Chapter 6: vernier bias

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 no bias, computing and plotting the evidence in each case.

%%% Model likelihoods and evidence

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

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

murange=[-2*sig 2*sig]; Dmu=diff(murange(1:2)); %don't expect any large biases

Nmu=101; mu2=linspace(murange(1),murange(2),Nmu);

mu1=0; Lpri1=0; Lpri2=-log(Dmu); %normalized log-priors

x=linspace(murange(1),murange(2),301); L=ones(size(x));

ind=[findnearestN(x,0,1) findnearestN(x,.05,1)];

for n=1:N, L=L.*npdf(dat(n)+.05,x,sig); end

figure(3); clf; hold on;

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

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

stem(.05,L(ind(2)),'k--','LineWidth',1.2); plot(.05,L(ind(2)),'ko','MarkerFaceColor',COL{2},'MarkerSize',12,'LineWidth',1);

xlabel(['location parameter (\mu)'],'FontName','Arial','FontSize',15)

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

Nx=31; xlist=linspace(.9*mu2(1),.9*mu2(end),Nx);

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

for i=1:Nx; xnow=xlist(i);

dnow=dat+xnow;

L1(i)=sum(log(npdf(dnow,mu1,sig)))+Lpri1; %single-mu model

L2tmp=zeros(size(mu2));

for n=1:length(dnow),

L2tmp=L2tmp+log(npdf(dnow(n),mu2,sig)); end %uncertain model

L2(i)=logsum(L2tmp+Lpri2-log(Nmu)); %last term is to approximate integration

%%% compute evidence

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

%%% plotting

figure(4); clf; hold on

ind=findnearestN(xlist,.05,1);

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

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

xlabel(['data mean'],'FontSize',15);

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

figure(5); clf; hold on

plot(.05*[1 1],[-10 EV(ind)],'k:')

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

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

plot(xlist([1 end]),[0 0],'k--','LineWidth',2);

axis([murange -10 100]);

xlabel(['data mean'],'FontSize',15);

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

Note that although we would observe only a single dataset in our experiment, such as the dataset whose mean was , we have plotted the evidence calculation for multiple datasets to get as sense of the large picture of how the computation turns out in different contexts. To remind ourselves that there is only a single result from our experiment, however, we have placed a vertical dashed line at the above-mentioned result on the evidence plot.