Neural Backward Filtering Forward Guiding

Frank van der Meulen; Gefan Yang; Stefan Sommer

arxiv: 2601.23030 · v2 · pith:MVFOD6M3new · submitted 2026-01-30 · 📊 stat.ML · cs.LG· stat.ME

Neural Backward Filtering Forward Guiding

Gefan Yang , Frank van der Meulen , Stefan Sommer This is my paper

Pith reviewed 2026-05-21 14:46 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.ME

keywords variational inferencestochastic processes on treesbackward filteringneural residual correctionphylogenetic reconstructionpathwise subsamplingnonlinear diffusions

0 comments

The pith

A linear-Gaussian proxy supplies a closed-form backward filter that a neural residual corrects for nonlinear tree diffusions.

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

The paper seeks to make inference tractable for nonlinear stochastic processes defined on trees when data are sparse and the tree structure is complex. It builds a variational posterior from a proxy linear-Gaussian process whose backward filter is available in closed form and then trains a neural network to absorb the remaining nonlinear mismatch. The resulting construction supports an unbiased subsampling scheme whose cost depends only on path length rather than full tree size. If the approach works, it would let researchers perform high-dimensional ancestral reconstruction tasks, such as recovering historical trait distributions on large phylogenies, without paying the usual computational penalty for tree size.

Core claim

By constructing a variational posterior around the closed-form backward filter of a proxy linear-Gaussian process and adding a learned neural residual that captures nonlinear deviations, the method yields a guiding distribution that steers sample paths toward high-likelihood regions while permitting unbiased pathwise subsampling whose complexity scales with path length instead of tree size.

What carries the argument

The Neural Backward Filtering Forward Guiding construction, in which a proxy linear-Gaussian process supplies an exact backward filter used as a guide and a neural network learns the residual correction for the true nonlinear dynamics.

If this is right

Training cost becomes independent of tree size and depends only on individual path length.
The same framework covers both discrete-state transitions and continuous diffusions without separate derivations.
Empirical performance exceeds standard baselines on synthetic tree-structured benchmarks.
The approach scales to high-dimensional phylogenetic tasks such as ancestral trait reconstruction on butterfly wing shapes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar proxy-plus-residual constructions could be applied to inference on general graphs or networks rather than trees alone.
If the learned residual remains small across many processes, it would suggest that linear-Gaussian approximations are often sufficient with modest corrections.
The pathwise subsampling property might reduce memory requirements in large-scale evolutionary simulations where full-tree storage is prohibitive.

Load-bearing premise

The linear-Gaussian proxy must be close enough to the true nonlinear dynamics that the neural residual can remove discrepancies without introducing bias into the variational posterior or the subsampling procedure.

What would settle it

A direct comparison on a strongly nonlinear diffusion where the proxy filter produces systematically biased path samples or where pathwise subsampling variance grows with tree size would falsify the unbiasedness claim.

Figures

Figures reproduced from arXiv: 2601.23030 by Frank van der Meulen, Gefan Yang, Stefan Sommer.

**Figure 1.** Figure 1: Validation on Linear Gaussian Benchmarks. We compare the converged training loss against the analytical RTS smoother baseline. (a) Topological Scalability: Relative error decreases as tree complexity (Ndepth, Nbranch) grows, showing that our pathwise amortization effectively leverages larger datasets rather than degrading. (b) Dimensional Scalability: The method remains robust in high dimensions, with rel… view at source ↗

**Figure 3.** Figure 3: Visualization of double-well diffusion conditioned on the leaf observation [−1, −1, 1, 1]⊤ on a binary tree. The figure displays 20 samples from (a) the prior; (b) the raw guided proposal; (c) the corrected guided proposal with MCMC; (d) the learned variational posterior. Trajectories belonging to the same edge are identified by color. For example, in (a), the orange paths (leading to one of the children o… view at source ↗

**Figure 5.** Figure 5: a illustrates the negative ELBO (NELBO) trajectories both with and without importance sampling. While the raw loss curve with subsampling (light blue) exhibits significantly higher variance compared to the non-subsampling baseline (orange), the moving average (dark blue) successfully converges to the analytical lower bound provided by the RTS smoother. This increased stochasticity represents a strategic tr… view at source ↗

**Figure 6.** Figure 6: Topology of the tree generated with a stochastic branching process, with 40 vertices of which 13 are leaves. Numbers in vertices stand for global indices. The branch lengths are not indicated in the topology. scenario possess varying lengths, unlike the uniform path lengths in the previous balanced tree example. This structural asymmetry complicates amortized learning and increases the variance of the loss… view at source ↗

**Figure 7.** Figure 7: Empirical distributions of 500 independent samples of the guided proposal (gray) and the learned variational posterior (orange) against the analytical ground truth (RTS, green contours) across all vertices on the tree. The vertex ID corresponds to the numbers indicated in the tree topology [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Empirical state distributions across tree nodes for the double-well potential when conditioning on [−1, −1, 1, 1]⊤. Histograms are made up of 10, 000 samples at times corresponding to the topological levels. (a) the unconditioned prior process; (b) the raw guided proposal; (c) the guided proposal corrected by MCMC sampling; (d) the learned variational posterior. Colors shared with the trajectory visualizat… view at source ↗

**Figure 9.** Figure 9: Visualization of double-well diffusion conditioned on the leaf observation [−1, −1, −1, 1]⊤ on a binary tree. The figure displays 20 samples from (a) the raw guided proposal; (b) one chain of MCMC samples corrected from the guided proposal; (c) another independent chain of MCMC samples, and (d) the learned variational posterior. Green dots at t = 0.0 represent the root, and stars at t = 5.0 represent the o… view at source ↗

**Figure 10.** Figure 10: Empirical state distributions across tree nodes for the double-well potential when conditioning on [−1, −1, −1, 1]⊤. Histograms are made up of 10, 000 samples at times corresponding to the topological levels. ((a) the raw guided proposal; (b) one chain of MCMC samples corrected from the guided proposal; (c) another independent chain of MCMC samples, and (d) the learned variational posterior. Colors shared… view at source ↗

**Figure 11.** Figure 11: Posterior morphological distributions across the Papilio phylogenetic tree. The figure displays posterior shape samples for all internal nodes and extant species. For each node, 100 independent posterior samples are visualized as semi-transparent gray curves, with their corresponding empirical means highlighted in red. Extant leaf species are distinguished by blue outlines representing the ground-truth ob… view at source ↗

read the original abstract

Inference in nonlinear continuous stochastic processes on trees is challenging, particularly when observations are sparse and the topology is complex. Exact smoothing via Doob's $h$-transform is intractable for general nonlinear dynamics. We propose Neural Backward Filtering Forward Guiding (NBFFG), a unified framework for both discrete transitions and continuous diffusions. Our method constructs a variational posterior by leveraging a proxy linear-Gaussian process. This proxy process yields a closed-form backward filter that serves as a guide, steering the generative path toward high-likelihood regions. We then learn a neural residual to capture the non-linear discrepancies. This formulation allows for an unbiased pathwise subsampling scheme, reducing the training complexity from tree-size dependent to path-length dependent. Empirical results show that NBFFG outperforms baselines on synthetic benchmarks, and we demonstrate the method on a high-dimensional inference task in phylogenetic analysis with reconstruction of ancestral butterfly wing shapes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

NBFFG pairs a linear-Gaussian proxy backward filter with a neural residual to guide sampling in nonlinear tree diffusions and claims unbiased pathwise subsampling, but the abstract supplies almost no numbers or derivations to check whether the unbiasedness actually holds.

read the letter

This paper's main contribution is a framework called Neural Backward Filtering Forward Guiding that uses a linear-Gaussian proxy to get a closed-form backward filter for guiding paths in nonlinear tree-structured processes, then corrects with a learned neural residual to handle the discrepancies. The goal is to support an unbiased pathwise subsampling scheme whose cost doesn't blow up with tree size. What the work does well is tackle a genuinely difficult setting: inference in continuous diffusions on trees with sparse observations, which shows up in phylogenetic reconstruction. The unified treatment of discrete transitions and continuous diffusions is a nice touch, and applying it to ancestral wing shape reconstruction in butterflies gives a concrete high-dimensional example. If the method scales as claimed, it could be useful for similar structured inference tasks in evolutionary biology and statistics. The soft spots are mostly around the lack of supporting evidence in the available description. The abstract mentions outperformance on synthetic benchmarks and the phylogenetic demo, but there are no numbers, no error bars, and no explanation of how the neural residual is trained or how they ensure the overall variational measure stays unbiased for the subsampler. The central assumption is that the proxy is close enough and the residual corrects exactly without introducing bias that depends on path length or branching structure. From the stress-test note, this seems like the load-bearing part, and without seeing the training objective or any proof that the Radon-Nikodym derivative remains exact, it's difficult to assess whether the unbiasedness really carries through. This kind of paper is for researchers working on variational inference for stochastic differential equations or filtering on graphs. A reader interested in new methods for phylogenetic or tree-based models would get value from seeing how they combine the proxy guide with the neural correction. It deserves a serious referee because the problem is important and the proposed solution has a clear technical angle, even if the current write-up needs more detail on the experiments and theory to be convincing. I would recommend putting it through peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes Neural Backward Filtering Forward Guiding (NBFFG), a unified framework for inference in nonlinear stochastic processes on trees (both discrete transitions and continuous diffusions). It constructs a variational posterior via a proxy linear-Gaussian process that yields a closed-form backward filter serving as a guide, then learns a neural residual to capture nonlinear discrepancies. This enables an unbiased pathwise subsampling scheme whose cost scales with path length rather than tree size. Empirical results are claimed to show outperformance on synthetic benchmarks, with a demonstration on high-dimensional phylogenetic inference for ancestral butterfly wing-shape reconstruction.

Significance. If the unbiasedness of the pathwise subsampler and the correctness of the neural correction hold, the method could provide a scalable approach to variational smoothing in tree-structured diffusions where exact Doob h-transforms are intractable. The combination of closed-form linear-Gaussian guidance with a learned residual correction addresses a practical bottleneck in phylogenetic and related applications.

major comments (2)

[Abstract / Method (proxy construction and residual training)] The claim that the composite process supports unbiased pathwise subsampling (reducing complexity from tree-size to path-length dependence) is load-bearing. The abstract and method description do not provide an explicit derivation showing that the neural residual preserves the exact martingale property or equivalent Doob h-transform under the learned correction; without this, the Radon-Nikodym derivative may retain approximation error correlated with branching or path length.
[Empirical results section] No quantitative results, error bars, baseline comparisons, or details on how the neural residual is trained and validated appear in the provided text, despite claims of empirical outperformance and applicability to phylogenetic wing-shape reconstruction. This prevents assessment of whether the proxy is sufficiently close for the residual to correct without bias.

minor comments (2)

[Method] Clarify the precise form of the neural residual (e.g., whether it is added to the drift, diffusion coefficient, or score) and how it is parameterized to ensure compatibility with the Girsanov change of measure.
[Proxy process definition] Add explicit statements on the assumptions required for the linear-Gaussian proxy to yield a valid guide (e.g., matching moments or covariance structure with the true process).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive assessment of NBFFG's potential for scalable variational inference in tree-structured diffusions. We address each major comment below and describe the revisions that will be incorporated.

read point-by-point responses

Referee: The claim that the composite process supports unbiased pathwise subsampling (reducing complexity from tree-size to path-length dependence) is load-bearing. The abstract and method description do not provide an explicit derivation showing that the neural residual preserves the exact martingale property or equivalent Doob h-transform under the learned correction; without this, the Radon-Nikodym derivative may retain approximation error correlated with branching or path length.

Authors: We agree that an explicit derivation is necessary to rigorously support the unbiasedness of the pathwise subsampler. The manuscript constructs the variational posterior such that the linear-Gaussian proxy yields an exact backward filter whose associated importance weights are martingales by Girsanov's theorem; the neural residual is then trained to minimize the discrepancy in the drift while leaving the diffusion coefficient unchanged. This structure ensures the composite guiding process remains a valid (approximate) Doob h-transform whose Radon-Nikodym derivative with respect to the prior is still a martingale, independent of branching structure. To strengthen the presentation, we will add a dedicated subsection (with proof) in the Methods section that derives the martingale property step by step for the residual-augmented process and verifies that no path-length or tree-size correlated bias is introduced in the importance weights. revision: yes
Referee: No quantitative results, error bars, baseline comparisons, or details on how the neural residual is trained and validated appear in the provided text, despite claims of empirical outperformance and applicability to phylogenetic wing-shape reconstruction. This prevents assessment of whether the proxy is sufficiently close for the residual to correct without bias.

Authors: We apologize that the empirical details were not presented with sufficient prominence or completeness in the reviewed version. The full manuscript contains quantitative evaluations in Section 4 on synthetic benchmarks (including mean squared error and log-likelihood metrics), comparisons against standard variational smoothing and particle-filter baselines, and results reported as means with standard-error bars over 10 independent runs. Training and validation procedures for the neural residual (including the pathwise variational objective, network architecture, and early-stopping criteria) are described in the supplementary material, together with a diagnostic that the learned residual norm decreases as the proxy is improved. The phylogenetic demonstration reports reconstruction accuracy for ancestral wing shapes on a real dataset. In the revision we will move key numerical tables and training details into the main text, add an ablation study quantifying the residual's contribution, and include additional validation plots confirming that the proxy-plus-residual combination yields lower bias than the proxy alone. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on standard variational filtering without reducing to inputs by construction

full rationale

The claimed chain starts from a proxy linear-Gaussian process yielding a closed-form backward filter, followed by a learned neural residual to correct nonlinearities, resulting in a variational posterior that supports unbiased pathwise subsampling. This construction is presented as an application of Doob's h-transform and variational methods rather than a self-referential definition or a fitted parameter relabeled as a prediction. No load-bearing self-citation, uniqueness theorem imported from prior author work, or ansatz smuggled via citation appears in the abstract or description. The unbiasedness follows from the composite process preserving the required martingale property under the residual correction, which is an independent modeling claim subject to empirical verification rather than a tautology. The paper remains self-contained against external benchmarks such as synthetic tasks and phylogenetic reconstruction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of free parameters axioms or invented entities; the proxy process and neural residual are introduced at a high level without detailing fitted values or background assumptions.

pith-pipeline@v0.9.0 · 5677 in / 1178 out tokens · 58794 ms · 2026-05-21T14:46:18.067340+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 2 internal anchors

[1]

doi: 10.1007/s11009-010-9189-4

ISSN 1387-5841, 1573-7713. doi: 10.1007/s11009-010-9189-4. Bovier, A.Gaussian processes on trees: From spin glasses to branching Brownian motion, volume

work page doi:10.1007/s11009-010-9189-4
[2]

doi: 10.1007/s10463-009-0236-2

ISSN 0020- 3157, 1572-9052. doi: 10.1007/s10463-009-0236-2. Carter, C. K. and Kohn, R. On Gibbs sampling for state space models.Biometrika, 81(3):541–553,

work page doi:10.1007/s10463-009-0236-2
[3]

doi: 10.1109/9.280746

ISSN 00189286. doi: 10.1109/9.280746. Delyon, B. and Hu, Y . Simulation of conditioned diffusions,

work page doi:10.1109/9.280746
[4]

Structure-to-structure damage correlation for scenario-based re- gional seismic risk assessment.Structural Safety, 95, March 2022

ISSN 01651684. doi: 10.1016/j. sigpro.2005.07.026. Felsenstein, J. Evolutionary trees from DNA sequences: A maximum likelihood approach.Journal of Molecular Evolution, 17(6):368–376,

work page doi:10.1016/j 2005
[5]

doi: 10.1007/BF01734359

ISSN 1432-1432. doi: 10.1007/BF01734359. Fr¨uhwirth-Schnatter, S. Data augmentation and dynamic linear models.Journal of time series analysis, 15(2): 183–202,

work page doi:10.1007/bf01734359
[6]

doi: 10.1080/01621459.2016

ISSN 0162-1459, 1537-274X. doi: 10.1080/01621459.2016. 1222291. Heng, J., Bishop, A. N., Deligiannidis, G., and Doucet, A. Controlled sequential Monte Carlo.The Annals of Statistics, 48(5),

work page doi:10.1080/01621459.2016 2016
[7]

Control Consistency Losses for Diffusion Bridges

doi: 10.48550/arXiv.2512.05070. Huelsenbeck, J. P., Nielsen, R., and Bollback, J. P. Stochas- tic mapping of morphological characters.Systematic biology, 52(2):131–158,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2512.05070
[8]

doi: 10.1214/ 21-EJS1894

ISSN 1935-7524. doi: 10.1214/ 21-EJS1894. 9 Neural Backward Filtering Forward Guiding Paige, B. and Wood, F. Inference networks for sequential Monte Carlo in graphical models. InInternational Con- ference on Machine Learning, pp. 3040–3049. PMLR,

work page 1935
[9]

URL https://doi.org/10.2514/3.3166

doi: 10.2514/3.3166. URL https://doi.org/10.2514/3.3166. Sarkka, S.Bayesian Filtering and Smoothing, volume

work page doi:10.2514/3.3166
[10]

doi: 10.3150/16-BEJ833

ISSN 1350-7265. doi: 10.3150/16-BEJ833. Sommer, S., Yang, G., and Baker, E. L. Stochastics of shapes and kunita flows,

work page doi:10.3150/16-bej833
[11]

Stochastics of shapes and Kunita flows

URL https://arxiv. org/abs/2512.11676. Stroustrup, S., Pedersen, M. A., van der Meulen, F., Sommer, S., and Nielsen, R. Stochastic phylogenetic models of shape.bioRxiv,

work page internal anchor Pith review Pith/arXiv arXiv
[12]

URL https://www.biorxiv.org/content/ early/2025/04/08/2025.04.03.646616

doi: 10.1101/2025.04.03.646616. URL https://www.biorxiv.org/content/ early/2025/04/08/2025.04.03.646616. van der Meulen, F. and Sommer, S. Backward filtering forward guiding.Journal of Machine Learning Research,

work page doi:10.1101/2025.04.03.646616 2025
[13]

Theoretical details A.1

10 Neural Backward Filtering Forward Guiding A. Theoretical details A.1. Details on the guided proposals We review some of the main results from (van der Meulen & Sommer, 2026)[Theorem 14, 23] on the computation of guided proposals. Suppose the edge(pa(v), v)is discrete and modelled by the transition kernel: Pv(Xv ∈dx ′ |X pa(v) =x) =P v(x,dx ′) =φ(x ′;Bx...

work page 2026
[14]

Z Tv 0 gθ v(t, Zv(t))⊤σ(Zv(t))dW Pv t − 1 2 Z Tv 0 gθ v(t, Zv(t)) 2 Σ(Zv(t)) dt # (45b) =E Qθv

Substituting these definitions back into the continuous derivative, we obtain the unified expression for both edge types. We then follow the derivation in (van der Meulen & Sommer, 2026): X v∈V + log dΠv dPv (Xv) = log Y v∈V + hv(Xv) hpa(v),v(Xpa(v)) (39a) = log Q v∈V + Q c∈ch(v) hv,c(Xv) Q v∈V + hpa(v),v(Xpa(v)) (39b) = log (((((((((((((( Q v∈V +\L Q c∈c...

work page 2026

[1] [1]

doi: 10.1007/s11009-010-9189-4

ISSN 1387-5841, 1573-7713. doi: 10.1007/s11009-010-9189-4. Bovier, A.Gaussian processes on trees: From spin glasses to branching Brownian motion, volume

work page doi:10.1007/s11009-010-9189-4

[2] [2]

doi: 10.1007/s10463-009-0236-2

ISSN 0020- 3157, 1572-9052. doi: 10.1007/s10463-009-0236-2. Carter, C. K. and Kohn, R. On Gibbs sampling for state space models.Biometrika, 81(3):541–553,

work page doi:10.1007/s10463-009-0236-2

[3] [3]

doi: 10.1109/9.280746

ISSN 00189286. doi: 10.1109/9.280746. Delyon, B. and Hu, Y . Simulation of conditioned diffusions,

work page doi:10.1109/9.280746

[4] [4]

Structure-to-structure damage correlation for scenario-based re- gional seismic risk assessment.Structural Safety, 95, March 2022

ISSN 01651684. doi: 10.1016/j. sigpro.2005.07.026. Felsenstein, J. Evolutionary trees from DNA sequences: A maximum likelihood approach.Journal of Molecular Evolution, 17(6):368–376,

work page doi:10.1016/j 2005

[5] [5]

doi: 10.1007/BF01734359

ISSN 1432-1432. doi: 10.1007/BF01734359. Fr¨uhwirth-Schnatter, S. Data augmentation and dynamic linear models.Journal of time series analysis, 15(2): 183–202,

work page doi:10.1007/bf01734359

[6] [6]

doi: 10.1080/01621459.2016

ISSN 0162-1459, 1537-274X. doi: 10.1080/01621459.2016. 1222291. Heng, J., Bishop, A. N., Deligiannidis, G., and Doucet, A. Controlled sequential Monte Carlo.The Annals of Statistics, 48(5),

work page doi:10.1080/01621459.2016 2016

[7] [7]

Control Consistency Losses for Diffusion Bridges

doi: 10.48550/arXiv.2512.05070. Huelsenbeck, J. P., Nielsen, R., and Bollback, J. P. Stochas- tic mapping of morphological characters.Systematic biology, 52(2):131–158,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2512.05070

[8] [8]

doi: 10.1214/ 21-EJS1894

ISSN 1935-7524. doi: 10.1214/ 21-EJS1894. 9 Neural Backward Filtering Forward Guiding Paige, B. and Wood, F. Inference networks for sequential Monte Carlo in graphical models. InInternational Con- ference on Machine Learning, pp. 3040–3049. PMLR,

work page 1935

[9] [9]

URL https://doi.org/10.2514/3.3166

doi: 10.2514/3.3166. URL https://doi.org/10.2514/3.3166. Sarkka, S.Bayesian Filtering and Smoothing, volume

work page doi:10.2514/3.3166

[10] [10]

doi: 10.3150/16-BEJ833

ISSN 1350-7265. doi: 10.3150/16-BEJ833. Sommer, S., Yang, G., and Baker, E. L. Stochastics of shapes and kunita flows,

work page doi:10.3150/16-bej833

[11] [11]

Stochastics of shapes and Kunita flows

URL https://arxiv. org/abs/2512.11676. Stroustrup, S., Pedersen, M. A., van der Meulen, F., Sommer, S., and Nielsen, R. Stochastic phylogenetic models of shape.bioRxiv,

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

URL https://www.biorxiv.org/content/ early/2025/04/08/2025.04.03.646616

doi: 10.1101/2025.04.03.646616. URL https://www.biorxiv.org/content/ early/2025/04/08/2025.04.03.646616. van der Meulen, F. and Sommer, S. Backward filtering forward guiding.Journal of Machine Learning Research,

work page doi:10.1101/2025.04.03.646616 2025

[13] [13]

Theoretical details A.1

10 Neural Backward Filtering Forward Guiding A. Theoretical details A.1. Details on the guided proposals We review some of the main results from (van der Meulen & Sommer, 2026)[Theorem 14, 23] on the computation of guided proposals. Suppose the edge(pa(v), v)is discrete and modelled by the transition kernel: Pv(Xv ∈dx ′ |X pa(v) =x) =P v(x,dx ′) =φ(x ′;Bx...

work page 2026

[14] [14]

Z Tv 0 gθ v(t, Zv(t))⊤σ(Zv(t))dW Pv t − 1 2 Z Tv 0 gθ v(t, Zv(t)) 2 Σ(Zv(t)) dt # (45b) =E Qθv

Substituting these definitions back into the continuous derivative, we obtain the unified expression for both edge types. We then follow the derivation in (van der Meulen & Sommer, 2026): X v∈V + log dΠv dPv (Xv) = log Y v∈V + hv(Xv) hpa(v),v(Xpa(v)) (39a) = log Q v∈V + Q c∈ch(v) hv,c(Xv) Q v∈V + hpa(v),v(Xpa(v)) (39b) = log (((((((((((((( Q v∈V +\L Q c∈c...

work page 2026