerSize',8,'LineWidth',1.7); end, end