Backward Filtering Forward Guiding

Frank van der Meulen; Moritz Schauer; Stefan Sommer

REVIEW 2 major objections 6 minor 46 references

Backward Filtering Forward Guiding

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Exact weighted smoothing on trees and DAGs is possible without transition densities, via a backward information filter followed by a forward guided process.

desk verdict Useful unification of guided inference on trees and DAGs, but the continuous-time exactness claim rests on an unverified martingale condition. read the letter →

arxiv 2505.18239 v1 pith:KPDGY35U submitted 2025-05-23 stat.ME math.PR

Frank van der Meulen , Moritz Schauer , Stefan Sommer This is my paper · ORCID

classification stat.MEmath.PR MSC 60J0560J2560J2760J6062M05

keywords backwardinformationfilterguidedprocesssmoothingDoobh-transformexponentialchangeofmeasuredirectedacyclicgraphbranchingdiffusionparticlefiltering

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

This paper introduces Backward Filtering Forward Guiding (BFFG), a smoothing method for latent stochastic processes on trees and directed acyclic graphs where only leaf-node values are observed. The central claim is that by combining a backward information filter that approximates the likelihood-informed potentials with a forward guided process built from those potentials, one obtains weighted samples that are exactly from the smoothing posterior. Exactness hinges on an importance-weight identity, Theorem 4.2, which corrects for the gap between the true h-transform and the tractable approximation g, so the method works even when transition densities are intractable or unavailable. If correct, BFFG extends smoothing to branching diffusions, continuous-time Markov chains, interacting particle systems, and other structured dynamics, and it admits integration with MCMC, particle filters, and probabilistic programming.

What carries the argument

The engine is Doob's h-transform run twice. The backward information filter computes, exactly where possible, functions $h_t(x)$ equal to the density of the leaf observations in the subtree below $t$ given $X_t=x$; when this is intractable, a user-selected family $g$ is substituted so that every operation — pullback under a Markov kernel, fusion at a branching vertex, and initialisation at a leaf — closes in closed form. On continuous edges the guided process is defined by an exponential change of measure with density $Z^g_u = \frac{g_u(X_u)}{g_0(x_s)}\exp\!\left(-\int_0^u \frac{Ag}{g}\,d\tau\right)$, and a function $g$ is called 'good' when $Z^g$ is a true martingale. The guided process then evolves with kernels $\kappa^\circ(x,dy) \propto g_s(y)\,\kappa(x,dy)$ on discrete edges and the corresponding generator on continuous edges, and the Radon-Nikodym weight $\exp\!\left(\int_0^{\tau_e} \frac{Ag}{g}\,du\right)$ is precisely the correction that appears in Theorem 4.2.

What would settle it

On a linear-Gaussian tree, where forward-filtering backward-sampling gives exact smoothing expectations, run BFFG with a deliberately misspecified $g$ and increase the number of guided samples; if the weighted average of a bounded test function does not converge to the FFBS expectation, the condition $P^\star \ll P^\circ$ or the weight identity in Theorem 4.2 fails. On a DAG with a collider, compute the weight in (38) numerically: if the denominator $\prod_{u\in\mathrm{pa}(s)} g_{u,s}(x_u)$ vanishes on a set of positive prior probability, the Radon-Nikodym derivative is infinite and the exactness claim breaks down.

Watch

Extended reading notes

Core claim

The central claim is that the smoothing distribution on a tree or DAG can be represented exactly by a weighted sample from a guided process that preserves the forward model's dependency structure. Theorem 4.2 gives the identity $E^\star f(X_S) = \frac{g_r(x_r)}{h_r(x_r)}\, E^\circ\!\left[ f(X_S)\, \prod_{s\in S\cup V} w_{\mathrm{pa}(s),s}(X_{\mathrm{pa}(s)}) \right]$, where $P^\star$ is the conditional law given leaf observations, $P^\circ$ is the law of the guided process, and the edge weights $w$ are ratios of kernel pullbacks under $g$ and the true $h$. The same identity extends to DAGs (Theorem 8.1) after redefining the denominator of the weights to run over all parents of each vertex. A direct corollary is that the guided weight forms a positive unbiased estimator of the likelihood $h_r(x_r)$, making BFFG usable inside SMC, MCMC, and likelihood-based parameter estimation even when no transition density is available in closed form.

Load-bearing premise

For every edge the user must supply a tractable function $g$ that approximates the true backward potential $h$ well enough, can be pulled back and fused in closed form, and yields a guided process that can be simulated and weighed; the paper states that this backward-filtering step is problem specific and not automatic.

Editorial extensions

If this is right

Any model whose forward process can be simulated, and for which a tractable approximation to the h-transform can be found, becomes amenable to smoothing; transition densities no longer need to be known.
The weight formula yields a positive unbiased estimate of the likelihood, so parameter estimation can proceed by plugging BFFG into existing SMC or MCMC samplers, including gradient-based schemes where resampling non-differentiability is avoided.
Because the guided process inherits the forward conditional-independence structure, the same program that simulates the unconditional process can be transformed to simulate the guided one, opening a route to automatic conditioning inside probabilistic programming languages.
The DAG extension (Theorem 8.1) makes smoothing possible on graphs with multiple parents, at the cost of an additional product-normalisation in the weights.
The framework unifies prior algorithms: the discrete SMC computations of Ju et al. and the diffusion-tree backward filtering of Stoltz et al. appear as special cases.

Reading between the lines

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

The practical fidelity of BFFG is set by the choice of $g$: if the true $h$ is far from the tractable family, the weights in Theorem 4.2 gain variance and the scheme degrades toward naive importance sampling, so methods that adaptively refine $g$ from the empirical weight distribution would make the approach self-tuning.
On DAGs the product-normalisation in the weights of Theorem 8.1 keeps the guided process Markovian, but its quality depends on the KL projection in Proposition 8.2; a testable extension is whether a learned non-product approximation of $g$ at colliders beats the product form.
The compositional property proved in the companion category-theory paper suggests that BFFG is a natural compiler primitive: any probabilistic program written as a composition of Markov kernels could be automatically conditioned by rewriting each kernel in its guided form, leaving the backward information filter as the only inference-specific component.
The butterfly-wing example positions BFFG as a candidate engine for phylogenetic trait-evolution models on high-dimensional shape spaces; a head-to-head comparison against particle marginal Metropolis-Hastings on the same phylogeny would quantify when the non-resampling likelihood estimator wins.

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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

Frank,

The BFFG paper is a serious piece of work. What is genuinely new: a single template covering discrete and continuous edges on trees and DAGs, with exact importance weights expressed through the good-function framework, plus a DAG weight formula (Theorem 8.1) I have not seen elsewhere. The paper also credibly recovers Ju et al. (2021) and Stoltz et al. (2021) as special cases. The theoretical core, Theorem 4.2 and its DAG variant, is derived cleanly; the telescoping in the density ratio is neat. The authors are also honest that the backward filtering step is problem-specific and not automatic. The numerical examples are self-contained but not benchmarked.

The main soft spot is exactly where the stress-test lands. In the continuous-time SDE example, the guided process is built from a log-quadratic g, and the forward and auxiliary diffusions differ (even ignoring the apparent typos with undefined theta2 and sigma2). Then Ag/g contains a quadratic term in the state, so it is unbounded. Proposition A.3 therefore cannot be applied, and the paper provides no other verification that Z^g is a true martingale. Since the weight in Proposition 4.4 is the exponential change of measure, a merely local martingale makes the weighted samples biased for the smoothing posterior. This is not a cosmetic check; it underpins the exactness claim for all continuous-edge examples. The discrete cases do not have this problem, and the DAG extension inherits it only when continuous edges are present.

The DAG section is more of a recipe than a fully worked algorithm: the pullback kernel in (41) requires conditionals pi(x_{-u} | x_u), which are introduced abstractly and never constructed for a concrete example. That is a real gap, though not a fatal one. The paper would also be stronger with at least one baseline comparison in the numerics, and the undefined theta2 and sigma2 notation should be fixed.

Bottom line: the paper deserves a serious referee. The formalism is useful, the discrete-side theory is solid, and the DAG weight formula is a contribution. But the continuous-time exactness needs either verified good-function conditions in the examples or an explicit statement that the method is approximate in that regime. I would send it out, with that as the main referee request.

Referee Report

2 major / 6 minor

Summary. The manuscript introduces Backward Filtering Forward Guiding (BFFG), a framework for smoothing in discrete- and continuous-time stochastic processes on trees and DAGs. The backward step computes likelihood-informed potentials g from the leaves toward the root, typically using a tractable approximate dynamics; the forward step simulates a guided process under an exponential change of measure built from g. The central claim is that this yields weighted samples from the posterior distribution over latent paths, with weights given in Theorem 4.2 for discrete edges and Proposition 4.4 for continuous edges. The authors provide closed-form backward filtering, simulation, and weight formulas for several families: nonlinear Gaussian kernels, finite-state Markov chains, branching particles, Gamma increment processes, SDEs with linear Gaussian guiding, continuous-time Markov chains, and a Wright-Fisher diffusion approximation. A DAG extension is given in Section 8, and numerical experiments on tree-structured diffusions and a Kunita flow are reported.

Significance. If the exactness conditions are met, this is a useful unification: it connects message-passing on graphical models with guided-diffusion importance sampling and makes the backward/forward computations explicit for several tractable families. The paper is also honest about the role of g as a user-supplied approximation that is corrected by weights, and it benefits from the authors' earlier work on guided diffusions. The discrete-time results are self-contained and appear correct, and the availability of code is a concrete strength. However, the continuous-time exactness claim rests on an unverified martingale property that appears to fail for the main numerical example, and the DAG extension contains a normalization error and relies on conditional distributions that are never constructed. These issues affect the central claims and require substantial revision.

major comments (2)

[Section 8 / Proposition 8.2 / Appendix B] In the continuous-time setting, exactness in Proposition 4.4 requires that the exponential local martingale Z^g be a true martingale. Proposition A.3 gives sufficient conditions: bounded g and bounded Ag/g, or bounded g and Ag with positive infimum. In the SDE example of Section 9.1 the potential is log-quadratic as in Section 7.1, and using the auxiliary linear Gaussian process, Remark 5 gives (L-L~)g/g = sum_i (b_i-b~_i)r_i + (1/2) sum_{ij}(a_{ij}-a~_{ij})(partial_j r_i + r_i r_j) with r_i = F_i - H_i x. Since the diffusion coefficients a and a~ are different constant matrices and b~ is linear while b is bounded, this integrand is unbounded on R^2; hence g is not in B(E) and Ag/g is not in B(E). The paper supplies no alternative verification, such as a Novikov-type condition or a direct density computation, that E[Z^g_T]=1. Without this, P^o is only a subprobability and the weight estimator g_r(x_r) exp(integral Ag/g du) is not shown to be unbiased for h_r(x_r), so the weighted samples are not shown to target P*. This is a load-bearing gap in the main continuous-time numerical demonstration.
[Section 8, Eqs. (40)-(41), Appendix B] The DAG extension is built on a KL projection onto product form, but the normalization in the proof of Proposition 8.2 is incorrect. The stationarity equation integrates to lambda_j c_j = integral pi(x)g(x) dx, not to lambda_j c_j = 1; the proof silently assumes integral pi g = 1. Consequently the minimizer formula g_i(x_i)=c_i^{-1} E_pi[g(X)|X_i=x_i] is only valid under this unstated normalization, and with the choice c_i=1 made in the text it generally violates the stated constraint when integral pi g is not 1. Moreover, the conditional distributions pi(x_-u | x_u) required in Eqs. (40)-(41) are never constructed or specified in any example, and no guidance is given for choosing the auxiliary pi. Until this is repaired, Theorem 8.1's claim that Theorem 4.2 applies to arbitrary DAGs is not established. The DAG version of Eq. (15) also writes g_0(x_0) in the denominator, although Figure 2 explicitly has two roots; the denominator should be a product over all roots.

minor comments (6)

[Section 1.1] There is a typo: "continuoos-time" should be "continuous-time".
[Section 3.1, Definition 3.1] The phrase "we call an edge... discrete if it is assumed that onethe probabilistic evolution" contains a textual error; "onethe" should be removed or corrected.
[Section 6.1] The sentence "just like in message passing algoritheorems in general" appears to contain the typo "algoritheorems".
[Section 9.1] The notation in the paragraph defining the auxiliary process is inconsistent: the model in Eq. (42) uses parameters (theta_0, theta_1, sigma_0, sigma_1), but B is written with theta_1 and theta_2 and sigma~ with sigma_1 and sigma_2. Please clarify which parameters are used.
[Figure 4 caption] The caption contains the typo "valews" for "values".
[Section 4.2] The text says Z^g_u is a "mean-one F_u-local martingale"; a local martingale is not automatically mean-one, and for a nonnegative local martingale the property E[Z^g_u]=1 is equivalent to being a true martingale. This should be stated as an assumption, not as a consequence of Corollary A.2.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the BFFG weights are a derived Radon-Nikodym identity, and the self-citations are not the source of the exactness claim.

full rationale

The central exactness claim, Theorem 4.2, is proved inside the paper from the definitions of the h-transformed kernels and the guided kernels: dP*/dP and dP°/dP are computed, and the weight is their ratio. The unbiasedness of g_r(x_r) times the weight product as an estimator of h_r(x_r) is a consequence of this Radon-Nikodym identity, not a fitted parameter renamed as a prediction. The user-supplied g is an approximation to h, and the weights correct for the discrepancy whenever the change of measure is valid; Proposition A.3 (from Palmowski and Rolski, an external source) gives sufficient conditions. The continuous-time formula of Proposition 4.4 is likewise the exponential-change-of-measure density, not an independent fitted object. Theorem 7.1, which gives closed-form quadratic g's for SDEs, is inherited from the authors' prior Mider et al. (2021), a published and parameter-free result; it supplies the proposal family but is not the basis of the exactness identity. The companion-paper citation (Schauer et al. 2025) concerns a side property (compositionality), and the Schauer et al. (2017) remark concerns a shorter proof of a known likelihood expression; neither is load-bearing for the smoothing theorem. The numerical sections are self-contained simulations rather than external benchmarks, which limits scope but is not circularity. The skeptic's concern that in Section 9.1 Ag/g is unbounded and Proposition A.3's conditions are unverified is a genuine correctness/martingale gap for that example, but a failed assumption would invalidate the theorem's hypothesis, not make the theorem's conclusion an input. Overall: minor self-citations, no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Markov-process assumptions plus the availability of a user-specified tractable approximation g; no fitted constants enter the core derivation.

free parameters (4)

Auxiliary linear dynamics (B_e, beta_e, sigma_tilde_e) for continuous edges = user-specified per edge
Defines the tractable approximation g whose guided process is simulated; the method requires this choice but does not estimate it.
Auxiliary kernel kappa_tilde (parameters Phi, beta, Q, or beta_tilde) for discrete edges = user-specified per edge
Chosen to make backward filtering closed-form; e.g., Gaussian kernels in Theorem 6.1 and the Gamma rate beta_tilde in Theorem 6.5.
Truncation order K in Chebyshev expansion = not stated
In the Stoltz et al. example (Section 7.3), K controls accuracy of g; the paper does not discuss the choice or its error.
Iterative scale schedule sigma_tilde_e^(i)(u) = defined recursively in Section 9.2
Ad hoc refinement for the butterfly shape example; convergence is not analyzed.

assumptions (5)

domain assumption Subtrees branch conditionally independently given the vertex value (Markov property on the tree).
Used throughout Section 3, Eq (7), to justify fusion and the recursive backward information filter.
standard math Leaf transitions admit densities k_{pa(v),v} with respect to a dominating measure.
Eq (4) in Section 3; enables the backward information filter and the likelihood ratio in Theorem 4.2.
domain assumption The chosen g is a good function, so Z^g is a true martingale and not merely a local martingale.
Section 4.2 and Proposition A.3 give sufficient conditions, but they are not verified for the numerical examples.
ad hoc to paper For DAGs, the conditional distributions pi(x_{-u} | x_u) needed for the KL projection exist and are computable.
Eq (40)-(41) in Section 8; no practical construction or numerical example is provided.
domain assumption Strong solutions to the forward SDEs exist in the numerical examples.
Assumed in Appendix C.1 to justify the pCN innovation update.

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Cite this review

Pith. "Pith review of Backward Filtering Forward Guiding." pith.science (2026). https://pith.science/paper/KPDGY35U

@misc{pith2026250518239,
  author       = {Pith},
  title        = {Pith review of: Backward Filtering Forward Guiding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPDGY35U}},
  note         = {Machine review of arXiv:2505.18239}
}

read the original abstract

We develop a general methodological framework for probabilistic inference in discrete- and continuous-time stochastic processes evolving on directed acyclic graphs (DAGs). The process is observed only at the leaf nodes, and the challenge is to infer its full latent trajectory: a smoothing problem that arises in fields such as phylogenetics, epidemiology, and signal processing. Our approach combines a backward information filtering step, which constructs likelihood-informed potentials from observations, with a forward guiding step, where a tractable process is simulated under a change of measure constructed from these potentials. This Backward Filtering Forward Guiding (BFFG) scheme yields weighted samples from the posterior distribution over latent paths and is amenable to integration with MCMC and particle filtering methods. We demonstrate that BFFG applies to both discrete- and continuous-time models, enabling probabilistic inference in settings where standard transition densities are intractable or unavailable. Our framework opens avenues for incorporating structured stochastic dynamics into probabilistic programming. We numerically illustrate our approach for a branching diffusion process on a directed tree.

Figures

Figures reproduced from arXiv: 2505.18239 by the authors.

**Figure 1.** Example of a tree with known root vertex r, with observations at vertices 5, 6 and 7. A continuous time stochastic process evolves on the branches (0, 3), (3, 4), (0, 1) and (1, 2) which are coloured blue. 17.4.3), (ii) for linear Gaussian systems the Kalman Smoother, also known as RauchTung-Striebel smoother (Murphy (2012), section 18.3.2) for the marginal distributions or its sampling version, where samples from … view at source ↗

**Figure 2.** A DAG with two leaves and two roots. □ Consider the DAG in [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Forward simulated paths from (42) with (θ0, θ1, σ0, σ1) = (0.0, 0.65, 0.1, 0.4) on a 5 level tree with 121 nodes of which 81 are leaf nodes. Only the values at the leaf nodes are observed. updates the unobserved paths conditional on θ and the observations, and θ conditional on the unobserved paths. Elements of θ were updated using random-walk MetropolisHastings steps. The missing paths were updated using the BFFG-a… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Traceplots for the parameters θ0, θ1, σ0, σ1. Traceplots for θ0 and σ0 are in blue; traceplots for θ1 and σ1 are in orange. Green horizontal lines indicate true valeus; red horizontal lines show posterior mean after removing burnin samples [PITH_FULL_IMAGE:figures/ful…

**Figure 5.** Figure 5: Densities after removing the first 2000 iterations which are considered burnin samples. in Baker et al. (2025). In evolutionary biology, trait evolution is commonly modelled with Brownian motions Felsenstein (1985), traditionally for low dimensional data, i.e. small nu…

**Figure 6.** Figure 6: Tree and observed shapes [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]

**Figure 7.** Figure 7: Trace and density plots for the parameters α, σ for the kernel used in the Kunita flow and the observation noise variance ϵ. Red lines show the mean values of the samples. References Ambrogioni, L., Lin, K., Fertig, E., Vikram, S., Hinne, M., Moore, D. and van Gerven, …

**Figure 8.** Figure 8: (left) Posterior samples of the root of the tree (leaf values in the background). (right) Posterior samples of the parent of the leaves representing butterflies papilio xuthus and papilio zelicaon (continuous lines). Baker, E., Yang, G. and Sommer, S. (2025). Stochasti…

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" * write output.state after.block = add.period write newline

ENTRY address archive author booktitle chapter collaboration edition editor eid howpublished institution journal key lastchecked month note number numpages organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence...
[46]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

work page

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page

[3] [3]

@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command n...

[4] [4]

\@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@first@sw \@firstoftwo \@ifundefined NAT@b*@#2 \@firstoftwo @num @NAT@ctr \@secondoft...

[5] [5]

@open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibsetup #1 @NAT@ctr @ @openbib .11em \@plus.33em \@minus.07em 4000 4000 `\.\@m @bibit...

work page

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Ambrogioni, L., Lin, K., Fertig, E., Vikram, S., Hinne, M., Moore, D. and van Gerven, M. (2021). Automatic structured variational inference

work page 2021

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and Desbouvries, F

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work page 2012