Epistemic Uncertainty-Driven Sensor Placement

Updated 4 December 2025

Epistemic uncertainty driven sensor placement is a method that quantifies and minimizes reducible uncertainty using Bayesian design and information theory.
It leverages advanced computational techniques such as Monte Carlo sampling, convex relaxation, and deep reinforcement learning to optimize sensor configurations.
Applications in structural monitoring, environmental sensing, and geophysical inversion demonstrate its effectiveness in reducing uncertainty and enhancing decision outcomes.

Epistemic uncertainty driven sensor placement refers to the systematic design of sensor networks such that the chosen locations maximally reduce the uncertainty in model parameters, predictions, or downstream decision variables resulting from incomplete knowledge about the system state, dynamics, or physical parameters. In contrast to strategies that ignore uncertainty or focus solely on coverage or observability under deterministic conditions, epistemic-uncertainty-driven approaches explicitly quantify and target the reducible (“epistemic”) component of uncertainty, often optimizing for maximum expected information gain, minimum posterior variance, or other information-theoretic and Bayesian criteria. Contemporary methods synthesize tools from Bayesian inference, optimal experimental design, information theory, deep learning, and combinatorial optimization to efficiently search large configuration spaces and accommodate complex sources of uncertainty.

1. Modeling and Quantification of Epistemic Uncertainty

Epistemic uncertainty arises from lack of knowledge about system parameters, initial/boundary conditions, or model inadequacy. In the context of sensor placement, its formalization typically involves:

Bayesian parameter/prior models: Assign a Gaussian or non-Gaussian prior $p(\theta)$ over uncertain parameters $\theta$ ; epistemic uncertainty is then posterior variance/entropy after incorporating data.
Process and input uncertainty: Inputs such as stochastic ground motion (Jabini et al., 2023), uncertain flow conditions (Sharma et al., 2018), or environmental fields (Eksen et al., 27 Nov 2025) are expressed as random variables or sampled from empirical distributions.
Predictive variance decomposition: In neural processes, predictive variance is split as $\sigma^2_{\mathrm{Var}}(x|C) = \sigma^2_{\mathrm{Ep}}(x|C) + \sigma^2_{\mathrm{Al}}(x|C)$ , isolating epistemic from irreducible aleatoric uncertainty (Eksen et al., 27 Nov 2025).
Uncertainty propagation: Through transfer operators (Sharma et al., 2018), Bayesian filtering (Poudel et al., 31 Jan 2025), or factor-graph models (Denniston et al., 4 May 2024), epistemic uncertainty is propagated in space and/or time to inform placement.

Monte Carlo sampling, Fisher information matrices, or closed-form Bayesian/posterior updates are used for tractable quantification, often leveraging Gaussian approximations.

2. Information-Theoretic and Bayesian Optimal Design Criteria

The driving metric in epistemic-uncertainty-based sensor placement is an information-theoretic or Bayesian design criterion that directly quantifies the expected reduction in uncertainty. Core formulations include:

Expected Information Gain (EIG) / Mutual Information: Maximize $I(\theta; y_s) = H(\theta) - H(\theta|y_s)$ , where $y_s$ is the observation vector for sensor subset $s$ (Bhattacharyya et al., 2019, Alexanderian et al., 31 Jan 2025).
Posterior entropy/variance reduction: Minimize $\mathbb{E}_{y_s}[H(\theta|y_s)]$ or $\operatorname{tr}(C_{\mathrm{post}})$ , i.e., the expected posterior entropy/variance (Jabini et al., 2023, Madhavan et al., 20 Feb 2025).
Fisher Information-based reward: Use the increase in the Fisher information matrix (e.g., $\Delta H(\theta) = \frac{1}{2}\log|F + P_0^{-1}| - \frac{1}{2}\log|P_0^{-1}|$ ) as the information gain at each placement (Jabini et al., 2023).
Acquisition functions for ML models: For deep neural processes, the acquisition is the expected reduction in epistemic uncertainty post-placement, e.g., $x_i^* = \arg\min_{x_i} \frac{1}{N_t}\sum_j \sigma^2_{\mathrm{Ep}}(x_j|C \cup \{(x_i, \hat{y}_i)\})$ (Eksen et al., 27 Nov 2025).
Downstream-oriented criteria: Minimize uncertainty in control/decision objectives (“control-oriented A-criterion”) as opposed to merely reducing parameter variance (Madhavan et al., 20 Feb 2025, Poudel et al., 31 Jan 2025).

The choice of design metric has a transformative impact on placement, focusing sensor resources on regions or modal directions where epistemic uncertainty is most critical for inference or control.

3. Algorithmic Methodologies for Epistemic-Driven Sensor Placement

Sophisticated algorithms underpin modern epistemic-uncertainty-driven sensor placement, exploiting both combinatorial optimization and machine learning.

Convex Relaxation and Continuous Optimization: By relaxing binary sensor selection to continuous weights within $[0,1]^m$ (with a sum constraint on weight), one obtains a convex optimization problem for mutual information maximization, often solved by Newton or interior-point methods (Bhattacharyya et al., 2019).
Greedy and Submodular Optimization: Greedy placement is widely used when the design criterion is submodular, yielding near-optimal solutions (within $(1-1/e)$ of optimal). Notable cases include set-cover in PF frameworks (Sharma et al., 2018), expected mutual information (Alexanderian et al., 31 Jan 2025), and context-relevant mutual information (CRMI, (Poudel et al., 31 Jan 2025)).
Bayesian Deep Reinforcement Learning: Sensor placement is cast as a Markov Decision Process (MDP) where states encode partial sensor masks, actions correspond to next placement, and the reward is the Monte Carlo-estimated information gain; deep Q-networks (DQN, DDQN) optimize the placement sequence (Jabini et al., 2023).
Neural Acquisition for Black-Box Models: For data-driven spatio-temporal fields, neural-process-based models use MDN heads to estimate epistemic/aleatoric variance and greedy selection over candidate points (Eksen et al., 27 Nov 2025).
Fast Linear Algebra and Matrix Sketching: Large-scale Bayesian designs use randomized trace/determinant estimation and column subset selection (CSSP) to compute D-optimal/EIG placement efficiently (Alexanderian et al., 31 Jan 2025).
Observability Coefficient Maximization: In hyperparameterized linear inverse problems, sensor sets are chosen to maximize a worst-case observability coefficient, with OMP-style greedy selection and surrogate modeling (Aretz et al., 2023).

Algorithmic choices are strongly governed by problem structure (e.g., model linearity, Gaussian assumptions, sensor budget, spatial domain size), with scalability and tractability driving method selection.

4. Application Domains and Empirical Findings

Epistemic-uncertainty-driven sensor placement methods have been adapted across domains, with distinct empirical insights:

Domain/Setting	Driving Uncertainty Source	Key Findings
Structural monitoring (Jabini et al., 2023, Bhattacharyya et al., 2019)	Gaussian stiffness/damping priors, stochastic excitation	Learned policies outperformed random/greedy by targeting the modes most sensitive to epistemic reduction
Indoor air quality & contaminant sensing (Sharma et al., 2018)	Uncertain occupancy/flows/boundary conditions	PF-based coverage maximization under uncertainty hedges against all scenarios, spreading sensors to “hedge”
Coupled path planning (Poudel et al., 31 Jan 2025)	Time/space-varying threat field	CRMI approach yields $\geq$ 50% reduction in required measurements, focusing on decision-critical regions
Environmental field modeling (Eksen et al., 27 Nov 2025)	Spatio-temporally variable SST fields, epistemic ML	Targeting epistemic (not total) variance accelerates error/nll improvement over random/total-variance policies
PDE-constrained optimal control (Madhavan et al., 20 Feb 2025)	Background source param/posterior uncertainties	cOED policies concentrate sensors on high-influence regions for the control target
Large-scale geophysical inversion (Aretz et al., 2023)	Family of admissible configurations, correlated observation noise	OMP greedy targeting worst-case epistemic uncertainty delivers scalable, competitive design

Empirical results consistently indicate that strategies focusing on epistemic uncertainty yield more efficient, robust, and decision-aligned sensor placements, either by accelerating error collapse or by minimizing variance in critical predictions and decision variables.

5. Computational Considerations and Scalability

Computation is frequently the bottleneck in high-dimensional or combinatorial settings. Notable approaches include:

Monte Carlo and Gaussian approximations: Replace analytically intractable information integrals by Monte Carlo estimation, leveraging closed-form Gaussian results for entropy and Fisher information (Jabini et al., 2023).
Precomputation and surrogate modeling: Surrogate or reduced models allow for rapid evaluation of information/observability metrics for large candidate sets (Aretz et al., 2023).
Matrix-free and randomized algorithms: SLQ, Nyström methods, and adjoint-free sketching facilitate scalable EIG and mutual information estimation without direct access to large Jacobians (Alexanderian et al., 31 Jan 2025).
Experience replay and batching: Deep RL methods amortize learning and sampling through batching and memory (Jabini et al., 2023, Eksen et al., 27 Nov 2025).
Efficient updating under correlated noise: Cholesky-based updates enable OMP-style selection with full-rank noise models (Aretz et al., 2023).
Submodular optimization guarantees: For set-cover and CRMI, greedy strategies exploit submodularity for provable near-optimality and polynomial time complexity (Sharma et al., 2018, Poudel et al., 31 Jan 2025).

A plausible implication is that computational innovations are essential for the practical deployment of epistemic-uncertainty-driven methods in real-world, high-dimensional scenarios.

6. Limitations and Directions for Extension

Key limitations and open directions include:

Assumptions of Gaussianity and linearity; most methods rely on analytical tractability of entropy, covariance, or information gain under these assumptions (Alexanderian et al., 31 Jan 2025, Bhattacharyya et al., 2019).
Scalability challenges as the number of uncertain inputs or environmental scenarios increases combinatorially, motivating sparse-grid and multifidelity approaches (Sharma et al., 2018).
Dynamic and adaptive sensor placement under time-varying or path-dependent uncertainty, requiring integration with control and feedback planning frameworks (Poudel et al., 31 Jan 2025, Denniston et al., 4 May 2024).
Robustness to model misspecification and non-Gaussianity; extensions to min-max design, Rényi-divergence criteria, and ensemble- or surrogate-based approximations are an active area (Alexanderian et al., 31 Jan 2025).
Incorporation of actuation (in emitter/sensor placement) and other multi-criteria objectives (cost, risk, latency) in design frameworks.

These limitations suggest ongoing cross-fertilization between optimal experimental design, Bayesian inverse problems, active learning, reinforcement learning, and control theory will continue to drive advances in epistemic-uncertainty-driven sensor placement.