Chapter 6: vernier setting precision

To understand how the evidence for a bias follows the size of the true bias, we will simulate a dataset and vary its mean from left to right of the no-bias point, computing and plotting the evidence in each case.

%%% Model likelihoods and evidence

sig0=.1; sigprime=2.5*sig0; dat=randn([10,1]); dat=dat-mean(dat); N=length(dat);

COL={.4*[1 1 1],.8*[1 1 1]};

murange=[-2*sig0 2*sig0]; Dmu=diff(murange); %don't expect any large biases

sigrange=[sig0/2 10*sig0/2]; dlsig=diff(log(sigrange));

Nmu=201; mus=linspace(murange(1),murange(2),Nmu);

Nsig=201; sigs=linspace(sigrange(1),sigrange(2),Nsig)';

lgpdf=@(x,munow,signow) -log(signow*ones(1,Nmu))-.5*log(2*pi)-.5*((x-ones(Nsig,1)*munow).^2./(signow*ones(1,Nmu)).^2);

Lprim=-log(Dmu); Lpris=-log(dlsig); %normalized log-priors

s=linspace(sigrange(1),sigrange(2),301); L=ones(size(s));

ind=[findnearestN(s,sig0,1) findnearestN(s,sigprime,1)];

for n=1:N, L=L.*npdf(dat(n)*sigprime,0,s); end

figure(3); clf; hold on;

plot(s,L,'k-','LineWidth',2.2);

stem(s(ind(1)),L(ind(1)),'k--','LineWidth',1.2); plot(s(ind(1)),L(ind(1)),'ko','MarkerFaceColor',COL{1},'MarkerSize',12,'LineWidth',1);

stem(s(ind(2)),L(ind(2)),'k--','LineWidth',1.2); plot(s(ind(2)),L(ind(2)),'ko','MarkerFaceColor',COL{2},'MarkerSize',12,'LineWidth',1);

xlabel(['dispersion parameter (\sigma)'],'FontName','Arial','FontSize',15)

ylabel('likelihood','FontName','Arial','FontSize',15)

Ns=31;

L1=nan(Ns,1); L2=nan(Ns,1); EV=nan(Ns,1);

slist=linspace(1.1*sigrange(1),.9*sigrange(2),Ns);

for i=1:Ns; snow=slist(i);

dnow=dat*snow;

L1tmp=zeros(1,Nmu); L2tmp=zeros(Nsig,Nmu);

for n=1:length(dnow),

L1tmp=L1tmp+lgpdf(dnow(n),mus,sig0); %single-sigma model

L2tmp=L2tmp+lgpdf(dnow(n),mus,sigs); end %uncertain-sigma model

L1(i)=logsum(L1tmp,[1 2]); %last term is to approximate integration

L2(i)=logsum(L2tmp+Lpris,[1 2]);

%%% compute evidence

EV(i)=10*(L2(i)-L1(i)); end

%%% plotting

figure(4); clf; hold on

ind=findnearestN(slist,sigprime,1);

plot(slist,L1,'ko','MarkerFaceColor',COL{1},'MarkerSize',8)

plot(slist,L2,'ko','MarkerFaceColor',COL{2},'MarkerSize',8)

xlabel(['dispersion'],'FontSize',15);

ylabel('model likelihood','FontName','Arial','FontSize',15);

figure(5); clf; hold on

plot(sigprime*[1 1],[-30 EV(ind)],'k:')

plot(slist(EV<0),EV(EV<0),'ko-','MarkerFaceColor',COL{1},'MarkerSize',9,'LineWidth',1);

plot(slist(EV>0),EV(EV>0),'ko-','MarkerFaceColor',COL{2},'MarkerSize',9,'LineWidth',1);

plot(sigrange,[0 0],'k--','LineWidth',2);

axis([sigrange -30 1000]);

xlabel(['dispersion'],'FontSize',15);

ylabel('evidence (db)','FontName','Arial','FontSize',15);