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.
Figures & tables
Figure 1: Schematic relation between prior support and likelihood visibility in physical source space. Prior support is determined by Σγ , whereas sensing visibility is determined by L . The prior-supported likelihood-null subspace is Uγ=Range(Σγ)∩Null(L) . Within the prior-supported likelihood-visible sector, measurement informativeness varies continuously from weakly to strongly informed directions. The four sectors provide a schematic geometric interpretation and need not form a direct-sum decomposition of RN .
Spectral regime
Retention fraction τ
Reduction fraction ρ
Interpretation
Strongly data-informed, κ≫1
τ≈0
ρ≈1
The measurements remove most of the prior variance along the direction.
Balanced information, κ=1
τ=1/2
ρ=1/2
Prior and likelihood precision are equal, so half of the prior variance remains and half is removed.
Weakly data-informed but visible, 0<κ≪1
τ≈1
ρ≈0
The likelihood contributes only weak information, so most of the prior variance remains after conditioning.
Exactly likelihood-null, κ=0
τ=1
ρ=0
The measurements provide no information along the direction, and its prior variance is preserved in the posterior.
Table 1: Interpretation of the spectral information value κ , variance-retention fraction τ , and variance-reduction fraction ρ .
Method
q
r
q−r
κi>1
0<κi<1
BMN with sLORETA
1284
204
1080
140
64
γ -MAP
11
11
0
10
1
Table 2: Spectral structure of the fitted models.
Figure 2: Normalized source reconstruction and aggregate posterior-covariance decomposition for the simulated example. Panel (a) shows the simulated sources, the posterior-mean reconstruction obtained with BMN with sLORETA normalization, and the reconstruction obtained with γ -MAP. Panel (b) summarizes the exact additive covariance identity at trace level. For BMN, approximately 2.8% of the conditional posterior trace is carried by the likelihood-visible block and 97.2% by the likelihood-null block; the total posterior trace is approximately 94.9% of the fitted prior trace. For γ -MAP, the likelihood-null block is zero within the pruned fitted support and the remaining conditional posterior covariance is entirely likelihood-visible, while its total trace is approximately 0.073% of the fitted prior trace. The latter statement is conditional on the learned support and does not imply that the original full source-space lead field is injective.
Figure 3: Spectral information and variance markers for BMN with sLORETA normalization and γ -MAP. The upper panels show κi=si2 with the reference level κi=1 . The lower panels show the corresponding fractions of prior variance remaining in the posterior, τi , and removed by the data, ρi . Exact likelihood-null directions of BMN with sLORETA are reported separately and are not included among the positive singular values.
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
Recently, deep generative models have been used for posterior inference in inverse problems, including high-stakes applications in medical imaging and scientific discovery, where the uncertainty of a prediction can matter as much as the prediction itself. However, posterior uncertainty is difficult to interpret because it can mix ambiguity inherent to the forward operator with uncertainty propagated through inference. We introduce a structural decomposition of posterior uncertainty that isolates intrinsic ambiguity. A cascade formulation makes this ambiguity accessible for calibration analysis, enabling qualitative diagnostics and simulation-based calibration tests that reveal failure modes that remain hidden when models are selected by reconstruction quality alone. We first validate the approach on a Gaussian example with analytical posterior structure, then illustrate the decomposition on accelerated magnetic resonance imaging (MRI), and finally apply the calibration diagnostics to electroencephalography (EEG) source imaging.
Yuxin Guo, Dongrui Deng, Pulkit Grover
1Neuroscience Institute, Carnegie Mellon University · 2Machine Learning Department, Carnegie Mellon University · Department of Electrical and Computer Engineering, Carnegie Mellon University
We propose a novel Bayesian framework for joint image reconstruction and uncertainty quantification from compressed sensing magnetic resonance imaging data. The problem is formulated as a linear inverse problem, where prior distributions are assigned to the unknown image parameters. Specifically, the image is assumed to be sparse in a given transform domain. We develop a general framework applicable to any sparsifying transform and demonstrate its performance using (1) a total variation transform based on image spatial gradients and (2) a wavelet-domain transform. Bayesian inference is performed using a split-and-augmented Gibbs sampler, while the resulting non-differentiable conditional distributions are efficiently sampled using a proximal Markov chain Monte Carlo method. The proposed algorithms are validated on both single-coil and multi-coil datasets using various k-space sampling patterns and acceleration factors. The results demonstrate that the proposed Bayesian methods consistently outperform their optimisation-based counterparts in image reconstruction while providing uncertainty estimates for the reconstructed images. Furthermore, the estimated uncertainty maps show a strong correlation with the true reconstruction errors and substantially outperformed deep learning-based uncertainty estimation methods.
Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander
Department of Computer Science, University of Exeter, Exeter EX4 4QF, U.K. · UCL Hawkes Institute, Department of Computer Science, University College London, London, U.K.