Fig 1.
A The generative model for the conditional dependencies graph and precision matrix. B The generative model for structural connectivity and the precision matrix, based on both BOLD time series X and probabilistic streamline counts N. Latent variables, observed variables and hyperparameters are indicated in white, yellow and grey, respectively.
Fig 2.
The evaluation procedure of the simulated fMRI data.
First, both the posterior distribution P(G, R ∣ X) and the point estimates (for the graphical LASSO or maximum likelihood estimate) are determined. Subsequently the error compared to the ground truth is computed for all samples in the approximated distribution as well as for the point estimates (see text for this procedure). These results are summarized by computing the z-score for the point estimate error relative to the distribution of errors obtained from the Bayesian approach. Finally, the z-scores are aggregated across the runs, resulting in a histogram of error z-scores for each simulation.
Fig 3.
The histograms for each of the 28 different simulations.
Positive error z-scores indicate that the point estimate was less effective in recovering the ground truth than the Gaussian graphical model, while the reverse is true for negative error z-scores. The red dashed lines indicate the interval outside of which the difference in performance is significant (p < 0.01, z-test). Note the different ordinate axes.
Fig 4.
A. Simulation details. First row: the ground truth connectivity of one run of simulation 1, as well as the constructions by the graphical LASSO (λ = 100) and the expectation of the Gaussian graphical model approach. Second row: estimated partial correlation for a true positive connection, a true negative connection with strong empirical correlation, and a true negative connection with weak empirical correlation. B. The same, but with stronger regularization for the graphical LASSO (λ = 10, 000). This time, the G-LASSO estimate is similar to the BGGM expectation for connection 1–4, but over-regularizes the true positive connection 1–5.
Fig 5.
Effect of different sample sizes in recovery of ground truth connectivity, for the BGGM approach as well as for the graphical LASSO with λ ∈ {5, 100, 1 000, 10 000}.
Error bars indicate one standard deviation over the 50 runs. For the BGGM approach, the error bars indicate one standard deviation over the expectations of the runs.
Fig 6.
Effect of small sample sizes, including n < p, in recovery of ground truth connectivity, for the BGGM approach as well as for the graphical LASSO with λ ∈ {5, 100, 1 000, 10 000}.
Error bars indicate one standard deviation over the 50 runs. For the BGGM approach, the error bars indicate one standard deviation over the expectations of the runs.
Fig 7.
Subcortical connectivity for one subject.
From left to right: the empirical correlation matrix, the mean posterior connection probability matrix and the mean posterior partial correlation matrix. The connections for the left hemisphere (LH) and the right hemisphere (RH) are separated by the dashed lines.
Fig 8.
Subcortical connectivity for one subject using the data fusion model.
From left to right: the empirical streamline log-counts, the mean posterior connection probability matrix and the mean posterior partial correlation matrix. Note the reduction in connectivity, in particular between the hemispheres, compared to Fig 7. The connections for the left hemisphere (LH) and the right hemisphere (RH) are separated by the dashed lines.
Fig 9.
Examples of differences in partial correlation estimates between the BGGM estimates and the data fusion approach.
Fig 10.
Subcortical connectivity for one subject using the informative prior.
From left to right: the prior probability of a non-independence, the mean posterior connection probability matrix and the mean posterior partial correlation matrix. The connections for the left hemisphere (LH) and the right hemisphere (RH) are separated by the dashed lines.
Fig 11.
Differences in posterior distribution shapes.
A. Entropy of the posterior distribution. B. Posterior probability of the mode . We refer to the prior distribution defined in Eq (8) as the vague prior.
Fig 12.
Scatter plot of the expectations of connection probabilities and partial correlations.
The top row shows the connection probabilities for the two model extensions versus the original model. The bottom row shows the same, but for partial correlations.
Fig 13.
Scatter plot of the variances of connectivity and partial correlations.
The top row shows the variance of connections for the two model extensions versus the original model. The bottom row is the same, but for the variance of partial correlations.