Chapter 6: evidence in the balanced case
When the prior probabilities of each model allow for only a single point in the parameter space, the evidence calculation reduces to a log-transform of the likelihood ratio based on the maximum attainable likelihoods under the two models. This was exactly the lever-pressing example from the rat experiment above.
%%% press-by-press calculation
rho1=.9; rho2=.8;
N=20; S=17; dat=permlist([ones(1,S) zeros(1,N-S)]);
EV=cell(2,1);
for i=1:length(dat),
datnow=dat(1:i); %for the first i data
B1=rho1.^sum(datnow).*(1-rho1).^sum(~datnow); %Bernoulli prob: high-rate
B2=rho2.^sum(datnow).*(1-rho2).^sum(~datnow); %Bernoulli prob: high-rate
EV{1}(i)=10*log10(B1/B2); end %the i’th evidence value
%%% all-at-once calculation
B1=rho1.^sum(dat).*(1-rho1).^sum(~dat); %Bernoulli prob: high-rate
B2=rho2.^sum(dat).*(1-rho2).^sum(~dat); %Bernoulli prob: high-rate
EV{2}=10*log10(B1/B2);
%%% compare the two calculations
isnear(EV{1}(end),EV{2})