First-level

Here we consider the first-level in isolation and derive the marginal posterior distribution for $\beta_k$, the vector of GLM height parameters for the first-level model fit. Equation 1 gives us the likelihood for a first-level model in isolation, $p(Y_k\vert\beta_k,\sigma_k^2)$. The joint posterior on all parameters in this model is then:

$$p(\beta_k,\sigma_k^2\vert Y_k) \propto p(Y_k\vert\beta_k,\sigma_k^2)p(\beta_k,\sigma_k^2)$$ (8)

where $p(\beta_k,\sigma_k^2)$ is the prior distribution on the regression and variance parameters. We use the Berger-Bernardo reference prior (see section 3.2), which is:

$$p(\beta_k,\sigma_k^2) = 1/\sigma_k^2.$$ (9)

However, equation 8 does not give the distribution of interest for inference. We would like to infer on the posterior distribution on the activation height parameters $\beta_k$ when the effect of estimating $\sigma_k^2$ is accounted for, i.e. we would like to infer on $p(\beta_k\vert Y)$. To get this distribution, we must marginalise the joint posterior (equation 8) over the parameter of no interest $\sigma_k^2$. This integral gives a multivariate non-central t-distribution for the posterior distribution on $\beta_k$ (18):

$$p(\beta_k\vert Y_k) \propto \int p(Y_k\vert\beta_k,\sigma_k^2)/\sigma_k^2 d\sigma_k^2$$

$$= \mathcal{T}(\beta_k;\mu_{\beta_k},\sigma_{\beta_k}^2\Sigma_{\beta_k},\nu_{\beta_k}).$$ (10)

where

$$\mu_{\beta_k} = (X_k^TX_k)^{-1}X_k^TY_k$$

$$\sigma_{\beta_k}^2 = (Y_k-X_k\mu_{\beta_k})^T(Y_k-X_k\mu_{\beta_k})/(T-P_K)$$

$$\Sigma_{\beta_k} = (X_k^TX_k)^{-1}$$

$$\nu_{\beta_k} = T-P_K.$$ (11)

Note that if inference is performed in the frequentist framework, the null distribution on $\beta_k$ is the multivariate central t-distribution with the exact same covariance structure $\sigma_{\beta_k}^2\Sigma_{\beta_k}$ and degrees of freedom, $\nu_{\beta_k}$, and the maximum likelihood estimate for $\beta_k$ is exactly $\mu_{\beta_k}$, the mean of the posterior distribution in the Bayesian framework.