Inference

The overall objective is to obtain the joint posterior distribution of all unobserved parameters in the model, given the observed data. Analytical approaches are not possible with complicated models such as those being considered in this work. Whilst it is possible to perform approximations to the distribution, it is difficult to assess the effect of these approximations on the inference performed. Therefore, the approach taken instead is to use Markov Chain Monte Carlo (MCMC) sampling from the full joint posterior distribution (see (22) and  (20) for texts on MCMC). We are able to use Gibbs sampling for all noise parameters. For the signal parameters (the activation height and HRF parameters) we use single-component Metropolis-Hastings jumps (i.e. we propose separate jumps for each of the parameters in turn). We use Normal proposal distributions with the mean fixed on the current value, and with a scale parameter $\sigma_k$ for each parameter that is updated every 30 jumps. At the $j^{th}$ update $\sigma_k$ is updated using:

$$\sigma_k^{j+1}=\sigma_k^{j}S\frac{(1+A+R)}{(1+R)}$$ (22)

where $A$ and $R$ are the number of accepted and rejected jumps since the last $\sigma_k$ update respectively, $S$ is the desired rejection rate, which we fix at $0.5$ (22). When appropriate, parameters are initialised using ordinary least squares. We use a burn-in of 2000 jumps, followed by 2000 further jumps of which every 4th is sampled. Observation of the chains with different (but still sensible) initial conditions confirmed that a burn-in of 2000 samples was sufficient. The HRF parameters are initialised to the middle of their ranges. Details of the sampling used for each of the different parameters are described in the appendix.