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.
Distributed magneto- and electroencephalographic (M/EEG) source reconstruction is a severely underdetermined linear inverse problem. Bayesian source imaging regularizes inversion through source-prior and sensor-noise covariances, but the posterior covariance does not directly distinguish uncertainty constrained by the measurements from uncertainty unobservable under the sensing geometry. We study this distinction for linear-Gaussian M/EEG models with diagonal source-prior and sensor-noise covariances, covering Type-I inference and Type-II inference conditional on learned covariance hyperparameters. After restricting the analysis to the prior-supported source space, standardizing the prior, and whitening the sensor noise, the complete singular spectrum of the forward operator separates likelihood-visible and likelihood-null directions. This yields an additive uncertainty decomposition of the conditional posterior covariance: uncertainty in likelihood-visible directions is reduced according to spectral information content, whereas likelihood-null uncertainty remains unchanged from the prior. The spectral representation also yields a prior-minus-reduction identity, showing that covariance reduction is confined to the likelihood-visible subspace. Spectral markers quantify likelihood information and prior-variance retention and reduction. Fixed-orientation MEG simulations with adaptive noise learning illustrate the framework. Dense Bayesian minimum-norm inference retains a large likelihood-null component, whereas sparse γ-MAP strongly reduces the fitted prior support. Within that support, the markers distinguish strongly informed directions from a weakly constrained direction. Thus, prior-support selection determines which source coordinates remain in the fitted model, while the spectral decomposition quantifies how strongly the retained directions are constrained by the measurements.
Classical sparse Type-II Bayesian methods for M/EEG brain imaging support joint estimation of source and noise hyperparameters, but rely on fixed iterative update rules. Although these updates are principled and interpretable, their dynamics are not data-driven. We propose to learn the update mechanism itself while preserving the underlying Bayesian structure by unfolding a classical joint hyperparameter-learning solver into a trainable neural network architecture whose layers mirror the original iterations. The resulting framework is initialized to recover the classical solver exactly before training and is enriched through progressively more expressive residual correction learning mechanisms, ranging from learnable biases to adaptive multilayer perceptron and attention-based contextual refinements. In this way, training does not replace Bayesian inference with a black-box predictor, but instead learns structured correction terms while retaining the interpretability and model-based character of the original update dynamics. Structured correction learning therefore aims to improve empirical reconstruction performance without replacing the original model-based inference mechanism. Experimental results show that the learned correction variants improve reconstruction performance and convergence behavior over the baseline unfolded solver while preserving its algorithmic transparency.
Marco Morik, Xiao Ruiting, Shinichi Nakajima +2
Berlin Institute for the Foundations of Learning and Data (BIFOLD), Germany · Technische Universität Berlin, Germany · RIKEN Center for Advanced Intelligence Project (AIP), Japan +2
Electroencephalography (EEG) records mixtures of brain-source activity. Even with a known anatomical forward model, experiments that excite only part of the source-state space leave the dynamics unidentified, and repetition cannot resolve the ambiguity. We show that unknown local mechanism changes can supply the missing information. We consider linear dynamics among fixed anatomical sources with known source-state initialization patterns. Changing one source's update rule for one transition leaves a rank-one, source-specific signature in subsequent EEG: subtracting matched baseline responses isolates it, and the forward model identifies the source and calibrates its response history. Combining these histories with initialization responses recovers source interactions without baseline reachability and without first identifying the intervention coefficients. We establish sufficient recovery conditions, a direct estimator, and a noise-sensitivity bound conditional on correct source labels. Simulated EEG on anatomy derived from magnetic resonance imaging confirms the information gain: with baseline excitation confined to four of twelve source coordinates, eight unknown changes recover all dynamics in 32/32 systems, whereas baseline realization, baseline regression through an invertible forward model, and changes that leave the tested states unexposed all fail, and explicitly constructed alternative dynamics reproduce every baseline mean. Where baseline information suffices, direct reconstruction is also more reliable than a matched-information spectral estimator. Nonlocal changes and forward-model error limit accuracy even when source labels are correct.
Ayana Mussabayeva, Jiaqi Sun, Anuar Aimoldin +2
Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE · Carnegie Mellon University (CMU), Pittsburgh, PA, USA