Null Data - Pseudo False Positive Rates

In this section we analyse some null/rest data using the full spatio-temporal model. The intention is to do a test that is similar to the investigation of false positive rates (FPR) in frequentist statistics. To investigate FPRs in a frequentist framework, a statistic is calculated a number of times from data conforming to the null hypothesis, to build up an experimentally obtained null distribution. The resulting null distribution should correspond to the known theoretical distribution under the null hypothesis. In Bayesian statistics we do not work with theoretical null hypothesis distributions for the statistic of a parameter, but instead we have a probability distribution over a parameter of interest. Nevertheless, we can still use artificial null data to ask: `If we threshold to assign voxels as ``positive'' when $p(a_i>0\vert y)>1-H$, then at what rate do we produce ``positives'' when we know $a_i=0$?' Unlike frequentist statistics we do not require this rate of ``false positives'' to approximate $H$, however, it is still interesting to investigate its properties.