To compare the goodness of fit for models with different numbers of parameters we used the integrated Bayes Information Criterion (iBIC) score. The iBIC score is related to the model log likelihood p(D|M) as:24logp(D|M)=∫dθp(D|θ)p(θ|M)25≈−12iBIC=logp(D|θML)−12|M|log|D|Where |M| is the number of fitted parameters of the prior, |D| is the number of data points (total choices made by all subjects) and iBIC is the integrated BIC score. The log data likelihood given maximum likelihood parameters for the prior logp(D|θML) is calculated by integrating out the individual session parameters:26logp(D|θML)=∑iNlog∫dhp(Di|h)p(h|θML)≈∑iNlog1K∑j=1Kp(Di|hj)Where the integral is approximated as the average over K samples drawn from the prior p(h|θML). Bootstrap 95% confidence intervals were estimated for the iBIC scores by resampling from the population of samples drawn from the prior.
