Exact Spectral Uncertainty Decomposition for Linear-Gaussian Bayesian M/EEG Inverse Problems

Ismail Huseynov; Stefan Haufe

cs.LGOct 21, 2026

Exact Spectral Uncertainty Decomposition for Linear-Gaussian Bayesian M/EEG Inverse Problems

Authors: Ismail Huseynov, Stefan Haufe

Organizations: Physikalisch-Technische Bundesanstalt, Berlin, Germany · Technische Universitat Berlin, Berlin, Germany · Charité–Universitätsmedizin Berlin, Berlin, Germany

Abstract

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

Panel of Figure 1 — Figure 1: Schematic relation between prior support and likelihood visibility in physical source space. Prior support is determined by $\bm{\Sigma}_{\bm{\gamma}}$ , whereas sensing visibility is determined by $\mathbf{L}$ . The prior-supported likelihood-null subspace is $\mathcal{U}_{\bm{\gamma}}=\operatorname{Range}(\bm{\Sigma}_{\bm{\gamma}})\cap\operatorname{Null}(\mathbf{L})$ . 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 $\mathbb{R}^{N}$ .

Spectral regime	Retention fraction $\tau$	Reduction fraction $\rho$	Interpretation
Strongly data-informed, $\kappa\gg 1$	$\tau\approx 0$	$\rho\approx 1$	The measurements remove most of the prior variance along the direction.
Balanced information, $\kappa=1$	$\tau=1/2$	$\rho=1/2$	Prior and likelihood precision are equal, so half of the prior variance remains and half is removed.
Weakly data-informed but visible, $0<\kappa\ll 1$	$\tau\approx 1$	$\rho\approx 0$	The likelihood contributes only weak information, so most of the prior variance remains after conditioning.
Exactly likelihood-null, $\kappa=0$	$\tau=1$	$\rho=0$	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

\kappa

, variance-retention fraction

\tau

, and variance-reduction fraction

\rho

Method	$q$	$r$	$q-r$	$\kappa_{i}>1$	$0<\kappa_{i}<1$
BMN with sLORETA	1284	204	1080	140	64
$\gamma$ -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 $\gamma$ -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 $\gamma$ -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 $\gamma$ -MAP. The upper panels show $\kappa_{i}=s_{i}^{2}$ with the reference level $\kappa_{i}=1$ . The lower panels show the corresponding fractions of prior variance remaining in the posterior, $\tau_{i}$ , and removed by the data, $\rho_{i}$ . Exact likelihood-null directions of BMN with sLORETA are reported separately and are not included among the positive singular values.

Uncertainty Quantification

Uncertainty quantification estimates how reliable predictions and scientific inferences are, and how uncertainty propagates into decisions.

See all 652 papers

Structure-Preserving Correction Learning for Sparse Bayesian Inference in Brain Source Imaging
Jun 5, 2026Marco Morik, Xiao Ruiting, Shinichi Nakajima +2Electroencephalography Bayesian Inverse Problems
Berlin Institute for the Foundations of Learning and Data (BIFOLD), Germany · Technische Universität Berlin, Germany · RIKEN Center for Advanced Intelligence Project (AIP), Japan +2
Separating Intrinsic Ambiguity from Estimation Uncertainty in Deep Generative Models for Linear Inverse Problems
May 14, 2026Yuxin Guo, Dongrui Deng, Pulkit GroverUncertainty Quantification Bayesian Inverse Problems
1Neuroscience Institute, Carnegie Mellon University · 2Machine Learning Department, Carnegie Mellon University · Department of Electrical and Computer Engineering, Carnegie Mellon University
Trustworthy MRI Reconstruction via Bayesian Uncertainty Quantification with Sparsity Prior Models
Jun 15, 2026Ahmed Karam Eldaly, Matteo Figini, Daniel C. AlexanderSparse Recovery Uncertainty Quantification
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.

Jun 5, 2026J/K move · Enter open · S save

Structure-Preserving Correction Learning for Sparse Bayesian Inference in Brain Source Imaging

Marco Morik, Xiao Ruiting, Shinichi Nakajima, Stefan Haufe+1

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 Bayesian Inverse Problems Bayesian Inference Algorithm Unrolling

Open paper AskPDF arXiv

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.

Abstract

Uncertainty Quantification

Explore similar work

Structure-Preserving Correction Learning for Sparse Bayesian Inference in Brain Source Imaging

Separating Intrinsic Ambiguity from Estimation Uncertainty in Deep Generative Models for Linear Inverse Problems

Trustworthy MRI Reconstruction via Bayesian Uncertainty Quantification with Sparsity Prior Models