Uncertainty calibration in Bayesian M/EEG source imaging
Authors: Ismail Huseynov, Mohammad Orabe, Srikantan Nagarajan, Stefan Haufe
Organizations: Physikalisch-Technische Bundesanstalt Braunschweig und Berlin, Berlin, Germany · Technische Universität Berlin, Berlin, Germany · University of California San Francisco, Biomagnetic Imaging Laboratory, Department of Radiology and Biomedical Imaging, San Francisco, CA, United States · Charité-Universitätsmedizin Berlin, Charitéplatz 1, Berlin, Germany
M/EEG source imaging requires accurate localization and reliable uncertainty estimates, yet posterior credible regions are rarely assessed for empirical calibration. We present a simulation-based framework for uncertainty quantification and calibration of empirical Bayesian inverse methods with closed-form conditional Gaussian posterior summaries. We compare Bayesian minimum norm (BMN) with sLORETA depth weighting and sparse Bayesian learning (SBL) with sparse basis field expansions (S-FLEX). S-FLEX allows retained coefficients to contribute nonzero marginal posterior variance at locations where voxel-wise SBL yields degenerate variances. For BMN, we derive joint source- and noise-variance updates through evidence maximization. We test whether adaptive noise learning yields near-calibrated uncertainties. Both methods retain residual miscalibration, motivating post-hoc isotonic recalibration of credible-region levels using empirical coverage. Simulations assess fixed- and free-orientation models across signal-to-noise ratios (SNR), source sparsity levels, and subject-specific head models, using oracle, baseline-estimated, and adaptively learned noise. Before recalibration, SBL+S-FLEX yields lower Wasserstein localization error, smaller posterior uncertainty, and coverage closer to nominal credibility than BMN+sLORETA. Isotonic recalibration improves both methods, with larger gains for conservative BMN+sLORETA posteriors; transfer across mismatched SNR, sparsity, and head models remains condition dependent. Noise specifications yield broadly consistent reconstruction and uncertainty trends within each estimator. In real auditory MEG data, SBL+S-FLEX provides more focal localization and tighter uncertainties than BMN+sLORETA, suggesting greater statistical power for identifying bilateral auditory sources.
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Figure 1: Unified workflow for simulation-based source reconstruction, posterior uncertainty estimation, calibration learning, and out-of-sample evaluation. In Stage 1, synthetic cortical activity is generated, the forward model is prepared, and sensor-space measurements are simulated under homoscedastic Gaussian noise. Source reconstruction is performed using BMN + sLORETA and γ -MAP + S-FLEX under oracle, baseline, and adaptive noise-learning models. Posterior means and covariances are used to construct credible intervals for fixed M/EEG, credible ellipses for free MEG, and credible ellipsoids for free EEG. In Stage 2, calibration operators are learned from repeated calibration runs using empirical coverage estimation and inverse isotonic recalibration. In Stage 3, the learned calibration is applied to held-out evaluation runs. Calibration metrics (MAD, MOD, MUD, and MSD) and reconstruction and uncertainty metrics (UNC and EMD) are computed in parallel and summarized through comparative visualizations.
Figure 2: Calibration curves derived from reduced free-MEG joint two-dimensional credible ellipses under adaptive noise learning, at αSNR=0.5 and NNZ=5 . A : BMN+sLORETA. B : γ -MAP+S-FLEX. Lines show mean empirical coverage over 100 evaluation runs; color-matched bands show plus or minus one run-level standard deviation. The dotted diagonal indicates ideal calibration (empirical coverage equal to nominal credibility). Pre denotes the raw results; post-oracle uses calibration simulations matched in head model, SNR, and NNZ; post-mismatch uses pooled-mismatch calibration simulations.
Figure 3: Reduced free-MEG SNR sweep with adaptive noise learning and NNZ=5 . The sweep uses the same benchmark and inverse-solver settings as the default configuration. The left column shows BMN+sLORETA and the right column γ -MAP+S-FLEX results. Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively. Each box contains 100 run-level observations at one αSNR value. Calibration regimes are pre, post-oracle, post-mismatch, and post-fixed. Post-fixed and post-oracle coincide at the default condition αSNR=0.5 .
Figure 4: Reduced free-MEG source-number sweep with adaptive noise learning and αSNR=0.5 . The sweep uses the same benchmark and inverse-solver settings as the default configuration. The column arrangement and panel pairs A–L match Figure 3 . NNZ takes values 1,3,5,10,100 and denotes the number of active sources. Each box contains 100 evaluation runs at one NNZ value. Calibration regimes are pre, post-oracle, post-mismatch, and post-fixed. Post-fixed and post-oracle coincide at the default condition NNZ=5 .
Figure 5: Fixed-orientation EEG reconstruction with adaptive noise learning on the fsaverage ico3 cortex ( N=1284 ), using a BioSemi64 montage, a three-layer boundary-element model, two bilateral auditory generators, and αSNR=0.5 . (a) Simulated EEG channels in gray, sensor RMS in red, and signed scalp-potential topography at 140 ms. The shaded interval denotes 120 – 160 ms. (b) BMN+sLORETA posterior means at the two generating locations, with the corresponding ground-truth waveforms shown as dashed curves and pointwise 95% posterior credible intervals. (c) Corresponding γ -MAP+S-FLEX reconstruction. Cortical maps show fixed-normal source-activity RMS over 120 – 160 ms in nAm; cyan markers indicate the generating locations. Separate color scales are used for the two cortical maps.
Figure 6: Measured auditory MEG and reduced free-orientation posterior summaries. (a) Evoked sensor fields, sensor RMS, and signed magnetic-field topography at 100 ms. (b) BMN+sLORETA signed directional posterior means and pointwise 95% credible intervals for the projected directional response at the selected left- and right-hemisphere sources, together with the cortical source-activity RMS map. (c) Corresponding γ -MAP+S-FLEX summaries. The curves are one-dimensional projections of the local vector posterior onto the fixed, source-specific leading SVD direction. Cortical color bars show source RMS in nAm and use separate ranges. The shaded response interval marks the response window.
Figure S1: Reduced free-MEG calibration curves derived from joint two-dimensional credible-ellipse containment under oracle noise at αSNR=0.5 and NNZ=5 . A : BMN+sLORETA. B : γ -MAP+S-FLEX. Lines denote mean empirical coverage over 100 evaluation runs; color-matched bands denote plus or minus one run-level standard deviation. The dotted diagonal indicates perfect calibration.
Figure S2: Reduced free-MEG SNR sweep under oracle noise, varying αSNR∈{0.1,0.3,0.5,0.7,0.9} with NNZ=5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S3: Reduced free-MEG source-number sweep under oracle noise, varying NNZ∈{1,3,5,10,100} with αSNR=0.5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S4: Reduced free-MEG calibration curves derived from joint two-dimensional credible-ellipse containment under baseline noise at αSNR=0.5 and NNZ=5 . A : BMN+sLORETA. B : γ -MAP+S-FLEX. Lines denote mean empirical coverage over 100 evaluation runs; color-matched bands denote plus or minus one run-level standard deviation. The dotted diagonal indicates perfect calibration.
Figure S5: Reduced free-MEG SNR sweep under baseline noise, varying αSNR∈{0.1,0.3,0.5,0.7,0.9} with NNZ=5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S6: Reduced free-MEG source-number sweep under baseline noise, varying NNZ∈{1,3,5,10,100} with αSNR=0.5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S7: Fixed-orientation calibration curves derived from scalar credible-interval containment under adaptive noise at αSNR=0.5 and NNZ=5 . A : BMN+sLORETA. B : γ -MAP+S-FLEX. Lines denote mean empirical coverage over 100 evaluation runs; color-matched bands denote plus or minus one run-level standard deviation. The dotted diagonal indicates perfect calibration.
Figure S8: Fixed-orientation SNR sweep under adaptive noise, varying αSNR∈{0.1,0.3,0.5,0.7,0.9} with NNZ=5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S9: Fixed-orientation source-number sweep under adaptive noise, varying NNZ∈{1,3,5,10,100} with αSNR=0.5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S10: Fixed-orientation calibration curves derived from scalar credible-interval containment under oracle noise at αSNR=0.5 and NNZ=5 . A : BMN+sLORETA. B : γ -MAP+S-FLEX. Lines denote mean empirical coverage over 100 evaluation runs; color-matched bands denote plus or minus one run-level standard deviation. The dotted diagonal indicates perfect calibration.
Figure S11: Fixed-orientation SNR sweep under oracle noise, varying αSNR∈{0.1,0.3,0.5,0.7,0.9} with NNZ=5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S12: Fixed-orientation source-number sweep under oracle noise, varying NNZ∈{1,3,5,10,100} with αSNR=0.5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S13: Fixed-orientation calibration curves derived from scalar credible-interval containment under baseline noise at αSNR=0.5 and NNZ=5 . A : BMN+sLORETA. B : γ -MAP+S-FLEX. Lines denote mean empirical coverage over 100 evaluation runs; color-matched bands denote plus or minus one run-level standard deviation. The dotted diagonal indicates perfect calibration.
Figure S14: Fixed-orientation SNR sweep under baseline noise, varying αSNR∈{0.1,0.3,0.5,0.7,0.9} with NNZ=5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Figure S15: Fixed-orientation source-number sweep under baseline noise, varying NNZ∈{1,3,5,10,100} with αSNR=0.5 . Panel pairs A–B , C–D , E–F , G–H , I–J , and K–L show EMD, UNC, MAD, MSD, MOD, and MUD, respectively, for BMN+sLORETA (left) and γ -MAP+S-FLEX (right). Calibration modes are pre, post-oracle, post-mismatch, and post-fixed.
Berlin Institute for the Foundations of Learning and Data (BIFOLD), Germany · Technische Universität Berlin, Germany · RIKEN Center for Advanced Intelligence Project (AIP), Japan +2