Classical and Bayesian Inference in fMRI - Advance Image Processing in Magnetic Resonance Imaging - page 541

that the effect is negligible. There are several shortcomings in this classical
approach. Firstly, one can never reject the alternate hypothesis (i.e., say that an
activation has not occurred) because the probability that an effect is exactly zero
is itself zero. This is problematic, for example, in trying to establish double
dissociations or indeed functional segregation; one can never say one area responds
to color but not motion and another responds to motion but not color. Secondly,
because the probability of an effect being zero is vanishingly small, given enough
scans or subjects one can always demonstrate a significant effect at every voxel.
This fallacy of classical inference is becoming relevant practically, with the thou-
sands of scans entering into some fixed-effect analyzes of fMRI data. The issue
here is that a trivially small activation can be declared significant if there are
sufficient degrees of freedom to render the variability of the activation's estimate
small enough. A third problem that is specific to SPM is the correction or adjust-
ment applied to the p values to resolve the multiple comparison problem. This has
the somewhat nonsensical effect of changing the inference about one part of the
brain in a way that is contingent on whether another part is examined. Put simply,
the threshold increases with search volume, rendering the inference very sensitive
to what it encompasses. Clearly, the probability that any voxel has activated does
not change with the search volume and yet the classical p value does.
All these problems would be eschewed by using the probability that a voxel
had been activated or indeed, that its activation was greater than some threshold.
This sort of inference is precluded by classical approaches, which simply give
the likelihood of getting the data, given no activation. What one would really like
is the probability distribution of the activation, given the data. This is the posterior
probability used in Bayesian inference. The posterior distribution requires both
the likelihood, afforded by assumptions about the distribution of errors, and the
prior probability of activation. These priors can enter as known values or can be
estimated from the data, provided we have observed multiple instances of the
effect we are interested in. The latter is referred to as empirical Bayes. A key
point here is that we do assess repeatedly the same effect over different voxels,
and we are, therefore, in a position to adopt an empirical Bayesian approach (29).

17.5.1 E MPIRICAL E XAMPLE

In this subsection, we compare and contrast Bayesian and classical inference
using PPMs and SPMs based on real data. The data set comprised data from a
study of attention to visual motion (30). The data used here came from the first
subject studied. This subject was scanned at 2-T to give a time series of 360
images comprising 10-block epochs of different visual motion conditions. These
conditions included a fixation condition, visual presentation of static dots, visual
presentation of radially moving dots under attention, and no-attention conditions.
In the attention, condition subjects were asked to attend to changes in speed
(which did not actually occur). This attentional manipulation was validated post-
hoc using psychophysics and the motion after-effect. Further details of the data

acquisition are given in the caption to Figure 17.8 . These data were analyzed