HRF priors

The question remains as to what priors to use on the HRF parameters. There are a number of possibilities. We could use Gaussian priors which probabilistically encode our prior belief about the expected range of shapes of the HRF. We can make the priors as tight or relaxed as we believe. In this work we choose to specify a relaxed range of shapes, using Uniform distributions over a sensible, but otherwise quite wide, range. This restricts us to sensible HRF shapes whilst giving the model full freedom to fit the HRF. This is desirable in our case as we are interested in investigating the HRF characteristics without biasing it with strong priors. The ranges used for the half cosine period parameters are:

$$m_{i1} \sim U(0s,6s)$$

$$m_{i2} \sim U(2s,14s)$$

$$m_{i3} \sim U(2s,14s)$$

$$m_{i4} \sim U(2s,14s)$$ (20)

It is not clear as to whether the data supports the existence of an initial dip or post stimulus undershoot. Hence, we would like to provide a mechanism for allowing the existence of these features to be determined automatically as part of inferring on the model. An approach to this has already been discussed in the context of autoregressive parameters in the noise model. There we use ARD priors, which can adaptively force a parameter to zero if there is no evidence to support it in the data. This is the approach we also take here for the parameters $c_1$ and $c_2$:

$$\sqrt{c_{i1}} \sim N(0,1/\phi_{c_{i2}})$$

$$\sqrt{c_{i2}} \sim N(0,1/\phi_{c_{i2}})$$ (21)