The Constitutional Filter: Bayesian Estimation of Compliant Agents

Benedict Flade; Devendra Singh Dhami; Felix Divo; Julian Eggert; Kristian Kersting; Simon Kohaut

REVIEW 3 major objections 5 minor 1 cited by

The Constitutional Filter: Bayesian Estimation of Compliant Agents

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A new Bayesian filter, CoFi, multiplies the standard measurement update by the probability that an agent satisfies a neuro-symbolic Constitution, and on real marine data this improves tracking accuracy whenever the learned trust ratio is…

desk verdict A novel and promising neuro-symbolic filter whose central empirical claim is currently undermined by a self-referential likelihood construction and in-sample trust calibration, but the idea deserves a serious referee. read the letter →

arxiv 2412.18347 v3 pith:YTPPMM5V submitted 2024-12-24 cs.RO

Simon Kohaut , Felix Divo , Benedict Flade , Devendra Singh Dhami , Julian Eggert , Kristian Kersting This is my paper · ORCID

classification cs.RO

keywords constitutionalfilterBayesianestimationneuro-symbolicsystemsprobabilisticlogicprogrammingparticlestatisticalrelationalmapmarinetraffictrackingtrustcalibration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a Bayesian tracker can be made more accurate by consulting an explicit, human-interpretable model of the rules the tracked agent is expected to follow, which it calls a Constitution. It introduces the Constitutional Filter (CoFi), which multiplies the usual measurement likelihood by a constitutional likelihood: the probability that the agent's state and sensor reading satisfy rules expressed in probabilistic first-order logic. On real marine traffic data, CoFi tracks vessels more accurately than a standard particle filter whenever a learned trust ratio is positive, and it degrades exactly to the baseline when the trust ratio is zero. The practical payoff is that expert knowledge about laws, physical limits, and preferences can be injected into any recursive Bayesian estimator without losing the safety net of the rule-free filter.

What carries the argument

The load-bearing object is the constitutional likelihood $p(c_t|x_t,z_t)$, obtained in three steps: (i) grounding a deep probabilistic first-order logic program (the Constitution, consisting of background knowledge, perception, and a StaR Map environment model) at a candidate state and measurement; (ii) computing the satisfaction probability via the sum-product of Eq. 11; and (iii) turning the discrete satisfaction values into a continuous density by kernel density estimation over samples from the filter's predictive prior. This density enters Eq. 15 as an additional multiplicative factor in the Bayes update, and Eq. 16 modulates it with the trust ratio $\tau$ so that $\tau = 0$ returns the standard filter.

What would settle it

Run CoFi on a trajectory with known ground truth where the agent systematically violates the Constitution while the process and measurement models are correct; if maximizing tracking accuracy ever selects $\tau > 0$ and the resulting error exceeds the $\tau = 0$ particle filter on that trajectory, the claim that CoFi guarantees baseline recovery is contradicted.

Watch

Extended reading notes

Core claim

The central discovery is a belief update (Eq. 15) in which the posterior is proportional to the product of the measurement likelihood, the standard predictive prior, and a constitutional likelihood $p(c_t|x_t,z_t)$ computed from a neuro-symbolic program. The Constitution encodes the agent's expected compliance as probabilistic first-order logic clauses over spatial relations from a StaR Map, background knowledge, and perceptual features; exact sum-product inference yields the probability that a state-measurement pair satisfies it. Because this probability is a discrete number, CoFi converts it into a density by kernel density estimation over samples drawn from the filter's own prior, and then blends it with a uniform distribution through a trust ratio $\tau$ (Eq. 16). The paper demonstrates on AIS vessel data that a trust-calibrated CoFi improves mean absolute tracking error as soon as $\tau > 0$, while $\tau = 0$ exactly recovers the unconstitutional particle filter, so an inaccurate Constitution cannot degrade performance.

Load-bearing premise

The update treats a kernel density estimate built from the filter's own predictive samples as a genuine likelihood for an unobserved 'constitution satisfied' event, even though those samples come from the very distribution the update is meant to correct.

Editorial extensions

If this is right

Any Bayes filter, whether particle, Kalman, or unscented, can host CoFi, because the Constitution step only adds one extra likelihood factor to the update.
For the roughly 89% of vessels in the dataset whose optimal trust ratio is positive, CoFi reduces tracking error; the remaining vessels fall back to the particle filter baseline.
Precomputing the constitutional likelihood as a static scalar field brings CoFi's per-update runtime close to the baseline particle filter (0.007 s versus 0.004 s in the experiments).
The Constitution is a symbolic, human-readable artifact, so the reasoning that guides tracking can be inspected and edited by a domain expert.
A false or ill-matched Constitution cannot hurt tracking below baseline, because trust calibration is allowed to set $\tau = 0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same construction should transfer to road traffic, pedestrian crowds, or airspace rules: any domain where a probabilistic first-order logic program can separate likely from unlikely states could show similar gains, with the StaR Map replaced by any uncertain semantic map.
The KDE step in Section IV-D samples from the filter's own predictive prior, so the resulting 'likelihood' is partly a function of the belief it is meant to correct; a testable consistency check would compare CoFi's posterior with one using an independent Monte Carlo estimate of $p(c_t|x_t,z_t)$.
Trust features could be learned online per agent instead of calibrated offline on historical data, letting CoFi adapt when an agent's compliance changes mid-trajectory, something the current time-invariant trust setting does not address.
If the Constitution is interpreted as a prior over states rather than a likelihood, Eq. 15 resembles a product-of-experts update; exploring that reading could clarify when the KDE approximation is valid and yield a more principled normalization.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read the CoFi paper with interest. The core idea is genuinely new: coupling hybrid probabilistic logic programs to recursive Bayesian filtering, with a trust ratio that lets the filter down-weight an unreliable constitution. That is a real step beyond the authors' earlier ProMis planning work, and the maritime example is well chosen. The paper is clearly written and they ship code, which I value.

What the paper does well: the Constitution encoding of AIS-based rules is concrete and interpretable, the trust ratio is a sensible safeguard, and the runtime analysis is honest about the cost of neuro-symbolic inference. The qualitative results in Figures 5-6 suggest the Constitution does capture real traffic patterns.

The soft spots are serious, and they land on the central claim. First, the constitutional likelihood in Section IV-D is built by KDE over samples drawn from the filter's own predictive prior. The stress-test note is right: for a fixed (x,z), P(C|x,z) is a deterministic scalar, and the KDE over prior-sampled values estimates a prior-predictive density, not a likelihood for an unobserved satisfaction event. Multiplying the measurement likelihood by this density double-counts the prior and can reinforce the filter's existing belief. A Bernoulli likelihood would be the natural construction, and the paper offers no consistency argument for Eq. 15 as a valid Bayes update. Second, the trust ratio tau is tuned on the same AIS data used in Figure 8, so the accuracy gain over the particle filter is in-sample fitting. Section V-D says "offline learning," but there is no train/test separation; the paper's headline claim - "more accurate tracking as soon as tau > 0" - is not a predictive validation. Third, the baseline is a plain particle filter; a map-aware filter using the same spatial relations (e.g., a potential field over depth or distance to waterway) would be a fairer test of whether the logic layer earns its keep.

These flaws are critical to the empirical claim as stated, but they are fixable. The self-referential likelihood can be replaced with a proper observation model for c_t; the trust calibration can be evaluated on a held-out set; a map-aware baseline would clarify the contribution. The underlying architecture is interesting and the open-source implementation is an asset.

My recommendation: this deserves a serious referee, not a desk reject. The current version is not ready to be accepted, but the idea is novel enough that a good review cycle, pushing for a corrected likelihood and proper validation, could make this a solid contribution. I'd bring it to a reading group to discuss the KDE issue if nothing else.

Referee Report

3 major / 5 minor

Summary. The paper introduces the Constitutional Filter (CoFi), an extension of recursive Bayesian estimation in which a neuro-symbolic "Constitution"—a probabilistic first-order logic program over states, measurements, and environmental features—contributes an extra likelihood factor p(c_t|x_t,z_t) to the belief update (Eq. 15). A scalar trust ratio τ (Eq. 16) blends this constitutional likelihood with a uniform term, with τ=0 recovering a standard particle filter. The method is evaluated on real-world AIS marine traffic data, where τ is calibrated per trust feature (vessel type, waterway binding, anchoring classifier) to maximize tracking accuracy; the paper reports that a trust-calibrated CoFi outperforms a baseline particle filter and recovers baseline performance when τ=0.

Significance. If the central construction were sound, CoFi would be a valuable contribution: it connects probabilistic logic programming and statistical relational maps with recursive state estimation, it is evaluated on real data, and the authors provide an open-source implementation. The interpretability of the Constitution and the idea of learned trust are appealing. However, the paper's core claim—that Eq. 15 is a valid Bayesian update using a constitutional likelihood—rests on an unjustified density-estimation step over the filter's own prior, and the reported performance gains are obtained by calibrating τ on the same data used for evaluation. These are load-bearing issues, not presentation problems.

major comments (3)

[IV-D, Eq. (15)] The constitutional likelihood is not derived as a likelihood. Given a fixed Constitution, P(C_t|x_t,z_t) from Eq. (11) is a deterministic function q(x,z) of the state and measurement. A valid likelihood for the unobserved event c_t=1 would be Bernoulli(q(x,z)), or its density q(x,z)^{c}(1-q(x,z))^{1-c} if c_t were observed. Instead, Section IV-D samples x_t from the filter's own predictive prior p(x_t|c_{1:t-1},z_{1:t-1}) and z_t from p(z_t|x_t), forms S={q(x_t^{(n)},z_t^{(n)})}, and applies KDE. The resulting density, evaluated at q(x_t,z_t), estimates the prior-predictive distribution of q, not the conditional likelihood p(c_t|x_t,z_t). Substituting this quantity into Eq. (15) multiplies the measurement likelihood by a factor derived from the same predictive distribution that the update is supposed to correct. No consistency argument is given, so Eq. (15) is not established as a Bayes update for p(x_t|c_{1:t},z_{1:t}).
[IV-F, Eq. (16)] The trust-ratio interpolation in Eq. (16) is dimensionally and semantically unclear. The paper alternates between treating P(C_t|x_t,z_t) as a probability (Eq. 11) and treating p(c_t|x_t,z_t) as a continuous density over KDE outputs (Section IV-D). The uniform term U(0,1) is a density for a continuous variable on [0,1], not a likelihood for a binary compliance event, and no observed value c_t appears anywhere in the data. Consequently, the mixture p_τ(ct|xt,zt)=τ p(ct|xt,zt)+(1−τ)U(0,1) does not have a clear probabilistic interpretation, and the claim that τ=0 reduces exactly to the standard Bayesian update is not supported by the definitions given.
[V-D, V-E, Figs. 7-8] The experimental evaluation is in-sample. Section V-D states that "CoFi chooses τ to maximize its tracking accuracy for agents with the respective trust features" and that this is computed as an offline task on historical AIS data. Figure 8 then reports errors at the optimal τ values selected for each vessel group. No training/test split, cross-validation, or held-out evaluation is described. Therefore the central claim of Section V-E—that "CoFi provides more accurate tracking as soon as τ>0"—is a statement about in-sample fitting, not a predictive statement about new trajectories. The claim that CoFi 'learns to trust' agents would require evaluation on data not used to choose τ.

minor comments (5)

[IV-C, Eq. (11)] The set J over which the sum-product is computed should be defined explicitly as the set of models satisfying the query constitution(X,Y); otherwise the sum over all models would not yield P(C_t|x_t,z_t).
[IV-D] The KDE bandwidth and kernel are not specified; the resulting density, and hence the behavior of Eq. (15), may be sensitive to these choices, so a sensitivity analysis or at least the chosen parameters should be reported.
[V-D] The paper assumes the trust features and the appropriate trust are time-invariant throughout a journey, but one of the trust features is an anchoring classifier that can change over time; this assumption should be justified or relaxed.
[V-E, Fig. 8] The y-axis label 'Relative Error' is undefined; the text mentions 'relative mean absolute error' but the reference value (e.g., relative to the particle filter error or to the true position scale) is not stated, and no absolute errors, standard deviations, or sample sizes are reported for the curves.
[IV-F, Eq. (16)] The notation for the trust ratio is inconsistent: Eq. (16) writes τ(ψ_t) but the surrounding text and experiments use τ without explicitly showing the dependence on ψ for each vessel group.

Circularity Check

2 steps flagged · score 7.0 of 10

CoFi's claimed gains reduce to: (1) a trust ratio τ fitted on the same AIS data used for evaluation and (2) a 'constitutional likelihood' estimated from the filter's own predictive prior; both make the central accuracy result partly self-confirming.

fitted input called prediction [Section V-D (Calibrating the Trust Ratio) and Section V-E (Constitutional Filtering), Figure 7/8]
"CoFi chooses τ to maximize its tracking accuracy for agents with the respective trust features. In our experiments, we perform this computation as an offline learning task on historical AIS data, comparing performance across discrete choices for τ. ... As Figure 8 shows, CoFi provides more accurate tracking as soon as τ > 0, leveraging the information provided by the Constitutional likelihood in CoFi's belief update."

The optimal τ values selected in V-D to maximize tracking accuracy on the AIS data are then used in V-E to report accuracy. No held-out split is described, and Figure 7 is explicitly computed over 'the entire dataset population'. Thus the relative-error curves in Figure 8 at τ>0 are the objective-function values of the τ-calibration, not an independent prediction. 'CoFi provides more accurate tracking as soon as τ>0' is therefore a restatement of the τ-selection criterion: τ was chosen to make that true. The τ=0 baseline is recovered by construction, but the claimed benefit is for τ>0, where the reported gain is in-sample.
self definitional [Section IV-D (Density Estimation of the Constitutional Likelihood) and Section IV-E (Constitutional Bayesian Belief Update), Eq. 12 and Eq. 15]
"S = { P(C_t|x_t^(n), z_t^(n)) }_{n∈{1,...,N}}, where x_t^(n) ∼ p(x_t|c_1:t−1,z_1:t−1) and z_t^(n) ∼ p(z_t|x_t). One can then approximate the density p(c_t|x_t,z_t) from S, for instance, using Kernel Density Estimation."

The 'constitutional likelihood' p(c_t|x_t,z_t) is defined by drawing x_t from the filter's own predictive prior p(x_t|c_1:t−1,z_1:t−1) and z_t from the measurement model, then KDE-smoothing the resulting scalar satisfaction probabilities. This estimates the prior-predictive density of q=P(C_t|x_t,z_t), not a conditionally independent likelihood for an observed constitution event. When this density is plugged into Eq. 15, the measurement likelihood is multiplied by a factor derived from the same predictive distribution the update is supposed to correct, so the posterior double-counts the prior. No observed c_t and no consistency argument justify the density as p(c_t|x_t,z_t); the 'constitutional information' is, by construction, an autocatalytic reweighting of the filter's own prior.

full rationale

The paper is not circular via self-citation: the cited prior work (ProMis, StaR Maps) supplies building blocks rather than the central result. The circularity is internal. First, the central performance claim is fitted rather than predictive: τ is calibrated per trust-feature group to maximize tracking accuracy on the AIS data, and the same data are then used to show that CoFi is more accurate for τ>0. Since no train/test split is described, Figure 8's improvement is the calibration objective, not an independent evaluation. Second, the constitutional likelihood itself is constructed from the filter's own predictive prior. Equation 12 samples x_t from p(x_t|c_1:t−1,z_1:t−1) and z_t from p(z_t|x_t), and the KDE over S yields the prior-predictive density of satisfaction probabilities. Using that as p(c_t|x_t,z_t) in Eq. 15 means the 'constitution step' re-weights the measurement update by a factor that already reflects the filter's prior beliefs, rather than by independent evidence. The τ=0 limit correctly recovers a standard particle filter, but the claimed benefit is specifically for τ>0, where both the fitted-trust and prior-derived-likelihood issues apply. Together these make the central accuracy result substantially self-confirming, though the underlying StaR Map and symbolic-constraint pipeline have independent engineering content.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The method relies on a small number of hand-assigned rule probabilities and distribution parameters, plus the crucial assumption that the KDE-based constitutional likelihood is legitimate. The trust ratio is the only parameter actually fitted to data, but it is fitted on the same data used to evaluate the method, which loads the performance claim onto a free parameter.

free parameters (3)

Trust ratio τ = Per trust-feature group, chosen from a discrete grid to maximize tracking accuracy
Selected in Section V-D on historical AIS data; the evaluation in Section V-E uses these same optimal values.
Constitution clause probabilities = 0.95 for safe, 0.90 for respects_waterways, 0.1 for over(x, anchorage)
Hand-assigned in Listing 1; the filter behavior depends on these choices.
Environment and perception distribution parameters = Means and variances in Listing 1 (e.g., distance to land normal(150,15), depth normal(20,1))
Provided by StaR Maps or expert assumptions; not fitted to tracking error but set the Constitutional likelihood.

assumptions (4)

standard math Markov property for states and measurements
Standard recursive filtering assumption, stated in Section III-A, Eqs. 5-6.
ad hoc to paper The KDE-estimated p(c_t|x_t,z_t) is a valid likelihood for an unobserved event
Section IV-D introduces this without proof; the samples used for KDE come from the filter's own prior.
domain assumption A static probabilistic logic program can capture agent compliance
Listing 1 fixes the rules and probabilities; the method does not learn or revise the Constitution.
domain assumption StaR Map moment estimates (Eqs. 7-8) represent true spatial relation distributions
Section III-B assumes the randomized map samples produce unbiased estimates used in the Constitution.

invented entities (1)

Constitution satisfaction event c_t
purpose: Acts as a pseudo-observation whose likelihood is multiplied into the Bayesian update (Eq. 15).
c_t is never directly observed; it is defined by the model. The only external check is tracking error on the same data used to calibrate τ, so there is no independent falsifiable handle.

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Cite this review

Pith. "Pith review of The Constitutional Filter: Bayesian Estimation of Compliant Agents." pith.science (2026). https://pith.science/paper/YTPPMM5V

@misc{pith2026241218347,
  author       = {Pith},
  title        = {Pith review of: The Constitutional Filter: Bayesian Estimation of Compliant Agents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTPPMM5V}},
  note         = {Machine review of arXiv:2412.18347}
}

read the original abstract

Predicting agents impacted by legal policies, physical limitations, and operational preferences is inherently difficult. In recent years, neuro-symbolic methods have emerged, integrating machine learning and symbolic reasoning models into end-to-end learnable systems. Hereby, a promising avenue for expressing high-level constraints over multi-modal input data in robotics has opened up. This work introduces an approach for Bayesian estimation of agents expected to comply with a human-interpretable neuro-symbolic model we call its Constitution. Hence, we present the Constitutional Filter (CoFi), leading to improved tracking of agents by leveraging expert knowledge, incorporating deep learning architectures, and accounting for environmental uncertainties. CoFi extends the general, recursive Bayesian estimation setting, ensuring compatibility with a vast landscape of established techniques such as Particle Filters. To underpin the advantages of CoFi, we evaluate its performance on real-world marine traffic data. Beyond improved performance, we show how CoFi can learn to trust and adapt to the level of compliance of an agent, recovering baseline performance even if the assumed Constitution clashes with reality.

Figures

Figures reproduced from arXiv: 2412.18347 by the authors.

**Figure 1.** The Constitutional Filter’s estimation cycle: A Constitution for a remote-controlled, unmanned aircraft system is shown on the left. The right-hand side shows the tracked drone and the likely compliant areas shaded in blue. neuro-symbolic model of its internal rules and perception of its state space. While such models have been employed as the basis for planning, e.g., mission design for Unmanned Aircraft Systems i… view at source ↗

**Figure 2.** StaR Maps parameters in an Advanced Aerial Mobility setting: Here, the expected values of two basic probabilistic spatial relations are shown in an urban environment, namely over and distance (used in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The Constitutional Filter’s architecture: CoFi extends recursive Bayesian estimation using a neuro-symbolic model of the tracked agent’s decision-making constraints, integrating background knowledge, perception, and a probabilistic environment representation. C. Exact Probabilistic Inference We now aim to compute the probability of a pair (xt, zt) satisfying the first-order logic program of Ct when solving for the… view at source ↗

Figures from the paper (5 more)

**Figure 1.** Figure 1: For CoFi, we assume Bt to be provided by a domain expert, i.e., by formalizing their knowledge as firstorder logic or by translating natural language descriptions through, e.g., a Large Language Model. Further, Pt encodes the semantics of the environment as experience…

**Figure 4.** Figure 4: Statistical relational maps for marine traffic: Expectations of StaR Map relations employed in Listing 1 at New York’s harbor, considering land, waterways, anchorage areas, and depth. Distances are shown in kilometers. CoFi’s prior of the state follows the standard for…

**Figure 5.** Figure 5: Marine traffic over constitutional probabilities: Here, we show real-world AIS traces from (a) often compliant cargo and (b) often incompliant towing vessels on top of the probabilities for each to satisfy the constitution. Cargo Towing SAR 0.0 0.2 0.4 0.6 0.8 1.0 E x…

**Figure 6.** Figure 6: shows how the probability of agents sticking to the Constitution depends on, e.g., their employment type. Note how, for example, towing vehicles are overall less likely to follow the Constitution, as seen in Figure 5b. We will revisit this observation in Section V-E wh…

**Figure 8.** Figure 8: CoFi adapting to appropriate trust: We show the relative mean absolute error, comparing CoFi’s performance with the baseline Particle Filter. Once a vessel profits from τ > 0, CoFi achieves greatly improved accuracy. VI. DISCUSSION We propose the Constitutional Filter …

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Forward citations

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