Chapter 2: gaussian sampling-likelihood surface

To generate the sampling-likelihood surface and the sliced sampling and likelihood plots (Fig. 2.7), we need to first compute a large set of probabilities for combinations of average datum (x-bar) and location parameter (mu), allowing us to plot the 2D surface. Then we select one average datum (corresponding to experimental observations) and compute the likelihood (taking a slice through the surface at the experimental observation). Similarly, we select one hypothetical location parameter (corresponding to the frequentist ‘null hypothesis’) and compute the sampling distribution (taking a slice through the surface at the null hypothesis).

up=.0017; dv=1; BR=-.75; %BRighten + or darken - the mesh

H0=50; murange=[15 85]; sig=10;

DatVals=45;

if length(DatVals)>1, sig=sqrt(var(DatVals,1)); Xbar=mean(DatVals); else Xbar=DatVals; end

while murange(2)/dv>60, dv=dv+1; end

N=round(murange(2)/dv);

figure(1); clf; hold on

x=linspace(murange(1),murange(2),N);

[X Y]=meshgrid(x,x); GA=[];

for n=1:length(x),

GA(:,n)=npdf(x,x(n),sig); end

mesh(X,Y,GA); colormap(bone); brighten(BR); view(-36,58);

r=axis; axis([murange murange 0 r(end)])

GA=npdf(x,H0,sig);

plot3(ones(size(x))*H0,x,zeros(size(x)),'k-','LineWidth',.9)

for n=length(x):-1:1, plot3(H0,x(n),GA(n)+up,'ko','MarkerFaceColor','w','MarkerSize',8,'LineWidth',.9), end

%plot3(ones(size(x))*H0,x,GA+up,'ko','MarkerFaceColor','w','MarkerSize',8,'LineWidth',.8)

figure(2); clf; hold on %stem(x,GA,'ko','MarkerFaceColor','w','MarkerSize',8,'LineWidth',.8)

GAi=interp1(x,GA,[x(1):1/(length(x)*5):x(end)],'pchip');

plot([x(1):1/(length(x)*5):x(end)],GAi,'k-','LineWidth',.9)

plot(x,GA,'ko','MarkerFaceColor','w','MarkerSize',8,'LineWidth',1.5)

GAt=interp1(x,GA,H0,'pchip');

plot(H0*[1 1],[0 GAt],'k-','LineWidth',1.3)

plot(H0,GAt,'ko','MarkerFaceColor','w','MarkerSize',8,'LineWidth',1.5)

GAt=interp1(x,GA,Xbar,'pchip');

plot(Xbar*[1 1],[0 GAt],'k--','LineWidth',1.3)

plot(Xbar,GAt,'ko','MarkerFaceColor','k','MarkerSize',8,'LineWidth',1.5)

r=axis; axis([murange 0 r(end)*1.05])

figure(1); GA=npdf(Xbar,x,sig);

plot3(x,ones(size(x))*Xbar,GA+up,'ko','MarkerFaceColor','k','MarkerSize',8,'LineWidth',.8)

plot3(x,ones(size(x))*Xbar,zeros(size(x)),'k-','LineWidth',.9)

figure(3); clf; hold on %stem(x,GA,'ko','MarkerFaceColor','k','MarkerSize',8,'LineWidth',.8)

GAi=interp1(x,GA,[x(1):1/(length(x)*5):x(end)]);

plot([x(1):1/(length(x)*5):x(end)],GAi,'k-','LineWidth',.9)

plot(x,GA,'ko','MarkerFaceColor','k','MarkerSize',8,'LineWidth',1.5)

GAt=interp1(x,GA,H0,'pchip');

plot(H0*[1 1],[0 GAt],'k--','LineWidth',1.3)

plot(H0,GAt,'ko','MarkerFaceColor','w','MarkerSize',8,'LineWidth',1.5)

GAt=interp1(x,GA,Xbar,'pchip');

plot(Xbar*[1 1],[0 GAt],'k-','LineWidth',1.3)

plot(Xbar,GAt,'ko','MarkerFaceColor','k','MarkerSize',8,'LineWidth',1.5)

r=axis; axis([murange 0 r(end)*1.05])