Cauchy prior distribution adjustment

In general, comparing activity in voxels between two conditions is the main objective in fMRI analysis. In many cases, researchers are interested in whether the activity in a task condition is significantly greater than that in a control condition. To adjust a prior distribution in these cases, the estimated X from (3) can be used to determine x in a probability distribution as a candidate for a threshold. Once we consider the cumulative probability of a prior distribution, we may assume that the cumulative probability between −∞ and X, Pr[ − ∞ ≤ x < X], becomes a specific amount, P. For instance, once we assume that P = 95%, then at X, the cumulative probability becomes Pr−∞≤x<X= ∫−∞Xfxdx=95%. Based on these assumptions, we can set a specific P for the prior distribution to be used. The percentile, P, in the scale adjustment process can be estimated as:

where f(x) is a Cauchy distribution, Cauchy(x₀ = 0, σ), with σ to be determined. This P determines the shape of the Cauchy prior distribution. If X is constant, then the greater P results in a smaller σ and a Cauchy distribution with a steeper peak at x₀. Once we expect that more incidences are situated <X (greater P), then there should be more incidences at x₀ (greater probability density) where the peak is. For instance, let us compare two cases, P = 95% and P = 80% (see Fig. 2 for an illustration). As demonstrated in the figure, setting P = 95% resulted in the Cauchy distribution with the steeper peak at x₀ = compared with setting P = 85% when X = .032.

Based on the determined P and X, with (4), we can numerically search for σ in order to adjust the Cauchy prior distribution. Here is one illustrative example. Let us consider a case when 100,000 voxels are analyzed. To acquire information to calculate X, it is required to know C, N, and R. Let us assume that a meta-analysis of relevant previous studies indicated that a total of 1,600 voxels were active, while the difference between the mean activity strength between significant versus non-significant voxels was 1.0, and the standard deviation of the activity strengths, the noise strength, was .50. In this case, C = 1.0, N = .50, and R=1,600100,000=1.60%. Based on these assumptions, X=CNR=1.0.501.60%=.032. If we intend to find a Cauchy distribution scale σ that suffices P = 95%, then we need to find a σ that suffices:

Once we numerically search for a σ that suffices in the equation above, then we can find that σ ≈ .0051 in this case. By using the same approach, we can calculate σs for different Ps, such as 80%, 85%, and 90%. When we calculate the aforementioned σs, they become .023, .016, and .010, respectively. When all other parameters, C, N, and R, are the same, the Cauchy prior distribution with the adjusted σ becomes more centered around zero with a steeper peak as P increases. Figure 2 demonstrates the different Cauchy prior distributions with different Ps when the parameters used in the prior example were applied. In general, as P increases, the resultant Cauchy distribution becomes more concentrated around zero. In general, use of a narrower Cauchy prior distribution centered around zero with the steeper peak resulting from a greater P is likely to produce a more stringent result in terms of fewer voxels that survived thresholding.