Dynamic Probabilistic Assurance

• Dynamic probabilistic assurance is a framework that quantifies risk and confidence with probabilistic metrics and updates these measures based on evolving evidence.
• It integrates methodologies like Bayesian updating, Gaussian processes, and model checking to connect runtime data with operational decisions.
• Applications span power systems, autonomous vehicles, and robotics, demonstrating improved risk mitigation and computational efficiency in real-time.

Dynamic probabilistic assurance denotes, in the surveyed literature, the quantification of confidence or risk in probabilistic terms together with mechanisms for revising that confidence as evidence, operating conditions, environments, or system configurations change. The concept appears across probabilistic security assessment in power systems, assurance measures for learning-enabled systems, probabilistic model checking for robotics and autonomous systems, runtime confidence updates in safety arguments, and dynamic or continuous assurance cases (Cremer et al., 2019, Asaadi et al., 2020, Dong et al., 2021, Herd et al., 21 May 2026, Calinescu et al., 2017). Across these strands, assurance is treated not as a static design-time artifact but as a quantity that can be computed, calibrated, propagated, monitored, and updated through runtime data, formal models, or structured assurance arguments.

1. Conceptual basis and scope

A central definition is the assurance measure, described as “a probabilistic quantification of confidence that an LES possesses system-level properties associated with functional capabilities and dependability attributes.” In the same vocabulary, an assurance property is a logical or probabilistic characteristic related to dependability attributes or functional capabilities, and an assurance claim applies such properties to a component or system to argue dependability (Asaadi et al., 2020). A closely related component-level view characterizes assurance for learning-enabled components by identifying relevant dependability attributes and quantifying those attributes and the associated uncertainty, using probabilistic techniques (Asaadi et al., 2023).

In power-system security assessment, the probabilistic perspective is expressed as the calculation and management of risk by explicitly incorporating the likelihood of operating conditions and contingencies and their associated outage costs, moving beyond deterministic approaches that treat all scenarios as equally likely or equally severe (Cremer et al., 2019). This broadens assurance from a Boolean secure/insecure judgment to a risk-aware decision process.

A further conceptual refinement comes from assurance-case research, which argues that confidence cannot be reduced to a single attribute or measurement. Instead, confidence is assessed from positive, negative, and residual doubt perspectives: positive support from evidence and argument, negative assessment through defeaters and challenges, and residual risk for unresolved doubts (Bloomfield et al., 2022). This suggests that dynamic probabilistic assurance is inherently multi-perspectival: it combines probabilistic support for claims with explicit treatment of uncertainty, challenges, and remaining risk.

2. Quantification of assurance, risk, and decision thresholds

A canonical probabilistic assurance measure is the probability that a safety-relevant variable remains within an acceptable envelope. In the autonomous taxiing example, the assurance property is expressed as

$$Pr(\text{Assured Taxi}) = Pr\left( |\mathrm{CTE}| < \text{offset} \right),$$

where CTE is cross-track error. The same framework allows thresholded runtime mitigation; for example, if

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$

then the associated assurance claim does not hold and mitigation is required (Asaadi et al., 2020).

In probabilistic dynamic security assessment for power systems, the transition from classifier output to operational assurance is made explicit. A calibrated classifier produces a probability estimate $\hat{p}^1(x_i)$ for the secure class, and the residual risks of predicting an operating condition as secure or insecure under contingency $c$ are

$$R^1_c(x_i) = C_c^{F1} \cdot p_c^C \cdot (1 - \hat{p}^1(x_i)),$$

$$R^0_c(x_i) = C_c^{F0} \cdot (1 - p_c^C) \cdot \hat{p}^1(x_i).$$

The corresponding cost-sensitive threshold is

$$z_c = \frac{C_c^{F1} p_c^C}{C_c^{F1} p_c^C + C_c^{F0} (1 - p_c^C)},$$

so prediction is “secure” iff $\hat{p}^1(x_i) > z_c$ (Cremer et al., 2019).

The same work defines aggregate security risk as

$$RISK^{SA} = \sum_{i \in \Omega^P} \sum_{c \in \Omega^C} p^I_i \, p^C_c \, S_{i,c},$$

with an ML-aware version

$$RISK^{ML} = \sum_{i \in \Omega^P} \sum_{c \in \Omega^C} p^I_i \, R_c(x_i),$$

and total risk

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$0

The operational consequence is a ranked allocation of effort: high-risk scenarios are assessed with conventional methods, while low-risk scenarios rely on ML prediction. Case studies reported up to 95% reduction in computational requirements for real-time DSA without increasing operational risk (Cremer et al., 2019).

A related large-scale power-system methodology extends the probabilistic dynamic security assessment setting to load and generation variability, N-2 contingencies, uncertain cascade propagation, and statistical stopping criteria for individual contingencies. It combines rigorous statistical indicators with optional contingency screening and interpretable machine learning for root-cause analysis and security enhancement (Sabot et al., 2 May 2025).

3. Runtime evidence, uncertainty quantification, and probabilistic updating

Several lines of work instantiate dynamic probabilistic assurance through runtime evidence models. For supervised learning-enabled components, a non-parametric Bayesian approach based on Gaussian process regression is used to model assurance measures such as sensor error. The predictive posterior is

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$1

and assurance properties such as “Error $\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$2” are evaluated through $\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$3 and Bayesian credible intervals (Asaadi et al., 2023). In this setting, abnormally high GP variance for new inputs is treated as a signal of dataset shift and can trigger failover or fault handling at runtime.

A different runtime update calculus is based on Subjective Logic. A binomial opinion about a binary claim is represented as

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$4

with evidence mapping

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$5

Runtime Safety Performance Indicators are attached to claims and evaluated over windows. The resulting update rule increases confidence in the absence of violations and imposes prompt penalties when violations occur; the method explicitly prioritizes safety-relevant responsiveness over exact classical Bayesian posterior updates (Herd et al., 21 May 2026).

Dynamic assurance can also be driven by operational safety measurement linked back to a risk model. In the safety-case framework for autonomous taxiing, development-time evidence is encoded as a Beta prior, operational data provide a Binomial likelihood, and the posterior is again Beta:

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$6

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$7

Risk change is then monitored through the risk ratio

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$8

This framework is intended to detect practical drift by revising barrier integrities and event probabilities from operational measurement (Denney et al., 2024).

For ML failure monitoring, SafeML combines runtime shift detection with Bayesian Networks. Distributional shift is assessed through the Wasserstein distance

$\Pr(|\mathrm{CTE}| \geq 2\mathrm{m}) \geq 0.3,$9

together with a bootstrapped p-value, and reliability is reduced to an ID/OOD decision:

$\hat{p}^1(x_i)$0

A Bayesian Network then fuses ML output, SafeML status, and contextual sensor readings through

$\hat{p}^1(x_i)$1

supporting dynamic safety evaluation and operational adaptation under uncertainty (Arshadizadeh et al., 7 Jun 2025).

4. Formal verification, prediction, and control-oriented assurance

Probabilistic model checking provides one of the most explicit formalizations of dynamic probabilistic assurance. For deep-reinforcement-learning-based robotics and autonomous systems, a Discrete-Time Markov Chain models risk and failure dynamics under disturbances, while dependability properties are formalized in PCTL. The framework evaluates safety, robustness, resilience, detection, and recovery, and the DTMC can be reconstructed or reparametrized from new evidence or environmental changes, enabling deployment-time assurance (Dong et al., 2021).

A complementary model-driven route begins from UML activity diagrams. QASCAD introduces a profile for probability, time, and quality annotations, interprets activity diagrams as DTMCs, MDPs, or CTMCs, and transforms them automatically into PRISM or Storm models. This permits fully automated verification and controller synthesis, with parametric constants left open for design-space exploration or later updates (Ye et al., 2024).

In control-oriented formulations, the adaptive probabilistic safety certificate introduces probabilistic invariance as a long-term safety notion for stochastic systems under uncertainty and user preference. The core guarantee is

$\hat{p}^1(x_i)$2

and the framework enforces a myopic safety condition at each step while accommodating Bayesian environment estimation and language-guided user preferences (Wang et al., 16 Nov 2025).

The Assurance Monitor Pattern provides a software-architectural view of runtime probabilistic assurance. Source agents, fusion agents, prediction agents, and check-violation agents together estimate the probability distribution of future trajectories and decide whether more data should be acquired under limited resources. The trajectory probability under control plan $\hat{p}^1(x_i)$3 is written as

$\hat{p}^1(x_i)$4

and the resulting distribution is used to determine whether an assurance constraint is likely to be violated (Duracz et al., 2020).

At a lower inference layer, dynamic sampling for Markov random fields maintains valid sampling-based inference while the graphical model changes. Under the Dobrushin-Shlosman condition, the method achieves $\hat{p}^1(x_i)$5 incremental time cost, rather than redrawing all samples after each model change (Feng et al., 2019). This suggests a generic computational substrate for assurance settings in which the probabilistic model itself evolves.

Formal theorem proving extends the same theme to failure logic. Dynamic fault trees have been formalized in HOL4 for AND, OR, Priority-AND, Functional DEPendency, Hot Spare, Cold Spare, and Warm Spare gates, including verified expressions such as

$\hat{p}^1(x_i)$6

The approach is distribution-agnostic and gives formally verified probabilistic analysis for dynamic failure behavior (Elderhalli et al., 2018).

5. Dynamic assurance cases and confidence propagation

Dynamic probabilistic assurance is frequently anchored in structured assurance arguments rather than in a single verifier or estimator. ENTRUST exemplifies this approach by partially instantiating assurance arguments at design time and filling evidence placeholders automatically at runtime as new verification or operational data become available. Its tool-supported instances use probabilistic model checking through PRISM and other verification engines, and each adaptation yields a new version of the assurance case (Calinescu et al., 2017).

Within Assurance 2.0, confidence is treated as a combination of soundness, defeater analysis, and probabilistic support. Clarissa operationalizes this through probabilistic valuations and confirmation measures such as

$\hat{p}^1(x_i)$7

$\hat{p}^1(x_i)$8

$\hat{p}^1(x_i)$9

together with automated propagation, what-if experiments, and defeater management (Bloomfield et al., 2022).

A newer probabilistic method for Assurance 2.0 makes the propagation rule depend on the decomposition pattern of the argument. For diverse and independent subclaims, the parent confidence is computed by product of doubts,

$c$0

whereas partitioned claims use weighted averages, containment cases use lower bounds such as $c$1, and cumulative cases use a chain rule. When dependence is uncertain, Fréchet bounds are used:

$c$2

$c$3

This method is explicitly presented as a complement to logical indefeasibility and dialectical examination, not as a replacement (Bloomfield et al., 21 Mar 2026).

Ontology-based management has brought these ideas into machine-readable infrastructure. OntoGSN provides a 1:1 formalization of the GSN Community Standard v3 in OWL with SWRL rules, together with classes such as gsn:ConfidenceArgument, reified gsn:RelationshipWithConfidence, and dialectic elements such as gsn:Defeater, gsn:inDoubt, and gsn:defeated. The ontology is queryable through SPARQL and is intended for automatic population, evaluation, and update of assurance cases (Momcilovic et al., 20 May 2025).

A closely related continuous-assurance workflow integrates RoboChart for functional correctness, PRISM for probabilistic risk analysis, and automatic regeneration of structured assurance arguments whenever formal specifications or verification results change. The framework uses explicit hooks such as <<DeferredEvidence>> and <<ConfidenceMonitor>> to connect formal verification, runtime monitoring, and assurance-case evolution (Abeywickrama et al., 17 Nov 2025).

6. Domains, benefits, and recurring issues

Dynamic probabilistic assurance has been demonstrated in power-system security assessment, autonomous taxiing, drone navigation, autonomous driving and platooning, self-adaptive software, agriculture robotics, and nuclear inspection robotics (Cremer et al., 2019, Asaadi et al., 2020, Duracz et al., 2020, Arshadizadeh et al., 7 Jun 2025, Adam et al., 24 Jun 2025, Abeywickrama et al., 17 Nov 2025). In an agricultural robotics example, probabilistic modelling of robot, object detection system, and human behavior through synchronized state machines and PRISM analysis quantified how risk mitigation changes injury probability during row transition: No/failed ODS gave probability of injury $c$4, Normal ODS reduced it to $c$5, and High-performance ODS reduced it by another order of magnitude to $c$6 (Adam et al., 24 Jun 2025).

A recurring misconception is that dynamic probabilistic assurance is equivalent to attaching a single probability to a top claim. The literature argues otherwise. Confidence is treated as structurally propagated, defeasible, and conditioned by dependence assumptions, while assurance cases must also represent negative evidence and residual doubts (Bloomfield et al., 2022, Bloomfield et al., 21 Mar 2026). Another recurring misconception is that runtime assurance is merely monitoring. The surveyed methods connect monitoring to Bayesian updating, probabilistic model checking, calibrated classification, safety certificates, or explicit assurance-case regeneration (Denney et al., 2024, Dong et al., 2021, Abeywickrama et al., 17 Nov 2025).

The literature also records substantive limitations. One quantified-assurance method for learning-enabled components is applicable to supervised learning where ground-truth output is available, but is not immediately applicable to reinforcement learning, end-to-end LECs, or contexts without clear labels (Asaadi et al., 2023). In DRL-based robotics, standard DRL training was found not to improve dependability properties, indicating the need for bespoke optimisation objectives (Dong et al., 2021). Ontology-based management of assurance cases supports confidence annotations and rule-based invalidation, but does not prescribe a built-in probability calculus (Momcilovic et al., 20 May 2025). Subjective-logic runtime updating deliberately departs from exact classical Bayesian posterior updates in favor of prompt penalties for violations (Herd et al., 21 May 2026).

Taken together, these results indicate that dynamic probabilistic assurance is not a single method but a family of methods unified by three commitments: probabilistic representation of confidence or risk, explicit update in response to evolving evidence or context, and traceable connection from quantitative results to operational decisions or assurance claims. The surveyed work suggests that the field is converging on integrated workflows in which stochastic modelling, runtime evidence, and structured assurance arguments are maintained as a single, evolving assurance fabric.