A finite-sample Rao-Blackwell diagnostic between Kalman and particle filtering

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A filtering distribution is the conditional law of an unobserved state given observations up to the current time. The Kalman filter propagates means and covariances exactly in a linear Gaussian state-space model. A particle filter approximates a filtering law by weighted random samples and updates those weights using observations. Nonlinear recursive filtering generally requires a specified transition and observation model.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The promised nonlinear recursive problem is not written in the source excerpt. No estimator, recursion, or finite-sample diagnostic is reconstructed from the catalogue title. The supplied catalogue marks this record open; its present research status was not independently checked. Its supplied reconstruction is labelled unverified. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this formulation as an unverified reconstruction; missing assumptions are not certified by the source link.

Statement recorded in the supplied source.

A.4 Haykin, Simon

For my talk, I will do the following: 1. Provide a brief background on the classical Kalman filtering algorithm, emphasizing its virtues and limitations. 2. Describe particle filtering, rooted in Bayesian estimation and Monte Carlo simula-tion. Here again, I will highlight the virtues and limitations of this second approach. 3. The background would then be set for describing a new approach that has the potential for making a significant difference to the literature. In particular, I will pose a nonlinear recursive problem that is currently the stumbling block. As such, to progress further with this approach, we have to solve this problem. It could be that the solution I am looking for will pop up in the supplemental technical breakout session following my talk on the second day. [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

The participant plans a comparison of Kalman and particle filters and mentions an unstated recursive obstacle.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001169 (AIM-COMPUTATION-0007). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-COMPUTATION-0007. https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20001169 and code AIM-COMPUTATION-0007 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The participant contribution in the primary AIM optimal-control workshop PDF was consulted. Its broad scope is retained above. The following explicitly chosen finite or smooth model tests one route suggested by the agenda; it is not a reconstruction of a missing theorem. The elementary arguments are not claimed novel, and no current-resolution audit of the whole field is asserted.

Research attempt: Moment closure fails in a scalar nonlinear observation model

The promised nonlinear recursive equation is absent from the source, so no specific recurrence can be solved faithfully. To test the potential shortcut "propagate only mean and variance as in Kalman filtering", choose the scalar observation $Y=X^2+Z$, with $Z\sim N(0,\tau^2)$ independent. A symmetric $\{ -1,1\}$ prior and a standard normal prior both have mean zero and variance one. Their predictive observation means are both one, but their predictive variances are respectively $\tau^2$ and $2+\tau^2$, because $E X^4$ equals one and three. The Gaussian fourth moment follows, for example, by twice integrating its density by parts using $\phi'(x)=-x\phi(x)$. Thus even predicting the next observation variance cannot be exact from the first two state moments alone for this nonlinear model.

For a standard normal prior, take $\tau=1$ and observe $Y=1$. Bayes' density is proportional to \[ \exp\{-x^2/2-(1-x^2)^2/2\} =\exp\{-1/2+x^2/2-x^4/2\}. \] Its log derivative is $x-2x^3$; the stationary points are zero and $\pm1/\sqrt2$. The second derivative $1-6x^2$ is positive at zero and negative at the other two, so the posterior is bimodal and not Gaussian. The quartic negative leading coefficient ensures the density is integrable. Linearizing $x^2$ at the prior mean gives derivative zero. A Kalman update based on that linearization has zero cross-covariance and makes no update, leaving a unimodal standard Gaussian. This supplies an exact mismatch, rather than an unsupported assertion that nonlinear filters can be difficult.

Particles can represent such a posterior in principle, but a finite sample does not guarantee both modes are represented. For $N$ independent exact posterior draws, symmetry gives probability $2^{1-N}$ that all fall on one side of zero. This follows from two disjoint events each of probability $2^{-N}$. It is only an elementary sampling diagnostic, not a particle-filter convergence theorem or the unpublished recursion sought by the participant.

Verification and adversarial audit

Predictive moments and all stationary points were computed explicitly. The posterior uses a specified positive-noise likelihood, avoiding conditioning on an impossible atom. The independent-draw diagnostic is not applied to dependent resampled particles without justification.

Audit corrections.

No recurrence was invented from the expanded title; a fully specified auxiliary model was used only to test a proposed closure mechanism.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source does not state its stumbling-block recurrence. Its solution, complexity and claimed new approach therefore remain unidentified. The first gap is that missing mathematical input; the displayed nonlinear diagnostic only rejects universal Gaussian moment closure.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.