Table 1. Quantification of the posterior collapse phenomenon for the regional variables θ^r as the dimensionality of the parameter space is increased.

Each column corresponds to a different model with varying numbers of regional parameters m_r; each cell shows the mean and the variance of Kullback-Leibler divergence between the approximate posterior and the prior, $\mathrm{KL}[q({\mathrm{\theta }}_j^r\mid \mathbf{y},\mathbf{u},\mathbf{c})\Vert N(0,1)]$, for all regions and all subjects. Because the inference process does not guarantee any specific order of parameter space dimensions, we order each column by the decreasing KL divergence. Values close to zero indicate that the approximate posterior matches the prior distribution. As the dimensionality is increased above m_r = 3, the KL divergences in the first three dimensions remain stable, while the values close to zero in the additional dimensions indicate that the additional dimensions are effectively unused.

mr	1	2	3	4	5
${\mathrm{\theta }}_1^r$	3.313 ± 0.268	2.946 ± 0.209	2.846 ± 0.135	2.744 ± 0.230	2.716 ± 0.133
${\mathrm{\theta }}_2^r$	–	2.266 ± 0.935	2.025 ± 0.786	2.238 ± 0.754	2.048 ± 0.714
${\mathrm{\theta }}_3^r$	–	–	0.917 ± 0.562	0.713 ± 0.601	0.881 ± 0.515
${\mathrm{\theta }}_4^r$	–	–	–	0.002 ± 0.002	0.004 ± 0.007
${\mathrm{\theta }}_5^r$	–	–	–	–	0.001 ± 0.000