{"id":"0e36207a-8be7-4d14-b1ae-46c8b503bf65","arxiv_id":"2512.25056","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"FBOVI sequentially approximates the joint state-parameter posterior by a parameter marginal times a conditional Gaussian state distribution, with error bounds and demonstrations on pendulum, Lorenz-96, and a 51D convection-diffusion system.","lead":"This paper introduces FBOVI, an online variational method that estimates both the hidden states and unknown parameters of a dynamical system from noisy, partial observations while quantifying uncertainty. It factorizes the approximate posterior into a parameter part and a state-given-parameter part, gives error bounds, and demonstrates competitive behavior on a pendulum, a chaotic Lorenz-96 system, and a 51-dimensional convection-diffusion system.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stage-2 NN training uses surrogate losses (Eq. 26) that do not control the KL terms appearing in Theorem 1; the theorem therefore does not support the implemented algorithm.","rationale":"The reader's weakest assumption correctly identifies the disconnect between the exact KL terms in Theorem 1 and the NN/surrogate-loss implementation of Stage 2. My stress-test confirms this is the most load-bearing concern: the theorem's bound is vacuous unless the KL terms are small, and the implemented training objective does not guarantee small KL. A second concern, the reliance on an imported theorem from the authors' own prior work, compounds the issue but is secondary because even a correct Theorem 16 would not close the algorithm-theory gap. The paper's framework and experiments are promising, but the advertised theoretical support does not extend to the practical algorithm. This leaves the paper in a CONDITIONAL state — requiring either a convergence/error analysis of the NN training or an empirical demonstration that the trained networks achieve small KL terms — which matches the reader's verdict, so no change is recommended.","tokens_in":30220,"tokens_out":4423,"duration_ms":51687,"concrete_test":"In the linear pendulum experiment, compute the exact Stage-2 KL loss in Eq. (25) using the Kalman-filter targets m*_k(θ), C*_k(θ) on held-out θ samples. After training, compare (i) the surrogate losses in Eq. (26) and (ii) the expected KL. If the achieved expected KL is not small (e.g., ≪1) while the joint posterior RMSE is small, Theorem 1 cannot be the explanation; if it is small, the concern is resolved. As a second check, independently verify Theorem 16 of [27] in the setting of Theorem 1; if it cannot be derived, the theorem's support is missing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical result, Theorem 1, bounds the joint posterior error by a sum of E_{θ∼ν_j}[KL(ρ_j(·|θ)∥ρ*_j(·|θ))] + KL(ν_j∥ν*_j). Stage 1 does target KL(ν_k∥ν*_k) within the chosen variational family. Stage 2, however, does not minimize the KL with respect to the online target: it trains the networks m_k(θ), C_k(θ) using the surrogate losses in Eq. (26) — mean squared error for m and Frobenius norm for C — rather than the expected Gaussian KL in Eq. (25). The surrogate losses do not control the KL: KL contains log|C*| − log|C| + Tr(C*^{-1}C) − n + δ^T C*^{-1} δ; small Euclidean/Frobenius errors can still yield large KL when C* is ill-conditioned or C has a small eigenvalue, and no convergence guarantee is given for the NN optimization. Since Theorem 1's bound is only useful when the KL terms are small, the theoretical guarantee does not transfer to the algorithm actually run in Section 4. The observed accuracy remains an empirical property of the trained networks, not a consequence of the theorem. Additionally, the proof of Theorem 1 relies entirely on Theorem 16 of the authors' own preprint [27], with no proof or independent verification in this manuscript; if that theorem has unstated conditions, the bound is unsupported. This gap is load-bearing because the paper advertises 'theoretical results on approximation accuracy' for the implemented method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes FBOVI, an online variational framework for joint state-parameter estimation in partially observed nonlinear dynamical systems. The approximate joint posterior is factorized as q_k(X_k, θ)=ρ_k(X_k|θ)ν_k(θ), with ν_k updated by maximizing an ELBO and ρ_k obtained by Gaussian filtering whose mean/covariance functions are represented by neural networks. The main theoretical contribution is Theorem 1, an upper bound on TV/Hellinger distance between the true posterior and q_k in terms of past KL divergences between the approximations and recursively defined online targets, and Theorem 2, an estimable version for linear systems. The framework is tested on a linear pendulum, a chaotic Lorenz-96 system, and a 51-dimensional convection-diffusion PDE, comparing with joint particle, unscented, and ensemble Kalman filters.","tokens_in":30718,"tokens_out":5000,"duration_ms":58186,"significance":"If the gap between the theoretical statement and the implemented algorithm were closed, this would be a useful contribution: the framework is modular, allows different Gaussian filters including the EnKF, and proposes an interpretable error bound that directly informs algorithmic choices. The numerical experiments are broad and show competitive accuracy, including robustness under model misspecification and high-dimensional state spaces. However, the advertised theoretical guarantee does not currently apply to the algorithm as implemented in the nonlinear/EnKF settings, and the proof of the central theorem is imported from an unpublished same-author preprint. These issues affect the central claim and should be resolved before acceptance.","major_comments":[{"comment":"Theorem 1 bounds the joint posterior error by sums of E_{θ~ν_j}[KL(ρ_j‖ρ*_j)] and KL(ν_j‖ν*_j). Stage 1 targets the second KL, but Stage 2 does not minimize the first KL when the surrogate losses of Eq. (26) are used. Small MSE/Frobenius errors do not control the Gaussian KL: if C* is ill-conditioned or C has a small eigenvalue, KL contains log|C*|-log|C|+Tr(C*^{-1}C)+δ^T C*^{-1}δ, which can remain large even when ‖C-C*‖_F and ‖m-m*‖_2 are small. The theorem therefore does not transfer to the algorithm actually run in Section 4 unless the authors either train with the KL loss, prove that the surrogate losses bound the KL under explicit conditions, or verify in each experiment that the trained networks drive the KL terms small.","section":"Sec. 3.2.2, Eqs. (25)-(26)"},{"comment":"The online parameter target ν*_k in Eq. (9) is defined using the exact predictive distribution ρ*^-_{k}(X_k|θ). In the nonlinear Stage 1, this distribution is replaced by the Gaussian approximation p_N(X_k; m^-_k(θ), C^-_k(θ)) in Eq. (15) and used to compute I(θ) in Eq. (18). Consequently the variational update minimizes KL to a modified target, not to ν*_k, and the KL(ν_j‖ν*_j) terms that appear in Theorem 1 do not account for this Gaussian/predictive approximation error. This is another unmonitored source of error in the bound, and it is especially relevant for strongly nonlinear or multi-modal predictive distributions.","section":"Sec. 3.1.2 Eqs. (15)-(18) vs. Eq. (9)"},{"comment":"The proofs of Theorems 1 and 2 rely entirely on Theorem 16 of the authors' preprint [27], which is imported without proof or even a statement of its precise hypotheses beyond the condition in Eq. (B.1)/(C.1). Since [27] is a same-author preprint and its theorem is the load-bearing ingredient, the manuscript should either include a self-contained proof or state and verify all conditions explicitly. As written, the reader cannot independently confirm that the bound is not contingent on unstated assumptions in [27].","section":"Appendices B and C, Theorem 3/4"},{"comment":"The bound in Eq. (24) contains the term  sqrt(E[Ψ_j] - r/2 log(2π) - 1/2 log C̃ - ε_j). The assumption L_i(ν_i) ≥ ε_i does not by itself ensure this radicand is nonnegative, especially for conservative choices of ε_i. Remark 1 claims the bound is estimable via Monte Carlo, but no experiment evaluates the bound or checks that the radicand is positive in practice. This leaves the practical usefulness of Theorem 2 under-specified.","section":"Theorem 2, Eq. (24)"}],"minor_comments":[{"comment":"The name 'Lorentz 96' appears several times (e.g., Sections 4.2 and figure captions); it should be 'Lorenz 96'.","section":"Title/headings"},{"comment":"The final sentence is a sentence fragment: '... convection-diffusion equation. outperforming the joint ensemble Kalman filter...' Should be joined to the previous sentence.","section":"Abstract"},{"comment":"The text says 'the mean m^-_k(θ) and the covariance m^-_k(θ)' where the second should be C^-_k(θ).","section":"Sec. 3.1.2, Eq. (16)-(17)"},{"comment":"The NN architecture, training epochs, optimizer, and hyperparameters for the networks m_k(θ), C_k(θ) are not reported, and no code is provided. Given that the method's practical performance depends on these choices, adding reproducibility details would be important.","section":"Sec. 3.2.2 and Algorithm 1"},{"comment":"The sign convention in the displayed KL expansion is easy to misread. It would help to explicitly state KL(ν_k‖ν*_k)=log Ẑ_k - L_k(ν_k) with L_k defined in Eq. (11).","section":"Eq. (10)"}],"recommendation":"major_revision","confidential_remarks":"The main theoretical claim is interesting, but the paper is not yet internally consistent: Theorem 1 bounds quantities that the implemented Stage 2 does not optimize when surrogate losses are used, and Stage 1's Gaussian predictive approximation introduces an additional untracked error. I do not see this as a reject, because the gap could be fixed by training on the KL loss or by adding a theorem showing the surrogate losses bound the KL under stated conditions. However, as written, the advertised 'theoretical results on approximation accuracy' do not apply to the algorithm in Section 4. The dependence on Theorem 16 of [27] should also be made self-contained or be independently verified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine contribution to online joint state-parameter estimation. The factorization into a variational parameter marginal and a parameter-conditional state filter is a natural design, and the paper extends the authors' prior work [11] in substantive ways: a general Gaussian-filter interface, an EnKF variant, a closed-form linear case, two error theorems, and experiments that include model misspecification and a 51D PDE. That is a real step beyond what was there before.\n\nThe main theorem, Theorem 1, is an honest bound: it says the joint approximation error is controlled by how well the parameter marginal and the conditional state distribution track their online targets. The proof is short and imports Theorem 16 of the authors' own preprint [27]. I did not find an obvious hole, but that imported theorem is not independently verified in this manuscript, and its assumptions are worth checking carefully. Theorem 2 for the linear case is more specific and is at least estimable in principle, though it assumes an ELBO lower bound that is not shown attainable in practice.\n\nThe stress-test note identifies a real gap, and I agree with its substance. In Section 3.2.2 the paper is upfront that the neural networks for m_k and C_k are trained with surrogate losses—mean squared error for the mean, Frobenius norm for the covariance—rather than the expected KL divergence of Eq. (25). Those surrogate losses do not control the KL terms that appear in Theorem 1; a small Euclidean/Frobenius error can still mean large KL when C* is ill-conditioned. So the theorem does not automatically cover the implemented algorithm. That is a soft spot, but not a fatal one. The framework would still be useful if the authors either used the KL loss when feasible, provided an analysis of the gap between the surrogate and the KL, or explicitly framed the theorem as a design guideline rather than a guarantee for the surrogate-trained networks. The same applies to the 51D experiment: it is one realization, evaluated visually, so the scalability claim rests on a thin base.\n\nWho is this for? Researchers in online data assimilation, digital twins, and parameter-state estimation. It deserves a serious referee, but a referee should ask for: a clear statement of what the theory covers vs. what the algorithm does; either a surrogate-loss analysis or a switch to KL training (or an acknowledgement of the gap); and quantitative results over multiple realizations for the high-dimensional case. I would send it to peer review with substantial revision expected.","headline":"A solid extension of the authors' earlier two-stage idea, with real experimental breadth, but the main theorem does not cover the surrogate-trained networks actually used in Stage 2, and the paper should be more explicit about that.","tokens_in":31102,"tokens_out":2348,"would_cite":true,"duration_ms":28677,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62M20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the intractable joint posterior of states and parameters can be approximated online by factorizing it into a parameter marginal and a state conditional, with a two-stage KL-minimization update that keeps total approxim","keywords":["Online data assimilation","Variational inference","Joint parameter-state estimation","Incomplete information","State-space models","Gaussian filtering","Ensemble Kalman filter","Bayesian filtering"],"falsifier":"Take a linear system where the exact Kalman-filter posterior is computable, run FBOVI over many steps, and compare the true total-variation or Hellinger distance between the exact joint posterior and the approximation with the right-hand side of the paper's estimable bound, evaluated by Monte Carlo. If the observed error consistently exceeds the estimated bound, or stays small when the estimated per-step KL terms are large, then the stability theorem or the ELBO-based step in the proof has failed.","tokens_in":1599,"feed_emoji":"⚙️","tokens_out":2095,"duration_ms":80884,"temperature":0.7,"pith_summary":"The paper aims to solve an online estimation problem: at each time step, given a new, possibly partial and noisy observation, update an approximation to the joint posterior of the hidden state and unknown model parameters without reprocessing past data. Its central proposal, FBOVI, factorizes the approximate joint posterior into a marginal distribution over parameters and a conditional distribution over states given those parameters, then updates each half by minimizing its Kullback-Leibler divergence from an 'online target'—the exact posterior that would arise if the previous approximations were exact. The paper proves a theorem bounding the total-variation and Hellinger distance between the true joint posterior and this approximation by a sum of per-step KL divergences, weighted by data-dependent constants. Because only the parameter marginal and the functional mean and covariance maps are propagated forward, per-step cost does not grow with time, and any Gaussian filter, including the ensemble Kalman filter, can be plugged into the state-conditional update. Numerical experiments on a linear pendulum, a chaotic system with model error, and a high-dimensional convection-diffusion PDE show the method tracking both parameters and states, and outperforming the joint ensemble Kalman filter in the high-dimensional case.","feed_headline":"Two-stage variational filter keeps posterior error provably small","feed_subtitle":"Factorizing the joint posterior lets parameters and states update online with a provable error bound.","key_machinery":"The engine is the two-stage KL-minimization against online targets. The approximate joint posterior is written q_k(X_k, θ) = ρ_k(X_k|θ) ν_k(θ), and the online targets ρ*_k and ν*_k are what the exact filter recursions would produce if the previous approximations were used in place of the true conditional and parameter posteriors. Theorem 1 shows that tracking these targets tightly controls the distance to the true joint posterior. The state conditional ρ_k is represented as a Gaussian with mean m_k(θ) and covariance C_k(θ), implemented as time-invariant neural networks trained on target values provided by a Kalman filter or another Gaussian filter; in the high-dimensional experiment, an ense","core_discovery":"The central claim is that the Bayesian joint posterior p(X_k, θ | Y^k) can be approximated online with controlled error by the factorization q_k(X_k, θ) = ρ_k(X_k | θ) ν_k(θ), where ν_k is an arbitrary variational distribution over parameters and ρ_k is a conditional Gaussian with mean m_k(θ) and covariance C_k(θ). Rather than matching the intractable exact posterior directly, each step matches two 'online targets': ρ*_k, the conditional filter that would result from applying the exact prediction-update operators to the previous approximations, and ν*_k, the corresponding parameter posterior. The paper proves that the total-variation and Hellinger distances between the true joint posterior a","pith_inferences":["An extension the paper leaves implicit: the same two-stage KL structure could be run with a non-Gaussian ρ_k, for example a normalizing flow, provided the filtering step can supply target samples; the theorem only requires the KL terms, not Gaussianity.","A practical diagnostic suggested by the estimable bound: monitor Monte Carlo estimates of the per-step KL terms and the ELBO; sustained drift would flag that the approximation has left the regime where the bound applies.","The data-dependent constants in the bound involve inverse powers of the minimal measurement-noise determinant, implying that accuracy guarantees degrade when measurements are weakly informative or nearly collinear—a testable prediction about when the method can be trusted."],"forward_implications":["If the theorem is right, practitioners can choose any variational family for ν_k and any Gaussian filter for Stage 2, and the joint approximation error is still controlled by how well each step tracks its online target.","Per-step computational cost stays bounded over time: only ν_k and the functions m_k(θ), C_k(θ) are passed forward, so historical data never needs to be replayed.","For linear systems, the paper derives a bound that can be estimated from the algorithm's own outputs—m_i, C_i, ν_i, and ELBO values—giving a practical, checkable diagnostic of approximation accuracy.","The every-step resampling variant lets the ensemble Kalman filter be used without carrying state particles forward, which is what allows the method to scale to a 51-dimensional discretized PDE system.","Because the factorization is not mean-field—states remain conditioned on parameters—the marginal state distributions can be non-Gaussian even when the conditional filter is Gaussian, as shown in the chaotic-system experiments."],"fun_headline_variants":["Factorized variational filter bounds joint posterior error","Online joint state-parameter filter with provable error bound","Two-stage variational update keeps posterior error small","Provably small error in online Bayesian state-parameter estimation","Variational framework guarantees error bound for joint filtering"],"cache_read_input_tokens":32384,"weakest_assumption_plain":"The practical error bound presumes that the neural networks trained in Stage 2 actually drive the expected KL divergences to small values; the theorem bounds the joint error only by those KL terms, and the paper does not prove that the trained networks achieve small KL—if the surrogate fitting is poor, the stated guarantee does not apply.","fun_headline_variants_meta":{"raw":{"variants":["Factorized variational filter bounds joint posterior error","Online joint state-parameter filter with provable error bound","Two-stage variational update keeps posterior error small","Provably small error in online Bayesian state-parameter estimation","Variational framework guarantees error bound for joint filtering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1330,"prompt_tokens":807,"completion_tokens":523,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":450}},"tokens_in":551,"tokens_out":523,"duration_ms":5925,"temperature":1.0,"reasoning_tokens":450,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:10:25.571359+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a linear system where the exact Kalman-filter posterior is computable, run FBOVI over many steps, and compare the true total-variation or Hellinger distance between the exact joint posterior and the approximation with the right-hand side of the paper's estimable bound, evaluated by Monte Carlo. If the observed error consistently exceeds the estimated bound, or stays small when the estimated per-step KL terms are large, then the stability theorem or the ELBO-based step in the proof has failed.","supporting_citations":[],"review_version":1}