Belief Space

In partially observable settings, agents must act without full knowledge of the world state and rely on uncertain state-estimation pipelines. Obtaining grounded and verifiable symbolic plans under such uncertainty remains a key challenge. Recent work has integrated Vision-Language Models (VLMs) to bridge perception and symbolic reasoning, following two main paradigms. The first, VLM-as-planner, maps images directly to action sequences, and the second, VLM-as-grounder, grounds observations into symbolic predicates used as the initial state by off-the-shelf planners. Both approaches ignore uncertainty in the planning process, compromising robustness. We introduce a third paradigm, VLM-as-probabilistic-grounder, a novel approach that captures the uncertainty of VLM predicate groundings as a probability distribution over symbolic states. This enables planning in belief space and producing robust plans under uncertainty. Experiments in simulated household robot settings show improved robustness and task success over deterministic grounding, underscoring how our approach leverages foundation models for reliable planning under uncertainty.

An agent chooses an action based on her private information and a recommendation from an informed but potentially misaligned adviser. With a known probability, the adviser truthfully reports his signal; with the remaining probability, he can send any message. We characterize optimal robust decision rules that maximize the agent's worst-case expected payoff. Every optimal rule is equivalent to a trust-region policy in belief space: the adviser's reported beliefs are taken at face value if they fall within the trust region but are otherwise clipped to the trust region's boundary. We derive alignment thresholds above which advice is strictly valuable and fully characterize the solution in both binary-state and binary-action environments.

Autonomous exploration missions require decision-making under sensor uncertainty and computational constraints, yet integrating scientific representations into POMDP planning has remained intractable due to high-dimensional observation spaces. Information-theoretic planners overcome this by assuming deterministic observations, sacrificing the principled uncertainty quantification that POMDPs provide. We introduce the Science Hypothesis Map POMDP (SHM-POMDP), which makes science-driven belief-space planning more tractable by branching on inferred physical properties rather than raw sensor data. This preserves full sensor information through learned observation models while enabling the planner to reason jointly about navigation and scientific properties under uncertainty. On an extended RockSample domain with 50-dimensional observations, SHM-POMDP achieves 18.6% higher rewards and 32.9% reduced computation time per step than continuous-observation baselines. On realistic geologic exploration using Cuprite hyperspectral data, SHM-POMDP achieves 2.5$\times$ higher information gain than the best information-theoretic baseline by maintaining beliefs and replanning adaptively---reaching 80% of oracle performance using only uniform priors. These results demonstrate that integrating hierarchical probabilistic models into belief-space planning enables tractable, principled autonomous science that outperforms both traditional POMDP methods and science-aware information-theoretic approaches.

Computational cognitive modeling seeks to infer latent cognitive mechanisms underlying observed behavior. Bayesian inverse planning provides a principled framework for such inference, but its success depends critically on the experimental environment. Existing approaches typically treat environments as fixed, leaving open the question of which cognitive experiments are most informative for cognition parameter inference. We formulate the design of cognitive planning experiments as a Bayesian Experimental Design (BED) problem, treating the experimental environment as the design variable. We establish an exact Monte Carlo BED benchmark and introduce an amortized Bayesian experimental design framework for efficient posterior inference and design evaluation. Experiments on the Mouselab-MDP process-tracing paradigm show that amortized BED closely matches the environment rankings of exact Monte Carlo BED while substantially reducing computational cost. We further show that no single environment is uniformly optimal across cognitive inference objectives, revealing trade-offs between expected information gain, posterior recoverability, and information efficiency. These results provide a principled framework for designing informative cognitive experiments for Bayesian parameter inference.

Sophisticated Inference is a variant of active inference often associated with recursive belief modeling and tree search. We argue that its central computational role is simpler: within a planning horizon, it makes active inference closed-loop by allowing future actions to depend on future states and observations. This closed-loop structure can be represented in the epistemic-prior variational free energy framework. Epistemic priors supply the active-inference objective, while a joint posterior over future states and actions supplies the state-contingent control structure. We evaluate this decomposition in the Reactivity Maze, a stochastic benchmark designed to separate epistemic incentive from inner-horizon closed-loop control. The comparison includes three variational objectives with the same state-action posterior family, an action-state factorized active inference objective, Sophisticated Inference, and standard Expected Free Energy planning. The results show that neither ingredient is sufficient on its own. Methods without an epistemic component do not seek information, while methods that prevent future actions from depending on future states cannot turn information into reliable goal-reaching. By contrast, both Sophisticated Inference and full-joint epistemic-prior active inference solve the environment by combining epistemic drive with closed-loop inference. These results show that the advantage associated with Sophisticated Inference need not be specific to tree search itself. It arises from the closed-loop form of active inference, and this form can be represented in epistemic-prior variational inference when the posterior keeps future actions dependent on future states.

Physical sensing and actuation noise floors should inform how much belief resolution a decision-making system can reliably use. We introduce Finite Reliability Representations (FRR), a framework for covering belief spaces by reliability cells: regions within which the optimal action-value function Q*(b,u) varies by at most a tolerance epsilon, uniformly over actions. The framework is formulated on beliefs rather than states and uses a cover rather than an equivalence quotient, because approximate decision-closeness is not transitive in general. A central technical point is that noisy Bayesian updates should not be treated as globally contractive on arbitrary beliefs. We therefore separate three objects: the fixed-observation filter map, the predictive observation law, and the controlled belief-transition kernel. For nonlinear continuous-state systems, FRR is obtained under a reachable-set Lipschitz modulus for the belief-transition kernel. For finite-state POMDPs, the same construction becomes exact on the belief simplex: prediction is linear, Bayesian correction is a normalized positive linear map, sensor noise enters through observation-distribution distinguishability, and actuation uncertainty enters through an action-execution channel. Under the corresponding action-value Lipschitz condition, an FRR cover supports a cell-constant policy whose suboptimality is bounded by 2 epsilon/(1 - gamma). We also introduce reliability entropy, the logarithm of the minimal number of reliability cells, as a measure of certified decision-relevant belief complexity. The framework distinguishes representation sufficiency from fundamental performance floors imposed by sensing, process, and actuation noise. It applies to finite POMDPs, linear-Gaussian filters, locally linearized nonlinear filters, and particle-filter implementations through analytic or empirical certification of reliability cells.

We address sampling-based motion planning for continuous-time stochastic systems under process and measurement uncertainty, with probabilistic guarantees on safety and performance. The robot dynamics are modeled as a continuous-time linear stochastic differential equation, while sensor measurements arrive at discrete time instants. We derive an offline hybrid belief propagation model in which the belief evolves according to continuous-time ODEs between measurements and undergoes discrete Kalman filter update jumps at measurement times. To ensure safety, we introduce a belief-barrier-function-based safety checker for segment-level probabilistic verification. This enables the planner to certify safety over entire continuous trajectory segments and detect inter-sample chance-constraint violations that are missed by conventional node-based checks. Together, these components provide a principled framework for sampling-based belief planning that accounts for both continuous-time uncertainty propagation and continuous-time safety requirements. We integrate the method with RRT and SST planners and evaluate it across multiple benchmark environments. The results show that the proposed method achieves high success rates and robust enforcement of chance constraints, including in narrow-passage scenarios where discrete-time counterparts fail due to missed inter-sample unsafe behavior.

Semantic navigation is a fundamental task for embodied agents operating in unseen environments, requiring both semantic understanding and long-term decision-making. Recent foundation models have empowered agents with rich semantic priors for this task. However, without structured global representations, decision-making often falls back on local observations and greedy strategies, resulting in inefficient exploration and myopic behaviors, especially in long-distance navigation. To address these challenges, we propose a zero-shot semantic navigation framework. Our method incrementally maintains an online Hierarchical 3D Scene Graph (HSG) to form a multi-granular semantic topology over objects, zones, and regions, serving as a compact state abstraction for global planning. Building on this memory, we introduce a hierarchical belief-based planning framework that fuses semantic priors with exploration evidence on the HSG, and performs finite-horizon rollouts on an HSG-based simulator to explicitly estimate the long-term expected returns of candidate macro-actions. This enables globally consistent decisions and reduces redundant backtracking. Extensive experiments in high-fidelity simulation environments across multiple tasks and datasets demonstrate that our method outperforms existing state-of-the-art methods, particularly in long-distance scenarios, where our approach improves SR and SPL by an average of 9.4% and 5.0%, respectively.

Belief-space planning under motion uncertainty and state and control constraints remains a fundamental challenge, largely due to the difficulty of establishing reachability guarantees in constrained belief spaces. Existing constrained belief-space planners rely on sampling to construct multi-query belief roadmaps and explicitly find feasible trajectories between sampled nodes to establish reachability. These methods often struggle to cover the belief space or use robust control techniques that improve coverage at the cost of indirect, high-cost trajectories; they also lack finite-time or finite-memory completeness guarantees. We propose PRISM, a multi-query motion planning algorithm for belief spaces with state and control constraints that targets both high coverage and low cost. We present a new result on controllability of the state covariance under constraints, which is used by PRISM to decompose belief-space planning into deterministic mean planning and covariance shrinking. PRISM further includes an online local optimization method that reduces the cost of feasible belief-space trajectories. Under mild assumptions on the start and goal distributions, we prove that PRISM guarantees full coverage (i.e. completeness) despite actuator and obstacle constraints. In challenging simulated scenarios, PRISM achieves substantially higher roadmap coverage than state-of-the-art belief-space planning methods while producing trajectories with lower mean cost and cost variance. For example, PRISM achieves 100% coverage in easy and medium-difficulty scenarios, and, in the hardest scenario, which violates PRISM's coverage assumptions, it still achieves 97-100% coverage, while all other methods achieve less than 45%.

Decision making in lithium production is challenging, whether from an investor's perspective or a strategic production standpoint. Determining which mines to open and when to open them involves not only geological and price uncertainties, but also complexities around the choice of extraction method, from direct lithium extraction to hard rock mining. Prior work explored models of this problem and different methods to optimize mining decisions; these models did not account for uncertainty in pricing, uncertainty in demand, or different mining technologies to extract lithium. Incorporating different pricing models and extraction technology into these models enables more robust strategies for determining not only when and where to open a mine, but also which method of production to pursue. We frame the problem as a partially observable Markov decision process (POMDP) and solve using belief state planning methods to get optimal decision making. In our study, we show that POMDP solvers outperform human inspired heuristics by dynamically adapting to shifting lithium price regimes (static, linear, exponential, and stochastic) through belief state planning and explicit uncertainty management. By optimally sequencing exploration, production, and technology choice, the framework achieves higher demand fulfillment and more balanced economic environmental outcomes over the projects lifetime in all different pricing and deposit scenarios.

Real-world robot task planning must operate under both stochastic action execution and partial observability, yet constructing Partially Observable Markov Decision Process (POMDP) models for real robotics domains remains difficult and labor-intensive. We introduce PO-PDDL, a symbolic formulation of POMDPs that preserves the relational structure and LLM-friendly syntax of the Planning Domain Definition Language (PDDL), while explicitly modeling partial observability, stochasticity, and beliefs. Building on this formulation, we propose a demonstration-driven pipeline for learning PO-PDDL models. The proposed method reconstructs latent symbolic state trajectories from real-robot execution videos, identifies partial observability via inconsistencies between inferred states and visual observations, and learns stochastic transition and observation models accordingly. The resulting PO-PDDL domains are reusable across tasks and enable online belief-space planning under both perception and execution uncertainty. Experiments on real-world long-horizon manipulation tasks show that our method consistently outperforms existing PDDL and POMDP model-learning approaches, achieving robust task planning under uncertainty with significantly lower planning cost.

Reliable trajectory planning under partial observability depends not only on computing a feasible geometric path, but also on whether the robot receives informative observations while executing that trajectory. Existing approaches usually keep the environment fixed and adapt the robot through belief-space planning, active localization, or added sensing, often incurring costly uncertainty propagation and brittle behavior in observation-poor regions. We flip this perspective and address the largely open problem of \emph{task-aware environment augmentation}: given a mapped environment, a planned task trajectory, and a small budget of visual fiducial markers, where should the environment be augmented so that the planned trajectory can be executed reliably under uncertainty? Our key observation is that useful marker layouts are defined by the localization support they provide along the task trajectory: a small number of well-timed observations can be sufficient to prevent uncertainty from accumulating in regions where state-estimation error would otherwise compromise control. Building on this observation, we present \tbp{SCoDA}, $\textbf{S}$hielded $\textbf{Co}$nditional $\textbf{D}$iffusion for Environment $\textbf{A}$ugmentation. \tbp{SCoDA} learns a conditional distribution over high-performing fiducial layouts from data, using the environment, planned trajectory, disturbance context, and desired execution profile as conditioning. Its shielded sampler reasons over where along the planned execution pose corrections should occur, and steers this distribution toward task-relevant, finite-budget augmentations. Across simulated benchmarks and hardware deployments, we show that \tbp{SCoDA} improves trajectory execution reliability and completion time over strong baselines. Code, models and dataset available at: \hyperlink{scoda-diffusion.github.io}{https://scoda-diffusion.github.io/}

Cancer treatment is at the core a sequential decision-making problem with partial observability, latent patient heterogeneity, and explicit constraints on the budget for medical measurements. Unlike standard Reinforcement Learning (RL) approaches that control state trajectories, cancer treatments permanently modify patients' transition dynamics, changing how states evolve over time. We model cancer treatment as a belief-space planning problem using active inference, deriving an expected free-energy objective that unifies goal-directed control and information acquisition under measurement budgets without. We implement this framework using real clinical cancer data from the AACR Project GENIE Biopharma Collaborative dataset. Results on clinical data demonstrate a simultaneous patient categorization and high treatment efficacy, under real measurement and treatment constraints.

Long-horizon interactions require language models to manage accumulating information: when to update their state, when to preserve their state, and what to ignore. We study this challenge as Contextual Belief Management (CBM): maintaining a predicted belief state aligned with formal evidence while isolating task-irrelevant noise. To make CBM measurable, we introduce BeliefTrack, a closed-world benchmark spanning Rule Discovery and Circuit Diagnosis, where a finite belief space and symbolic verifiers enable exact turn-level evaluation. BeliefTrack diagnoses three failures: Failed Stay, Failed Update, and Failed Isolation. Across multiple LLMs, vanilla models exhibit severe CBM failures, while explicit belief-tracking prompts provide limited gains. In contrast, reinforcement learning with belief-state rewards reduces failure rates by 70.9% on average. Further probing reveals latent belief-state dynamics behind these failures, and representation-level steering reduces failure rates by 46.1% across two tasks (Code is available at https://github.com/zjunlp/CBM).

World models for partially observed environments must imagine multiple compatible hidden futures and steer between them under counterfactual actions. Joint Embedding Predictive Architectures (JEPAs) do this in latent space, but a vector-valued latent has no internal structure for carrying the belief over hidden continuations through blind rollout. We introduce the Unitary World Model JEPA (UWM-JEPA), a JEPA world model with a density-matrix latent on a joint system-environment space and a learned unitary predictor. The construction preserves the joint-state spectrum exactly during rollout, so the predictor itself cannot dissipate the represented uncertainty. On a hidden-velocity indicator task requiring five-step forward simulation under a given action sequence with the target observation masked, UWM-JEPA reaches 0.77 accuracy and degrades monotonically as actions are perturbed; a parameter-matched LSTM-JEPA trained under the same counterfactual-target objective and action head collapses to majority-class accuracy (0.53) under every action condition. Under blind rollout, UWM-JEPA loses fewer than ten points of probe R^2 at short horizons while vector-latent baselines lose forty-one and sixty-eight; both nevertheless tie on a held-out context probe, locating the separation in the predictor rather than the encoder. Action sensitivity itself requires training against counterfactual rather than teacher-forced targets, a finding that applies beyond the unitary parameterisation. For JEPA world models to imagine under partial observability, latent geometry and predictor dynamics matter, not frozen context-encoding capacity alone.

A "path-based sensor" produces a single observation along a continuous path. For example, a boolean path-based sensor returns a single "1" if an event of interest is detected at any point along the path and a "0" otherwise. Notably, a "1" provides no direct information about where along the path the event(s) may have occurred. Previous work has demonstrated that observations from multiple path-based sensors can be fused to create a Bayesian belief map over the spatial locations of the underlying event or phenomenon. Moreover, path planning can employ Shannon information theory to accelerate the rate of convergence of the belief map. In this paper, we present a new method to update the belief map based on a path-based sensor observation, and then plan paths to increase information gain. In contrast to prior work that approximates the posterior by averaging over the alternative event histories, we introduce a Bayesian Network (BN) formulation that models the probabilistic relationships between the latent variables and path-based sensor measurements, enabling a more principled Bayesian belief update. We consider static hazard detection in a communication-denied environment as a representative problem setting. The event of a robot returning from its path corresponds to a path-based hazard sensor reading of "0" (hazard not detected), while a robot failing to return corresponds to a reading of "1" (hazard detected). We consider false positives and false negatives. We find that the new method leads to quicker convergence of the belief map than prior work in both single- and multi-robot cases.

Many safety-critical control problems are modeled as risk-sensitive partially observable Markov decision processes, where the controller must make decisions from incomplete observations while balancing task performance against safety risk. Although belief-space planning provides a principled solution, maintaining and planning over beliefs can be computationally costly and sensitive to model specification in practical domains. We propose a lightweight risk-gated reinforcement learning approximation for risk-sensitive control under partial observability. The method constructs a compact finite-history proxy state and learns an action-conditioned predictor of near-term safety violation. This predicted candidate-action risk is used in two complementary ways: as a risk penalty during value learning, and as a decision-time gate that interpolates between optimistic and conservative ensemble value estimates. As a result, low-risk actions are evaluated closer to reward-seeking estimates, while high-risk actions are evaluated more conservatively. We evaluate the approach in two safety-critical partially observable domains: automated glucose regulation and safety-constrained navigation. Across adult and adolescent glucose-control cohorts, the method improves overall glycemic tradeoffs and substantially reduces runtime relative to a belief-space planning baseline. On Safety-Gym navigation benchmarks, it achieves a more favorable reward-cost balance than unconstrained RL and several standard safe-RL baselines. These results suggest that action-conditioned near-term risk can provide an effective local signal for approximate risk-sensitive POMDP control when full belief-space planning is impractical.

Embodied agents in household environments must plan under partial observation: they need to remember objects, track state changes, and recover when actions fail. Existing benchmarks only partially test this ability. Egocentric video datasets capture realistic human activities but remain passive, while interactive simulators support execution but rely on synthetic scenes and hand-crafted dynamics, introducing a sim-to-real gap and often assuming fully observable state. We introduce Ego2World, an executable benchmark that turns egocentric cooking videos into executable symbolic worlds governed by graph-transition rules. Built on HD-EPIC, Ego2World derives reusable transition rules from video annotations and executes them in a hidden symbolic world graph. During evaluation, the simulator maintains the hidden world graph, while the agent plans over its own partial belief graph using only local observations and execution feedback. This separation forces agents to update memory and replan without observing the true world state. Experiments show that action-overlap scores overestimate physical-state success, and that persistent belief memory improves task completion while reducing repeated visual exploration -- suggesting that belief maintenance should be a first-class target of embodied-agent evaluation.

Multi-agent debate has emerged as a promising approach for improving LLM reasoning on ground-truth tasks, yet current methodologies face certain structural limitations: debate tends to induce a martingale over belief trajectories, majority voting accounts for most observed gains, and LLMs exhibit confidence escalation rather than calibration across rounds. We argue that the genuine value of debate, and dialectic systems as a whole, lies not in ground-truth tasks but in defeasible domains, where every position can in principle be defeated by better reasoning. We present the Council of Hierarchical Agentic Language (CHAL), a multi-agent dialectic framework that treats defeasible argumentation as an engine for belief optimization. Each agent maintains a CHAL Belief Schema (CBS), a graph-structured belief representation with a Bayesian-inspired architecture, that facilitates belief revision through a gradient-informed dynamic mechanism by leveraging the strength of the belief's thesis as a differentiable objective. Meta-cognitive value systems spanning epistemology, logic, and ethics are elevated to configurable hyperparameters governing agent reasoning and adjudication outcomes. We provide a series of ablation experiments that demonstrate systematic and interpretable effects: the adjudicator's value system determines the debate's overall trajectories in latent belief space, council diversity refines beliefs for all participants, and the framework generalizes across broad fields. CHAL is, to our knowledge, the first framework to treat multi-agent debate as structured belief optimization over defeasible domains. Further, the auditable belief artifacts it produces establish the foundation for dedicated evaluation suites for defeasible argumentation, with broader implications for building AI systems whose reasoning and value commitments are transparent, aligned, and subject to human oversight.

Large Language Models (LLMs) update their behavior in context, which can be viewed as a form of Bayesian inference. However, the structure of the latent hypothesis space over which this inference operates remains unclear. In this work, we propose that LLMs assign beliefs over a low-dimensional geometric space - a conceptual belief space - and that in-context learning corresponds to a trajectory through this space as beliefs are updated over time. Using story understanding as a natural setting for dynamic belief updating, we combine behavioral and representational analyses to study these trajectories. We find that (1) belief updates are well-described as trajectories on low-dimensional, structured manifolds; (2) this structure is reflected consistently in both model behavior and internal representations and can be decoded with simple linear probes to predict behavior; and (3) interventions on these representations causally steer belief trajectories, with effects that can be predicted from the geometry of the conceptual space. Together, our results provide a geometric account of belief dynamics in LLMs, grounding Bayesian interpretations of in-context learning in structured conceptual representations.

In many real-world robotic tasks, robots must generate dynamically feasible motions that reliably reach desired goals even under uncertainty. Yet existing sampling-based kinodynamic planners typically optimize accumulated trajectory costs and treat goal reaching as a feasibility check, rather than explicitly optimizing terminal-state quality, such as goal preference or goal-reaching reliability. In this work, we introduce a terminal-cost formulation for kinodynamic planning that allows terminal-state quality to be optimized alongside accumulated trajectory cost. We prove that AO-RRT, an asymptotically optimal kinodynamic planner, preserves its asymptotic optimality under this augmented objective. We further extend the formulation to belief space and prove that minimizing the Wasserstein distance between the terminal belief and the goal improves a lower bound on the probability of reaching the goal region. The resulting planner, KiTe, uses this terminal-cost objective to encode goal preferences and improve reliability under uncertainty. To support systems without analytical uncertainty models, we learn dynamics and process uncertainty directly from data and integrate the learned belief dynamics into planning. Experiments on Flappy Bird, Car Parking, and Planar Pushing show that KiTe consistently improves goal-reaching success under uncertainty. Real-world Planar Pushing experiments further demonstrate that KiTe can plan effectively with learned dynamics and uncertainty. Source code is available at https://github.com/elpis-lab/KiTe.

Learning-rate steps are usually treated as hyperparameters. This paper isolates a local beliefspace calculation: when an update is modeled as a projected forward step on the probability simplex, admissibility means contractivity in the natural KL/Bregman geometry. Under this model, the upper bound of an admissible step is not a tuning slogan but a formula.

This paper explores the benefits of computing arborescent trajectories (trajectory-trees) instead of commonly used sequential trajectories for partially observable robotic planning problems. In such environments, a robot infers knowledge from observations, and the optimal course of action depends on these observations. \revise{Trajectory-trees, optimized in belief space, naturally capture this dependency by branching where the belief state is expected to evolve into multiple distinct scenarios, such as upon receiving an observation. Unlike sequential trajectories, which model a single forward evolution of the system, trajectory-trees capture multiple possible contingencies.} First, we focus on Model Predictive Control (MPC) and demonstrate the benefits of planning tree-like trajectories. We formulate the control problem as the optimization of a tree with a single branching (PO-MPC). This improves performance by reducing control costs through more informed planning. To satisfy the real-time constraints of MPC, we develop an optimization algorithm called Distributed Augmented Lagrangian (D-AuLa), which leverages the decomposability of the PO-MPC formulation to parallelize and accelerate the optimization. We apply the method to both linear and non-linear MPC problems using autonomous driving examples. Second, we address Task And Motion Planning (TAMP), and introduce a planner (PO-LGP) reasoning on decision trees at task level, and trajectory-trees at motion-planning level. This approach builds upon the Logic-Geometric-Programming Framework (LGP) and extends it to partially observable problems. The experiments show the method's applicability to problems with a small belief state size, and scales to larger problems by optimizing explorative policies, which are used as macro-actions in an overarching task plan.

{Closed-loop inverse source localization and characterization (ISLC) requires a mobile agent to select measurements that localize sources and infer latent field parameters under strict time constraints.} {The core challenge lies in the belief-space objective: valid uncertainty estimation requires expensive Bayesian inference, whereas using fast learned belief model leads to reward hacking, in which the policy exploits approximation errors rather than actually reducing uncertainty.} {We propose \textbf{Distill-Belief}, a teacher--student framework that decouples correctness from efficiency. A Bayes-correct particle-filter teacher maintains the posterior and supplies a dense information-gain signal, while a compact student distills the posterior into belief statistics for control and an uncertainty certificate for stopping. At deployment, only the student is used, yielding constant per-step cost.} {Experiments on seven field modalities and two stress tests show that Distill-Belief consistently reduces sensing cost and improves success, posterior contraction, and estimation accuracy over baselines, while mitigating reward hacking.}

Contact variability, sensing uncertainty, and external disturbances make grasp execution stochastic. Expected-quality objectives ignore tail outcomes and often select grasps that fail under adverse contact realizations. Risk-sensitive POMDPs address this failure mode, but many use particle-filter beliefs that scale poorly, obstruct gradient-based optimization, and estimate Conditional Value-at-Risk (CVaR) with high-variance approximations. We instead formulate grasp acquisition as variational inference over latent contact parameters and object pose, representing the belief with a differentiable Gaussian mixture. We use Gumbel-Softmax component selection and location-scale reparameterization to express samples as smooth functions of the belief parameters, enabling pathwise gradients through a differentiable CVaR surrogate for direct optimization of tail robustness. In simulation, our variational neural belief improves robust grasp success under contact-parameter uncertainty and exogenous force perturbations while reducing planning time by roughly an order of magnitude relative to particle-filter model-predictive control. On a serial-chain robot arm with a multifingered hand, we validate grasp-and-lift success under object-pose uncertainty against a Gaussian baseline. Both methods succeed on the tested perturbations, but our controller terminates in fewer steps and less wall-clock time while achieving a higher tactile grasp-quality proxy. Our learned belief also calibrates risk more accurately, keeping mean absolute calibration error below 0.14 across tested simulation regimes, compared with 0.58 for a Cross-Entropy Method planner.

We study finite-horizon quadratic control of linear systems with bilinear observations, in which the control input affects not only the state dynamics but also the partial observations of the state. In this setting, the separation principle can fail because control inputs influence the future quality of state estimates. State estimation requires an input-dependent Kalman filter whose gain and error covariance evolve as functions of the control inputs. To address this challenge, we propose a belief-space model predictive control ($\texttt{B-MPC}$) method that plans directly over both the estimated state and its error covariance. In particular, $\texttt{B-MPC}$ plans with a deterministic surrogate of the belief evolution defined by the input-dependent Kalman filter. Through numerical experiments in two synthetic settings, we show that $\texttt{B-MPC}$ can outperform both the separation-principle controller and its MPC variant in favorable regimes, and that these gains are accompanied by lower estimation covariance and more uncertainty-aware action choices.

Many safety-critical control systems operate under latent uncertainty that sensors cannot resolve at decision time. Such uncertainty, arising from unknown physical properties, disturbances, or unobserved geometry, affects dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider systems with measured state and an unknown, time-invariant parameter represented by a belief distribution. We propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) controller that plans under this belief, regularizes performance using Conditional Value-at-Risk (CVaR), and imposes a CVaR constraint on a trajectory safety margin over the horizon. For the exact risk-constrained formulation underlying this controller, we establish three properties: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the objective risk weight tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In contact-rich MuJoCo simulations of vision-guided dexterous stowing, where a manipulator inserts a grasped object into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared with 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces. Project page: https://clintonenwerem.com/belief-cvar-mppi/.