Covariance-Based Map Uncertainty

• Covariance-based map uncertainty is a probabilistic representation where uncertainty is captured through covariance structures that encode variances, cross-correlations, and uncertainty propagation.
• It underpins diverse applications such as Bayesian inverse problems, map-based localization, and online mapping, enabling efficient uncertainty quantification and improved estimator consistency.
• The approach employs factorized and low-rank approximations to manage computational complexity while integrating physical constraints and enhancing decision-making in mapping scenarios.

Covariance-based map uncertainty denotes the representation of a map, map-conditioned state, or map-derived latent field as a probabilistic object whose uncertainty is carried by a covariance structure rather than by a deterministic geometry alone. Across Bayesian inverse problems, map-based localization, online vectorized mapping, occupancy mapping, and Gaussian-process surface reconstruction, the central quantity is a posterior or predictive covariance that encodes marginal variances, cross-correlations, and uncertainty propagation under measurement, model, and pose uncertainty (Saibaba et al., 2014, Dutoit et al., 2016, Gogoi et al., 20 Mar 2026, Jadidi et al., 2017, Zou et al., 2022). The term therefore covers both dense covariance operators over spatial fields and structured or factorized surrogates—such as low-rank updates, sparse Cholesky factorizations, Low-Rank plus Diagonal decompositions, and tangent-space covariances on Lie groups—used when exact dense storage is prohibitive or geometrically inappropriate.

1. Core meaning and problem classes

In the geostatistical approach to inverse problems, the unknown parameter field is modeled as a Gaussian random vector with prior covariance $\Gamma_{\text{prior}}$, and uncertainty quantification involves the posterior covariance $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$ at the maximum a posteriori point under a Laplace approximation (Saibaba et al., 2014). In map-based localization, the corresponding object is the prior map uncertainty and its cross-correlations with the online device state, represented by a joint covariance

$$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$

where neglecting $P_{MM}$ or the device–map cross-covariances underestimates the innovation covariance and leads to overconfident and often biased state estimates (Dutoit et al., 2016).

In online autonomous-driving maps, covariance-based uncertainty is attached directly to vectorized map elements. One formulation assigns to each map vertex $v^{(i)}$ a bivariate Gaussian with mean $\mu^{(i)} \in \mathbb{R}^2$ and covariance $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$ parameterized by $(\sigma_1,\sigma_2,\rho)$, so that the uncertainty ellipse can align with local road geometry (Zhang et al., 24 Jul 2025). A more structured formulation models an entire polyline element $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$ as a multivariate Gaussian with covariance $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$, explicitly encoding spatial correlations along the element (Gogoi et al., 20 Mar 2026).

In continuous occupancy mapping and building reconstruction, covariance-based map uncertainty appears as Gaussian-process predictive variance. For GP occupancy maps, the map is a continuous random field with predictive mean and variance at each query location, and pose uncertainty alters kernel evaluations through expected kernels or expected sub-maps (Jadidi et al., 2017). For uncertain building models, the primary outputs are the posterior mean depth $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$0 and covariance $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$1 of an implicit surface $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$2, where $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$3 is a Gaussian Mixture Model prior and $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$4 is a GP residual (Zou et al., 2022).

These formulations share a common interpretation: covariance is not merely an error bar on isolated map points, but a structural object that determines how uncertainty couples directions, neighboring vertices, device and map states, or tangent-space perturbations.

2. Posterior covariance in Bayesian inverse problems

For a Bayesian inverse problem with forward map $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$5, Gaussian prior $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$6, Gaussian observational noise, and Gauss–Newton data-misfit Hessian $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$7, the posterior covariance at the MAP point is

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$8

Introducing a prior square-root $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$9 such that $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$0 gives the exact identity

$$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$1

When the data are informative in a low-dimensional subspace, the spectrum of $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$2 decays rapidly, and the posterior covariance admits the low-rank update

$$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$3

where the generalized eigenmodes satisfy $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$4 (Saibaba et al., 2014).

The practical consequence is that covariance-based uncertainty can be evaluated without forming or storing a dense $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$5 posterior covariance. The storage reduces from $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$6 to $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$7, and the cost of applying the covariance or extracting pointwise variances scales as $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$8 plus low-rank terms, with the rank $$P = \begin{bmatrix} P_{RR} & P_{RM}\ P_{MR} & P_{MM} \end{bmatrix},$$9 typically independent of the dimension of the unknown parameter vector space (Saibaba et al., 2014). In the synthetic travel-time tomography and hydraulic tomography examples, eigenvalues of the generalized eigenproblem decay rapidly; with cutoff $P_{MM}$0, the number of retained modes depends on measurement count and prior smoothness, not on $P_{MM}$1 (Saibaba et al., 2014).

This representation supports scalar uncertainty measures directly from the retained eigenpairs. Pointwise variances obey

$P_{MM}$2

while the variance of a linear functional is

$P_{MM}$3

The same low-rank form yields A-optimality, D-optimality, E-optimality, information gain, and posterior correlation maps (Saibaba et al., 2014).

Two adjacent formulations broaden this posterior-covariance viewpoint. In linear-Gaussian data assimilation, perturbing both prior means and observations and recomputing ensemble MAP solutions produces analyses with covariance equal to the posterior covariance $P_{MM}$4 in expectation, so posterior variances of map functionals $P_{MM}$5 can be estimated from sample variances without forming $P_{MM}$6 (Stanley et al., 2023). In equality-constrained MAP and ML estimation, the estimate covariance is not the inverse Hessian of the augmented merit alone; it is obtained from the appropriate submatrix of the inverse KKT matrix

$P_{MM}$7

or equivalently by projection onto the tangent space of the constraints (Dutra, 2020).

3. Prior-map uncertainty in localization and visual odometry

Consistent map-based localization requires explicit treatment of prior map uncertainty and device–map cross-correlations. In the Schmidt-Kalman formulation, the innovation covariance contains not only the device term $P_{MM}$8 and the map term $P_{MM}$9, but also the cross terms $v^{(i)}$0 and $v^{(i)}$1:

$v^{(i)}$2

If the map is treated as perfect or if the estimator ignores device–map cross-covariances, $v^{(i)}$3 becomes underestimated, and repeated use of the same mapped scene makes subsequent updates inconsistent (Dutoit et al., 2016).

The Cholesky-Schmidt-Kalman filter replaces the dense map covariance $v^{(i)}$4 with the sparse Cholesky factor $v^{(i)}$5 of the map Hessian $v^{(i)}$6, using sparse triangular solves to compute

$v^{(i)}$7

The device–map cross-covariance is maintained in factorized form as $v^{(i)}$8, yielding a factorized joint covariance

$v^{(i)}$9

This avoids explicit formation of $\mu^{(i)} \in \mathbb{R}^2$0, gives memory requirements typically linear in the size of the map, and permits a relaxation to overlapping independent submaps in the sC-SKF to bound computation on mobile hardware (Dutoit et al., 2016).

The empirical comparison is explicit. On a Project Tango tablet with VICON ground truth, position RMSE was $\mu^{(i)} \in \mathbb{R}^2$1 cm for C-SKF single map, $\mu^{(i)} \in \mathbb{R}^2$2 cm for sC-SKF with two submaps, $\mu^{(i)} \in \mathbb{R}^2$3 cm for the perfect-map approximation with inflated $\mu^{(i)} \in \mathbb{R}^2$4, and $\mu^{(i)} \in \mathbb{R}^2$5 cm with no map updates (Dutoit et al., 2016). This directly supports a common methodological point: heuristic measurement-noise inflation does not reconstruct the missing cross-correlation structure in $\mu^{(i)} \in \mathbb{R}^2$6 and the Kalman gain, and therefore cannot guarantee consistency (Dutoit et al., 2016).

A related covariance-based construction appears in stereo visual odometry. MAC-VO predicts per-pixel flow and depth uncertainties and propagates them through the pinhole model to a full $\mu^{(i)} \in \mathbb{R}^2$7 spatial covariance for each 3D keypoint. Because $\mu^{(i)} \in \mathbb{R}^2$8 and $\mu^{(i)} \in \mathbb{R}^2$9 are both proportional to depth, the resulting covariance contains nonzero $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$0, $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$1, and $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$2 terms, and these off-diagonal terms are used in Mahalanobis-weighted residuals

$\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$3

for two-frame pose graph optimization (Qiu et al., 2024). The ablation is numerically sharp: on TartanAir v2 Hard, removing off-diagonals increased $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$4 from $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$5 to $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$6 and $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$7 from $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$8 to $\Sigma^{(i)} \in \mathbb{R}^{2\times 2}$9 (Qiu et al., 2024). This suggests that covariance-based map uncertainty is not only a storage device for confidence, but also a weighting mechanism whose correlation structure materially changes estimator behavior.

4. Structured covariance in online vector maps and trajectory prediction

In mapless trajectory prediction, the local HD map is generated online from sensor data and is partial and noisy due to occlusions, distance, sensor artifacts, and ephemeral scene changes. A covariance-based formulation therefore models each map vertex in bird’s-eye-view coordinates as a 2D Gaussian

$(\sigma_1,\sigma_2,\rho)$0

with learned $(\sigma_1,\sigma_2,\rho)$1 so that the uncertainty ellipse aligns with local map geometry (Zhang et al., 24 Jul 2025). The same construction admits an equivalent tangent–normal interpretation,

$(\sigma_1,\sigma_2,\rho)$2

where $(\sigma_1,\sigma_2,\rho)$3 and $(\sigma_1,\sigma_2,\rho)$4 is the tangent heading (Zhang et al., 24 Jul 2025).

The training objective replaces $(\sigma_1,\sigma_2,\rho)$5 regression with a 2D Gaussian negative log-likelihood

$(\sigma_1,\sigma_2,\rho)$6

and the predicted covariance is integrated into trajectory prediction by encoding $(\sigma_1,\sigma_2,\rho)$7 at the vertex level (Zhang et al., 24 Jul 2025). The paper further introduces Proprioceptive Scenario Gating, a six-layer MLP that fuses a baseline stream and an uncertainty stream using weights obtained by temperature softmax on predicted trajectories (Zhang et al., 24 Jul 2025). Empirically, uncertainty helps when kinematics change abruptly and can hurt when kinematics are steady; for stable states with $(\sigma_1,\sigma_2,\rho)$8, the average $(\sigma_1,\sigma_2,\rho)$9, otherwise $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$0 (Zhang et al., 24 Jul 2025).

The reported gains are substantial. On nuScenes, selected results include HiVT + MapTR improving from $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$1 to $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$2 in minADE/minFDE/MR, and DenseTNT + MapTRv2-CL improving from $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$3 to $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$4, with up to $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$5 Miss Rate reduction over the uncertainty baseline (Zhang et al., 24 Jul 2025).

A denser covariance model extends this point from per-vertex anisotropy to intra-element dependencies. In "structured probabilistic online mapping," each vectorized element is a Gaussian random vector with covariance

$m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$6

where $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$7 is diagonal, $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$8 is low rank, and experiments use $m_k = (x_1,y_1,\dots,x_N,y_N)\in\mathbb{R}^{2N}$9 (Gogoi et al., 20 Mar 2026). Woodbury identities and the Matrix Determinant Lemma make likelihood evaluation tractable:

$$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$0

This LRPD parameterization is presented as a tractable alternative to full covariance, diagonal covariance, and block-diagonal point-wise correlation (Gogoi et al., 20 Mar 2026).

The map-quality and motion-prediction numbers quantify the effect. For map generation on nuScenes, MapTR deterministic mAP $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$1 became $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$2 with LRPD, MapTRv2 deterministic $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$3 became $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$4, and MapTRv2-CL deterministic $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$5 became $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$6 (Gogoi et al., 20 Mar 2026). For HiVT with $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$7 modes, MapTRv2-CL LRPD achieved minADE6 $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$8, minFDE6 $$\Sigma_{\phi,k} = D_{\phi,k} + \kappa L_{\phi,k} L_{\phi,k}^T$$9, and MR6 $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$00, while the GT map lower bound was $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$01, $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$02, and $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$03 (Gogoi et al., 20 Mar 2026). A recurring theme is therefore that diagonal or factorized uncertainty can be too weak a model for road geometry, whereas covariance aligned with the map and correlated along the element yields smoother and more physically plausible map samples (Zhang et al., 24 Jul 2025, Gogoi et al., 20 Mar 2026).

5. Gaussian-process occupancy maps and uncertain building models

In continuous occupancy mapping, the latent occupancy field is modeled by a Gaussian process

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$04

with standard predictive mean and variance

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$05

When robot poses are uncertain, inputs are modeled as Gaussian random variables and the kernel must be integrated over the input distributions, yielding an expected kernel

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$06

For a Squared-Exponential kernel, this has a closed form involving $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$07, and replacing $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$08 and $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$09 by their expected versions propagates pose uncertainty directly into posterior mean and covariance (Jadidi et al., 2017).

The same paper introduces expected sub-maps and Warped Gaussian Processes. Expected sub-maps construct local deterministic maps in the robot frame and then fuse them under sampled pose distributions, with mixture moments

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$10

Warped Gaussian Processes further handle non-Gaussian observation noise by learning a monotonic transform $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$11 and computing observation-space predictive moments through inverse-warp integrals (Jadidi et al., 2017). Empirically, accounting for pose uncertainty increases map uncertainty, while WGPOM improved map quality relative to GPOM across expected-kernel and expected-sub-map settings (Jadidi et al., 2017).

For uncertain building models, covariance-based map uncertainty is organized around an implicit surface relative to a Gaussian Mixture Model prior. The modeled surface depth field is

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$12

where $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$13 is the expected plane depth from GMM and $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$14 is a latent GP residual (Zou et al., 2022). The posterior mean and variance of the depth are

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$15

and these are stored as the map’s primary outputs (Zou et al., 2022). Occupancy probability can then be derived via a probit link,

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$16

The method reduces computation by combining extracted facets from a Gaussian Mixture Model with local GP-block techniques. Only nonplanar or slope points and GMM outliers are used as GP residual targets, and the façade domain is partitioned into local blocks based on a covariance threshold derived from the Squared Exponential kernel (Zou et al., 2022). Compared to OctoMap, GPOM, BGKOctoMap, LARD-HM, and GPIS, the reported method achieved a higher Precision-Recall AUC for the evaluated buildings (Zou et al., 2022). A direct implication is that covariance here serves both as a geometric uncertainty descriptor and as a selection variable for downstream localization or registration, since uncertain regions can be down-weighted by inverse variance (Zou et al., 2022).

6. Geometric propagation, physical constraints, and limitations

When poses and map elements live on nonlinear manifolds such as $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$17 or $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$18, covariance must be represented intrinsically on the Lie group rather than in a Euclidean parameterization. On unimodular matrix Lie groups with surjective exponential maps, the mean is defined by a zero log-moment condition and the covariance lives in the tangent space at the mean:

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$19

Exact continuous-time propagation follows from the Fokker–Planck equation, and a closed-form second-order propagation formula is derived in terms of Lie-group Jacobians and adjoint operators (Ye et al., 2023). The same work states the left/right covariance transformation

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$20

which is central for frame-consistent map and pose uncertainty on $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$21 (Ye et al., 2023).

A different form of covariance-based map uncertainty appears in physically constrained covariance inflation from location uncertainty. There, the forecast step applies a small random diffeomorphism

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$22

to the tensorial representation of the state, so that uncertainty arises as random transport rather than as algebraic rescaling (Zhen et al., 2022). The induced stochastic dynamics preserve user-selected invariants by construction—mass, energy, vorticity, or helicity, depending on the tensor association—and yield an inflation operator shaped by the displacement fields $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$23 instead of a scalar multiplicative inflation (Zhen et al., 2022). This provides a physical counterpart to the map-uncertainty idea: the covariance reflects structured displacement of features, not only pointwise amplitude noise.

Several recurrent limitations are explicit across the literature. Strong nonlinearity may invalidate the Gaussian posterior approximation and require full uncertainty quantification by sampling such as MCMC (Saibaba et al., 2014). In mapless trajectory prediction, single-vertex Gaussian uncertainty may miss multi-modal or topological ambiguity such as forks, and no explicit temporal filtering of covariances is reported (Zhang et al., 24 Jul 2025). In LRPD online mapping, inter-element correlations are not modeled and overly small diagonal terms can produce overconfidence (Gogoi et al., 20 Mar 2026). In GP occupancy mapping, large pose uncertainty can smooth or fade maps, and WGP inference remains approximate (Jadidi et al., 2017). In localization, submap independence in sC-SKF is conservative by theorem but relaxes cross-submap information and slightly degrades accuracy (Dutoit et al., 2016).

A further methodological issue is that the covariance function itself may be uncertain. For stationary processes whose covariance functions agree on a finite discrete set of lags, the maximal discrepancy outside that grid can be upper-bounded by a finite-dimensional convex optimization problem,

$\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$24

where $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$25 and $\Gamma_{\text{post}} = (H + \Gamma_{\text{prior}^{-1})^{-1}$26 is a trigonometric-polynomial subspace (Elvander et al., 2021). This does not replace map covariance estimation, but it formalizes the uncertainty of the covariance model itself.

Taken together, these results define covariance-based map uncertainty as a family of probabilistic representations in which the decisive modeling choice is not merely whether uncertainty is present, but how its covariance is structured: low-rank or dense, diagonal or correlated, Euclidean or Lie-group intrinsic, local or cross-state, deterministic proxy or posterior object. The literature consistently treats those structural choices as consequential for consistency, calibration, tractability, and downstream decision-making (Dutoit et al., 2016, Gogoi et al., 20 Mar 2026, Ye et al., 2023).