f-contrasts

Variational Bayes gives us an approximation to the posterior distribution, $p(\beta,a,\phi_{\epsilon}\vert y)$. From this we can obtain the approximate marginal posterior distribution, $q(\beta,\bar{\beta}\vert Y)$, as being a multivariate Normal distribution (equation 25). If we write this as:

$$\left[\! \begin{array}{c} \beta_i \\ \bar{\beta_i} \end{array}\! \right]\quad\vert\quad y \sim MVN\left( \left[\! \begin{array}{c} \mu_{\beta_i} \\ \mu_{\b... ...beta_i}}& \Lambda_{\bar{\beta}_i} \end{array}\! \right] \right)$$ (32)

We can marginalise to get the marginal distribution over the regression parameters, $q(\beta_i\vert Y)$, as:

$$\beta_i\vert y \sim MVN\left( \mu_{\beta_i}, \Lambda_{\beta_i} \right)$$ (33)

We can now use the marginal distribution in equation 33 to perform inference. In this paper we take the approach of using the f-contrast framework traditionally used with basis functions in the frequentist GLM framework (Josephs et al., 1997). If $c$ is a $(N_eN_b)\times J$ vector representing an f-contrast, we can use the f-contrast framework to compute the normalised power explained by the f-contrast:

$$f = \mu_{\beta_i}^T c(c^T\Lambda_{\beta_i}c)^{-1}c^T\mu_{\beta_i}/J$$ (34)

with degrees of freedom $J$ and $\infty$. As with the use of basis functions in the frequentist GLM framework (Josephs et al., 1997), we lose directionality when doing an f-test. That is, we only investigate the power explained by linear combinations of the basis function parameters, regardless of the direction (i.e. whether it is an activation or deactivation). This means that we only look at the positive tail of the f-distribution to find both activations and deactivations. The alternative to doing a test on $\beta_i$ would be to ask ``What is the probability that $\bar{D}_{ie}$ is greater than zero?''. Recall from equation 11 that $\bar{D}_{ie}$, represents the HRF size (activation height). Can we recover the activation height, $\bar{D}_{ie}$, from the parameters we infer upon, $\beta_{ie}$ and $\bar{\beta}_{ie}$? To do this, we can rewrite equations 20 and  21 to get:

$$\bar{D}_{ie} = \bar{\beta}_{ie} \left( \sum_b{\frac{\beta_{ie}^2}{N_b\bar{\beta}_{ie}^2}} \right)^{1/2}$$

$$= (\bar{\beta}_{ie})\left(\sum{(\beta_{ie}^2/N_b)}\right)^{1/2}$$ (35)

Note that the term $\left(\sum{(\beta_{ie}^2/N_b)}\right)^{1/2}$ is the power we are testing when we do the f-test on $\beta_{i}$. Equation 35 tells us that we can use the sign of $\bar{\beta}_{ie}$ to give the sign of $\bar{D}_{ie}$, and therefore the direction of the activation. We could look to derive the posterior probability of the normalised power explained by the f-contrast in equation 34. Instead, the approach we take in this paper is to convert them to pseudo-z-statistics and then perform spatial mixture modelling on the spatial map of pseudo-z-statistics as described later. The f-to-z transform is carried out by doing an f-to-p-to-z transform (i.e. by ensuring that the probabilities in the tails are equal under the f- and z-distributions for the f- and z-statistics). We refer to them as pseudo-z-statistics as they are not necessarily Normally distributed with zero mean and standard deviation of one under the null hypothesis. This is because they have been obtained by performing Bayesian inference. Whether or not Bayesian inference produces the same null distribution as that in frequentist inference will depend on the form of prior used. As we shall see in section 4 and as we would expect (Penny et al., 2003), if we use noninformative priors we do get approximate equivalence between frequentist and Bayesian inference. However, when we use constrained HRF shape priors we get a different distribution under the null hypothesis. We will see later how we can adjust to this different inference, and take advantage of the extra sensitivity it offers, by using spatial mixture modelling.