Causal Score Conditioning for Multi-Resolution Latent Systems
Authors: Xuechun Li
Shan Gao
Susu Xu
Venue: ICLR 2026 Poster (2026)
Keywords: Causal Score Conditioning
Variational causal inference
Probabilistic graphical models
Multi-resolution observations
Score-based diffusion models
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Complex causal systems with interdependent variables require inference from heterogeneous observations that vary in spatial resolution, temporal frequency, and noise characteristics due to data acquisition constraints. Existing multi-modal fusion approaches assume uniform data quality or complete observability -- assumptions often violated in real-world applications. Current methods face three limitations: they treat causally-related variables independently, failing to exploit causal relationships; they cannot integrate multi-resolution observations effectively; and they lack theoretical frameworks for cascaded approximation errors. We introduce the Score-based Variational Graphical Diffusion Model (SVGDM), which integrates score-based diffusion within causal graphical structures for inference under heterogeneous incomplete observations. SVGDM introduces causal score decomposition enabling information propagation across causally-connected variables while preserving original observation characteristics. Diffusion provides a natural way to model scale-dependent sensing noise, which is common in remote-sensing, climate, and physical measurement systems, while the causal graph encodes well-established mechanistic dependencies between latent processes. We provide theoretical analysis and demonstrate superior performance on both synthetic and real-world datasets compared to relevant baselines.

Published as a conference paper at ICLR 2026
CAUSAL SCORE CONDITIONING FOR MULTI-RESOLUTION LATENT SYSTEMS
Xuechun Li
Shan Gao
Department of Civil and Systems Engineering Johns Hopkins University
Department Civil and Systems Engineering Johns Hopkins University sgao 650 juu edu
x113590 jhu.
Susu Xu
Department of Civil and Systems Engineering Johns Hopkins Data Science and Al Institute Johns Hopkins University susuxue jhu edu
ABSTRACT
Complex causal systems with interdependent variables require inference from het- erogeneous observations that vary in spatial resolution, temporal frequency, and noise characteristics due to data acquisition constraints. Existing multi-modal fu- sion approaches assume uniform data quality or complete observability. assump- tions often violated in real-world applications. Current methods face three limita- tions: they treat causally-related variables independently, failing to exploit causal relationships; they cannot integrate multi-resolution observations effectively; and they lack theoretical frameworks for cascaded approximation errors. We intro- duce the Score-based Variational Graphical Diffusion Model (SVGDM), which integrates score- based diffusion within causal graphical structures for inference under heterogeneous incomplete observations. SVGDM introduces causal score decomposition enabling information propagation across causally-connected vari- ables while preserving original observation characteristics. Diffusion provides a natural way to model scale-dependent sensing noise, which is common in remote- sensing, climate, and physical measurement systems, while the causal graph en- codes well-established mechanistic dependencies between latent processes. We provide theoretical analysis and demonstrate superior performance on both syn- thetic and real-world datasets compared relevant baselines.
INTRODUCTION
Complex causal systems, where multiple interact- ing variables influence each other through inherent causal dependencies, are ubiquitous in real world, such earth systems, epidemiological systems, etc. A wealth of spatio-temporal data are available for these systems and have led an increasing develop- ment and application of machine learning to estimate spatio-temporal states of latent variables. However, data acquisition constraints limit observation quality, leading to heterogeneity in spatial resolution, tempo- illlustrates gen-
Causal Structure
Latent Variable
Multi-resolution Observations
ral frequency, and noise. Figure [ eral scenario where causally-cone conneceed latent vari- ables must be inferred from partial observations that vary systematically in spatial detail, temporal sam- pling, and measurement noise. A key challenge that these variables correspond to different physical processes and are observed at different resolutions and qualities. Thus, the problem is not multi-view
Causal structure with multi- resolution observations. Latent variables (circles) are connected through known causal relationships and observed through measure- ments that vary in spatial resolution, noise levels, and measurement precision (grids).
Figure 1:
"Corresooniing author.

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fusion of single object, but joint inference across distinct variables linked by causal mechanisms. Latent graphical models formalize these dependencies and enable information to propagate along causal edges, allowing high-quality observations of one variable to improve inference for others. This causal information transfer is crucial for integrating heterogeneous, multi-resolution data in complex systems. Joint inference of multiple latent spatio-temporal variables from such heteroge- neous and incomplete data is thus funndaeental to applications including natural hazard monitoring, climate modeling, epidemiological surveillance, and financial system analysis (He & Cha 2018 Xu
et al. 2022b Li & Xu 2025 Nowack et al., 2020; Reich et al., 2021 Wang et al. 2024 Yu et al.,
2024
Most existing multi-moda or multi data fusion approaches assume (1) explicit systems with
closed-form dependencies among variables (Zhao et al. 2024; Ravi et al. 2025 al. 2025b), (2) observed data of uniform quality (e.g., resolution, modality, noise level) (Li et al. 2024 Xinde et al.)
2024 Li et al. 2025a), or (3) complete observability of all variables- assumptiosss often violated in real-world applications. For instance, in earth systems, remote sensing observations of key variables such as soil moisture, precipitation, and geophysical properties range from 30 m (Vergopolan et al. 2021) to 5 km in resolution and exhibit distinct noise characteristics due to different measurement principles and satellite capabilities. Several works explore using diffusion processes to assimilate
multi-resolution data, but mainly focus on single-variable systems (Batchu et al. 2022 Tu et al. 2025).
We use diffusion models because their forward SDE naturally encodes scale-dependent noise, central property of remote sensing, InSAR, radar, and optical imagery where noise accumulates as resolution coarsens. This makes diffusion uniquely suited for multi-resolution inference, enabling resolution-aware score functions that normalizing flows or standard variational inference cannot capture. Our problem setup is therefore to estimate multiple causally related physical processes from observations that are (i) multi-resolution, (ii) uneven in quality, and (iii) incomplete, by lever- aging partially known causal structure to propagate information across variables and scales. This corresponds directly to real-world systems such as earthquake cascades, wildfire dynamics, and climate teleconnections. We introduce a novel framework that enables effective inference for inter- dependent systems with implicit and complex causal dependencies and heterogeneous, incomplete observations. Specifically, we introduce diffusion processes to approximate complex causal de- pendencies among variables and between variables and observations. Integrating diffusion process modeling with complex causal structure substantially improves the ability to model interdependent systems. However, effectively inferring large numbers of latent variables remains challenging due to the coupled complexity of intractable causal paths and diffusion processes.
To address the challenges, we introduce the Score-based Variational Graphical Diffusion Model (SVGDM), a framework that integrates known causal graphical structures with score-based diffusion processes for latent variable inference under multi-resolution observations. The causal graph is pro- vided by established physical or mechanistic knowledge (e.g., earthquake + landslide - damage), allowing information to propagate across variables that are observed at incompatible spatial resolu- tions. Diffusion serves as a natural model for scale-dependent, resolution-dependdett sensing noise, such as speckle, atmospheric delay, and resampling artifacts, offering advantages over flow-based methods that do not explicitly model noise accumulation across spatial scales. We address these limitations through causal score decomposition, enabling joint inference while preserving original resolutions and borrowing strength across causal structures. We make the key contributions:
1. We introduce the Score-based Variational Graphical Diffusion Model (SVGDM), which integrates probabilistic graphical models (Koller & Friedman, 2009) with score-based dif- fusion processes (Song & Ermon 2019 [onn et 20000)) Unlike existing approaches that treat variables independently. SVGDM leverages causal dependencies to propagate information across variables with heterogeneous observation quality.
2. We develop a causal score decomposition that respects graphical structure while maintain- ing the efficiency of score-based inference. This enables coherent integration of multi- resolution observations without information loss or artifacts, borrowing strength across causally-connected variables rather than using ad-hoc preprocessing.
3. We provide theoretical analysis of our approach, establishing conditions under which our locally Gaussian approximation remains valid and deriving error bounds for the cascading effects in our inference procedure.
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4. We demonstrate SVGDM's effectiveness on synthetic causal systems and real-world appli- cations where causal relationships are well-established and multi-resolution observations are prevalent, showing consistent improvements over variational inference and deep learn- ing baselines, with superior performance on both synthetic and real-world datasets.
2 PRELIMINARIES
We briefly review the foundations of graphical models, score-based diffusion models, and their limitations for multi-variable inference with known causal structure and heterogeneous observations.
Probabilistic graphical models (PGMs) (Jordan 2003 Koller & Friedman 2009) provide a structured representation of joint distributions over high-dimensional
Probabilistic Graphical Models:
variables by encoding conditional dependencies in graph (Jordan 2003 Koller & Friedman 2009
Pearl 2009; Wainwright et al., 2008). Let X = (X1.... Xx) denote a collection of random
variables. directed acyelic graph (DAG) specifies factorization of the joint distributions as
P(X) = Ih= (P)).. where PXX;) denotes the parent nodes of X,, This representation enables efficient inference and learning in complex systems, but often requires approximation when dependencies are nonlinear, stochastic, or latent. In dynamic settings, latent states evolve in time ac- cording transition models, leading to state-space formulations of graphical models. While PGMs provide a general factorization formalism, existing PGM-based inference methods for continuous systems typically assume aligned spatial supports and compatible noise models across variables. In practice, most message-passing and variational inference algorithms break down when different nodes are observed at vastly different resolutions, noise scales, or spatial supports, as they require a common discretization or homogeneous likelihood families (Zhang et al. 2021 Wang et al. 2018 Akbayrak et al. 2022). Our method addresses precisely this limitation.
Diffusion Models and seore-based Methods: Diffusion models have emerged powerful tools for generative modeling of complex and high- sigh--imensional data (Dhariwal & Nichol 2021 |202 Ho et al., 2020 Dhariwal & Nichol 2021 Song et al.) 2020b; Kingma et al.) 2022 Rombach et al. 2022). A forward process gradually corrupts the the data with Gaussian noise through a stochastic differential equation (SDE), while training reverse-time SDE recover the distribution. Central this frame- work is the score function so (0(x,1) R V log (zr). where (r( denotes the marginal distribution of the perturbed data at time Current score-based approaches treat variables independently during training and inference: (1) No causal structure awareness: Score functionn are typically learned for each variable isolation, ignoring known causal dependencies that could improve estimation qual- ity. (2) Independent gradient estimation: The score Vz log p(z) is approximated without leveraging information flow through causal pathways, missing opportunities to borrow strength from well- observed variables to improve poorly-observed ones. (3) No multi-resolution coherence: Existing score matching objectives cannot handle observations of the same latent phenomena at different scales, requiring artificial preprocessing that discards information.
Neither PGMs nor standard score-based methods can address the fundamental challenge of joint inference with known causal structure and heterogeneous observations. Causal score decomposition is necessary because: (1) It enables score functions to respect graphical structure, allowing infor- mation propagation through causal pathways during inference. (2) It provides a principled way to integrate multi-resolution observations by borrowing statistical strength across causally-connected variables. (3) maintains the computational efficiency of score-based methods while incorporating the structured dependencies of PGMs. Variational inference (Blei et al. 2017 Hoffman et al., (0113) provides a scalable framework for approximating intractable posteriors p(((x) via tractable fami- lies 4 (2()),,, but standard variational families may fail to capture complex posterior geometries in high-dimensional nonlinear systems. Recent advances suggest that score- based diffusion processes can serve as flexible variational distributions, motivating the development of causal based approaches that unify structured dependencies with generative flexibility.
3 SVGDM: SCORE-BASER VARIATIONAL GRAPHICAL. DIFFUSION MODEL.
Problem Formulation: Given
1. A known causal DAG G = (V,E) with N latent variables {2)(t) == where each variable
evolves according to dzz(t)= (p(o + gt(tdW where P(i) denotes the causal parents of variable i.
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2. Multi-resolution observations Y {yt (1.k) I) where indexes spatial locations, k indexes resolution/mnodalityy related latent variables via vf of (2p,) + EL C1,b (0, E1,), where Pt S are the latent parents of observation yf.
The assumption of a known prior graph G is realistic in many scientific and engineering domains where latent variables represent concrete physical processes with well-established causal relation- ships. Such dependencies arise from decades of experiments. mechanistic modeling, and field ob- servations and are standard in spatio-temporal data assimilation (Reich & Cotter 2015} Crrsssee Wikle 2011). Importantly, our method does not require the DAG to be perfectly known. Even partial causal directionality provides useful information pathways for integrating heterogeneous, observations (Evaluation is shown in Appendix By.
We follow the standard diffusion convention where z(t) denotes latent variable i diffusion time t E [0, 1}, with t 0 representing the true latent state and larger injecting noise, while our Abservation-constraiinn
SDE (Equation [ keeps z(1) anchored observations rather than col- lapsing to N N(0,7 I) as in classical DDPMs.
Goal: Find the posterior distribution: p(Z p(Y]Z)p(Z) while (1) preserving orginal observa- tion resolutions, (2) leveraging causal dependencies to propagate information from well-observed to poorly-observed variables, and (3) providing theoretical guaranteex for inference quality.
Assumption (Observation noise). For the heterogeneous observations Y, assume (1) additive Gaussian noise with variance schedule of. (2) conditional independence of mulli-resolutiin obser- vations given latent variables, and (3) smooth log- densities permitting interchange of gradient and expectation. These assumptions align with prior work on diffusion probabilistic models and causal Bayesian inference.
Assumption 2 (Lipschitz and boundedness). For each E V, the drift f&(-) is Lipschitz contin- uous and has most linear growth, and the diffusion ge(t) is bounded and strictly positive. The observation operators ot are measurable and bounded on compact domains.
Theorem (Existence and locality of node-wise causal SDEs). Under Assumption
2 the system
dz,(t) = fi (z;(1,, (t)) dt + ()) (t. iE V,
(1)
with parent-dependent drift functions fo and independent Brownian motions Wt(t). which is an independent standard Wiener process associated with node i. All Wiener processes W, mutually independent, admits a unique strong solution. Moreover, the infinitesimal generator of the joint process decomposes as
L, Σ Lis
fis: Se(x, 02. (1)
(2)
iEV
so that each local operator Li,t depends only on 2; and its parents 2P(1) in the causal graph G. This expresses locality of the generator, not conditional independence of pt. the initial distribution po factorizes according to G as
po(z) II po 2p()),
then the factorization holds only at t O by definition of the initial condition. For O, the marginal law Pt of the SDE solution need not preserve this global factorization; in general it may exhibit additional dependencies induced by the dynamics. In our framework, we use the generator- level decomposition Lt E; Li,t as an architectural prior for an approximate score-based causal decomposition (Theorem 2) rather than as an exact distributional Markov property of pt-
This result follows from standard SDE existence-and- existenceeamd--uiiqeness theory applied to each node-wise equation, together with the independence of the driving Wiener processes and the conditional de- pendence structure encoded by G. For general SDE preliminaries, see Särkkä & Solin (2019).
To address our problem, each latent variable is constrained by heterogeneous observations through:
(1) = f((#1, ZP()) l)dl + Σ (0lol ( z (1) + ((((dW
(4)
where (-) maps observation y to the latent space and (t) controls the influence of resolu- tion i obsevations. This formulation naturally handles information flow across different resolutions through the diffusion process while preserving orginal observation characteristics.
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Assumption (Time-reversal regularity). The forward diffusion satisfies: (i) non-degenerate diffu- sion coefficient a(t) > O. (ii) drift f is locally Lipschitz, and (iti) the marginal density pt exists and is continuously differentiable in 2.
Under Assumption
3 classical results on time-reversed diffusion (Anderson 1982 Bhattacharyal
& Waymire, 2009) and the score-based SDE formulation (Song et al. 2020b) guarantce that the
reverse-time process exists with well-defined drift f.
Lemma 1 (Node-wise reverse SDE with observations). For each E V, the reverse-time SDE incorporating multi-resolution observations is:
dz; [[i((z,
E (t)o 2P() (.) - (t) log log g:
where W₂ is a standard Wiener process in reverse time.
(5)
The key challenge in our problem formulation that standard score-based methods cannot leverage causal structure during inference. Theorem 1 establishes that we have valid causal SDE system, but the reverse-time SDE (Lemma) requires computing the causal score V., Pr (2)))))) This causal score is crucial because it (T) enables information flow through causal pathways during in- downstream
ference (2) allows well-observed variables to improve inference of poorly-obsere variables; and (3) maintains causal consistency while preserving multi-resolution observation char- acteristics. Without causal score decomposition, we would be forced to either treat variables in- dependently (losing causal information) downsample all observations to common resolution (losing fine-graine the-grained information). The following theoretical development shows how causal score decomposition addresses this fundamental limitation.
The central quantity in the reverse drift is the causal score Zz, (o This causal score essential for problem formulation because it enables joint inference while preserving orginal observation resolutions, the core challenge identified in our problem definition.
3.1 CAUSAL SCORE DECOMPOSITION VIA MARKOV BLANKETS
Recent work in score-based generative modeling for sequences (Rozet & Louppe 2023) demon-
strates that for Markov chains FI.L, the global score can be decomposed using Markov blanket properties: V,, log p( = V.,. æs log Ir.)), To.) where 2h, represents the Markov blanket of z;- For first-order Markov chains, this blanket consists of immediate neighbors: Ih, = -x 11
In our setting, we extend this principle by recognizing that the causal parents P(i) form natural "causal blanket" for each variable 2;- The key insight is that the causal Markov property ensures:
2; T NonDescendants(23) P(())
This conditional independence structure provides the foundation for our score decomposition.
Theorem (Causal Blanket Preservation under Diffusion). Under Assumptions | 3 for diffusion-- perturbed variables {2; ( p the causal blanket relationship is approximately preserved:
V (e) log.e(z1:N xe V ait(t) log (p((())))))((()))
The diffusion process preserves the conditional independence structure of the original causal graph up to noise-dependent approximation errors. As t - O, the approximation becomes exact since 2t(t) + (0) and the original causal structure is recovered. For > O, the approximation quality depends the noise level (t) and the strength of causal dependencies. Full proof in Appendix C.1 The local nature of this decomposition ensures computational scalability even for large causal score decomposition idea
graphs, as detailed in Appendix D This extends the sseudo-blanketo
of Rozet & Louppe 2023) to the causal graphical setting.
Proposition 1 (Causal score decomposition). For any E V and the causal blanket (p)))());
Zz, (ogpp()//p(a)) = log v., logp((zp())22))
This decomposition separates two influences:
(1) Marginal term Vz((1) log (2:(t)) captures the unconditional score of (t); (2) Causal consistency term V (0 log (e (t)) ensures com- patibility with causal parents. The second term is not backwards causation but rather represents the constraint that (1) must remain consistent with the joint distribution over the causal family
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(t)) During reverse diffusion, this term helps maintain causal structure by ensuring that updates to (() preserve plausible relationships with its parents (Proof in Appendix C.2)
(p()))
The causal score decomposition enables coherent integration of observations different resolutions because the marginal term V log (2;) incorporates direct observational evidence for variable 2; from all available resolutions. The causal consistency term F. log Deggp(Ppp (z) |2;) allows observations of parent variables to inform child variable inference through causal pathways. Besides, this de- composition preserves orginal resolutions by avoiding the need to downsample observations to a common scale, instead leveraging the natural information flow through the causal structure
3.2 SCORE FUNCTION ESTIMATION
We estimate the first term in Equation 8 using continuous time denoising score matching (DSM)
(Hyvärinen Dayan 2005 Vincent, 2011 Song et al. 2020b). Given samples 2; 20) po(=:),
the forward SDE generates (t) with known perturbation kernel pr(zalz:(0)). We approximate the marginal score with neural network sp, (2,, () trained via:
LDSM
(0).... (t) Vat(o (ogp((=()))=(()))
(9)
where A(t) > 0 is weighting function. This corresponds to the continuous-time DSM objective
Hyvärinen & Dayan 2005 Vincent 2011 Song & Ermon 2019).
Proposition 2 (DSM consistency). Under Assumption the population minimizer satisfies $2 (2..t) = =. log pr(2) (2;). (Proof is shown in Appendix C.3
For the second term in Equation 8 we adopt locally Gaussian approximation that avoids the =(0) dependency inconsistency. We model:
(10)
Theorem 3 (Validity of Local Gaussian Approximation), The locally Gaussian approximation in Eq. is valid when the true conditional p=(zp(e) (2)) is log-concave in a neighborhood of &,(1), the
diffusion noise dominates higher-order nonlinearities such that o (t)2 > || log Pogp((2(()))))))) and the Tweedie reconstruction error remains bounded as [|2;(t)
(t)] I 5 for small 5.
Under these conditions, the approximation error satisfies:
V (og - Vz log.ooNNeeee Ex)l = 0 + (t)-
The approximation is most accurate during early diffusion stages (small f) when noise levels are low, and deteriorates gracefully as f + 1. While our theoretical analysis uses Gaussian and locally log- concave noise assumptions for tractability, the implementation does not rely on these assumptions. We empirically tested heavy-tailed and skewed noise regimes, and SVGDM remains stable with only modest degradation, indicating that the assumptions are sufficient for analysis but not required in practice (see Appendix E). The posterior mean estimate (t) in the Gaussian model is obtained via Tweedie's formula:
2t (t) + (2)))) (z.(1).
(11)
This follows Tweedie's formula (Efron 2011 Kim & Ye, 202T) and the decomposition of the poste-
rior score into prior and likelihood components (Song et al.) 2020b Chung et al. 2022). Under this model, the causal consistency score becomes:
V (0 log (()(())))) OlogN((p Ex) Ope.
02,(1)
(12)
0pe
d2t(t) 02,(t)
The validity of Eq. 12 requires that saa()) is differentiable in a neighhorhood of (t),,, the Gaussian model provides a reasonable local approximation with KL(px ( (4)
Ex) S c for
small e, and score estimates are sufficiently accurate such that ||sw, (z:, 4) V log p. < M. When these conditions
violated, employ adaptive regularization: if ||V т, we add penalty Aveg / I2= to the causal loss. This formulation ensures complete consistency between training and inference. The causal model parameters Pes Σg are trained forward samples (see Appendix F for implementation details). This formulation resolves training consistency through iterative refinement procedures described in Appendix G

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3.3 STOCHASTIC VARIATIONAL INFERENCE
Given the learned reverse-time SDE dynamics, we perform posterior inference over the latent vari- ables {2,
V) from multi-resolution observations Y = =v/² (1, k) E Т}, where I indexes spatial locations and & indexes observation resolution modality. The variational poste-
rior qu (Z ) is implicitly defined as the distribution of samples generated by the reverse SDE with posterior score:
log qe (21122()) Y) se. (z)(t), V log (Y
(13)
The variational parameters vo {1:))=11 directly parameterize the reverse drift through the learned score functions. By Jensen's 's inequality:
log 2 (vr(gw) E [Yogp(Y())) + (log p(z) E [log qv (Z|Y))
(14)
Each observation y¹t depends on latent parents Pi C V via observation map (.). yf 2p,
( (zp,), ( This formulation explicitly connects multi-resolution observations to the causal structure, enabling the information propagation mechanisms described in our causal score decomposition. The likelihood contribution becomes:
Egg llo p (Y)
1
Σ (1,))EZ
Egg [lla (f +C
(15)
The forward SDE induces graph-structured prior Eg [log p(Z]] Bay Lievlog
Since our reverse SDE defines a normalizing flow from noise to data, we leverage the flow Jacobian for stable entropy computation (Rezende & Mohamed, 2015
Kingma et al. 2016
K
A [[w]] = Hl[((T)] Eng [log (ot(
(16)
F(((2)) represents the discretized reverse SDE step. For computational efficiency, we use Hutchinson's 's trace estimator (Hutchinson, 1989 Pearlmutter 1994 Chen et al.] 2018):
where 2k-1
log (et ~ e" (8Fs/ /2zz),, where e ~ N0, 7) is random probe vector.
Combining all components, our final objective
Liaal A1
Σ LDSM. (voe) + A2 Σ Cemusal (O,s) + XzLV1
(17)
where L is the score matching loss (Equation 9); Leassal is the causal model training loss (Ap- pendix Fb; FvzLyz is the variational bound; X's are weighting hyperparameters.
The "normalizing flow" interpretation refers to the deterministic probability-flow ODE associ- ated with the reverse SDE, while in practice our implementation uses the stochastic reverse-SDE predictor-corrector sampler for numerical stability (Appendix H)
ERROR ANALYSIS AND APPROXIMATION CASCADE
Our method involves a sequence of approximations whose errors may compound. We provide systematic analysis of how these errors propagate and establish conditions under which the method remains stable.
4.1 APPROXIMATION HIERARCHY
Let E E2, E3, E4, Es denote the approximation errors from: (1) E1: Marly-aa SDE discretiza-
tion (Kloeden Platen 1992] (2) E2: Neural score function approximation (DSM) (Hyvärinen &
Dayan | Vincent
2011 Song et al. (020b): (3) E3: Locally Gaussian conditional model (Tier-
ney & Kadane 1986 Rue al. 2009: Wainwright al. 2008); (4) e4: Tweedie's formula posterior
estimation (Robbins, 1992 Efron 201 Song et al. 2021 (5) E5: KDE entropy estimation (Joc) 2014).
1989
Tsybakov| 2008) Singh Póczos
Theorem 4 (Cascade Error Bound). Under appropriate regularity conditions, the total approxima- tion error in the posterior score satisfies:
V a(t) log go (z(t)l)) - V log ((=(((())) v)ll2 I CCe1 + C282 + C3es + C4ea O(e253) (18)
where O are problem-dependent constants and the cross-term O(e2E3) captures the interaction between score approximation and Gaussian modeling errors (Li el al. 2023a).
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4.2 INDIVIDUAL ERROR ANALYSIS
The main error contributions of our framework can be decomposed into five parts. First, the Euler-Maruyama discretization introduces an eπror = 0( for step size At, following standard SDE discretization theory, which directly affects the perturbation kernel Pejo (z))()())):
Second, the neural score approximation yields an eттоr 22 = O(1/VN + >reg), where N is the training set size and Areg is the regularization strength; the DSM objective ensures this vanishes as N + cc. Third, when replacing true conditionals with local Gaussian approximations, the error Eg O(IIv log (x (o))) o?) arises, which remains small if the conditional distribution is approximately log-concave Fourth, Tweedic's formula introduces an error E1 = =(((t)² E2), showing that score approximation errors amplified at higher noise levels (t).. Finally, entropy estimation via kernel density estimation contributes Es O(h (log Ny4/2 /VN), where flects the smoothness of the underlying density and d the dimensionality.
4.3
CRITICAL ERROR INTERACTIONS
The most significant error interaction occurs between score approximation and Gaussian conditional modeling. Since Tweedie reconstruction &; depends on the approximate score Sy., errors in score estimation directly impact Gaussian conditional parameters:
||,((; + =26) - (2)2)ll2 S 1,
(19)
where L,, is the Lipschitz constant of Pe-
4.4 ERROR PROPAGATION ANALYSIS
The chain rule computation in Eq. 12 involves multiple approximation stages whose errors can compound. The total error decomposes as Etoral =
EGasssian + Eweedic + EChain + Elnteraction- where Gaussion V logptlla Pol o²) represents Gaussian approximation eπrоr, Tweediic = (()(t)² Escore) captures Tweedie formula error, EChain = O(L 'ETwediie) accounts for chain rule propagation, and Interactionn O GGaussian) represents cross-term interactions.
The interaction term Einteraction can dominate when both score approximation and Gaussian modeling poor. Our iterative training procedure mitigates this by improving score quality first, then refining causal parameters, ensuring errors do not compound catastrophically.
4.5 STABILITY CONDITIONS Condition 1 (Bounded Propagation). The score approximation error remains bounded during re- verse diffusion:
(2) v., log M <0
(20)
for all t E [0, 1] and in the support.
Condition (Lipschitz Causal Functions). The conditional mean functions satisfy [|pe(z') Me(z)ll2 S Llla² 2|12. ensuring that score errors don't explode in the causal consistency term.
Condition 3 (Variance Scheduling). The noise schedule satisfies o1) (02/)((() S K for some constant K, preventing Tweedie amplification from becoming unbounded.
4.6 PRACTICAL ERROR MITIGATION
Use smaller Marer-Maruyama steps A/ regions where the score gradient is large Aladpttive min(Al Ahase /// || (=((), ()) Train score networks multiple noise levels to improve robust- ness Luulti Ee (t)M with weights w(t) emphasizing critical time regions. Add regular- ization to prevent overfitting in the Gaussian conditional model Leausll = LCSM + A/|V P
Theorem (Convergence under Error Control). each approximation error E; can be made arbi- trarily small, then limee..e.esss..rs -0 -0 (2lg) = p(zly). in the weak sense, ensuring that SVGDM converges to the true posterior as approximation quality improves.
RESULTS
Evaluation Metrics: We adopt task-appropriate evaluation metrics depending on the estimation setting. For synthetic datasets with ground truth latent trajectories, we use Normalized Root Mean Square Error (NRMSE), Mean Absolute Percentage Error (MAPE), and Continuous Ranked Proba- bility Score (CRPS) assess latent state reconstruction quality and posterior calibration. For real- world seismic hazard modeling (landslides. liquefaction, and building damage), we use Area Under the ROC Curve (AUROC) measure discrimiorginal performance across classification thresholds.
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For wildfire spread prediction, which is formulated spatiotemporal binary classification prob- lem, we follow prior work (Gerard et al., 2023) and report FI Score and Average Precision (AP). This flexible choice of metrics ensures that each task is evaluated with the most informative and widely accepted measures.
Table I: Evaluation on synthesized data. We report mean + std for MAPE, NRMSE, and CRPS across multiple runs (lower is better).
Setting
Variable
MАРE
NRMSE
CRPS
21
0.0526 : 0.0029
0.0683
0.0037
0.039 = 0.0021
VFO
22
0.0991 + 0.0056
0.1239
0.0068
0.0756 0.0049
0.0763 + 0.0043
0.1031
0.0059
0.0567 0.0036
0.0756 + 0.0048
0.0922 + 0.006S
0.0572 + 0.0039
LFO
0.1214 1 0.0079
0.1557 1. 0.0084
0.0919 1 0.0067
0,1451 +: 0.0086
0,1814 1 0.0098
0.1088 + 0.0074
0.1067 1 0.0061
0.1227 1 0.0077
0,0810 = 0.0048
LPO
22
0.1470 + 0.0088
0.1730 + 0.0109
0.1107 = 0.0071
23
0.1961 0.0117
0.2228 0.0129
0.1515 0.0096
Evaluation Synthetic Data: We evaluate our framework on synthetic datasets that replicate multi-resolution observation patterns found in real-world systems while providing ground truth for quantitative assessment. We construct synthetic causal systems with known DAG structures where latent variables evolve according to mean-reverting stochastic differential equations, incorporating realistic spatial correlations through FFT FFT-generated Gaussian random fields.
We create observations y with varying information content by sampling from different stages of the forward diffusion process process-earlie earlier stages provide high-resolution observations (less noise, more spa- tial detail), while later stages yield low-resolution observations (more noise, less detail). This sim- ulates realistic scenarios where different variables are observed with different measurement quality and spatial resolution. We evaluate three scenarios: (1) Varying-resolution Full Observation (VFO): each variable observed at different resolution levels test information flow through causal structure; (2) Low- resolution Full Observation (LFO): all variables observed uniformly at low resolution; (3) Low-resolution Partial Observation (LPO)): only subset of variables observed at low resolution.
Table demonstrates our multi-resolution approach's is effectiveness, with VFO achieving the best performance and systematic degradation from VFO LFO to LPO. This validates that causal struc- ture enables effective information propagation between variables with different observation quali- ties. sigure@Jshows SVGDM substantially outperforms all baselines, achieving 2-3x better perfor- mance than domain-specific methods (VBCI, DisasterNet) and 10 -0 improvements over general variational inference methods. The poor VI baseline performance (MAPE > 60%) highlights the importance of incorporating causal structure for effective latent variable inference. Figure in Ap- pendix I.T provides visual confirmation that our reconstructions closely match ground truth while substantially outperforming baselines.
We additionally conducted synthetic experiments with 10-15 latent variables under both sparse and dense causal graphs. Accuracy remains stable as the system size increases, while runtime scales approximately linearly with the number of causal edges [E]. Full results, including all settings (VFO/LFO/LPO)), are provided in Appendix D.I
To further clarify the relation to multi-modal / multi-view methods, we also evaluate family of multi-view VAES (JMVAE, MMVAE, MoPoE-VARE VAE)) that treat all observations different views of a single shared latent representation. As reported in Appendix J SVGDM achieves substantially lower NRMSE and MAPE across all latent variables. This gap quantifies how much accuracy is lost when collapsing causally linked, multi-resolution physical processes into single shared latent and ignoring both the causal graph and resolution-specifee noise diffusion.
To evaluate contributions of individual loss components (local DSM score, causal-blanker score, and intervation-conisttenn term), we conducted an ablation on the synthetic 3-node system; removing any single component leads to consistent degradation (Appendix K
Evaluation on Real-World Disaster Systems: Multi-Hazard Seismic Assessment. We evaluate our model three major earthquake events: the 2020 Puerto Rico, 2021 Haiti, and 2023 Turkey- Syria earthquakes Xu et al. (2022b a). These events provide diverse geological settings and data availability conditions for comprehensive evaluation. Our approach simultaneously estimates mul- tiple earthquake-indee induued hazards, including landslides (21.5), liquefaction (21,F), and building dam- age (ZBD).

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MAPE (59) [lower betterr) NAMSE (lower is bebeen
2.23
1.09
0.75
000
40-
60-
40
20-
Our Model
DisasterNet
SIVI
NUTS
VICI
HSVI
AUVI
Figure 2: Comparison of model performance on synthesized data. Bars show normalized root mean squared error (NRMSE: top row) and mean absolute percentage error (MAPE: bottom row) across three target variables (z1, 22, z3). Our proposed model (red, bold outline) consistently achieves the lowest error across variables, substantially outperforming existing baselines.
Table 5 in Appendix 1.2 presents AUROC scores across all three events. For the 2020 Puerto Rico earthquake, our method achieves AUROC scores of 0.9331, 0.9317, and 0.9512 for the three hazards respectively, demonstrating consistent performance across interconnected phenomena. Compared to variational inference baselines (BBVI, ADVI, NUTS), our approach shows substantial improve- of 14-21%. %. The performance advantage over recent deep learning methods (VCBI, Disaster- Net) further validates the effectiveness of our causal modeling approach. Results generalize across events: the 2021 Haiti earthquake yields AUROC scores 0.9550 (landslides) and 0.9587 (building damage), while the 2023 Turkey-Syrin earthquake achieves 0.948% for building damage estimation. Performance remains robust even when ground truth is partially available, demonstrating practical applicability under real-world constraints.
Evaluation on Real-World Disaster Systems: Wildfire Spread Prediction. We evaluate our tem- poral extension large-scale wildfire spread prediction, following the benchmark setup of (Gerard ct al. 2023). Our method achieves an Fl score of 0.5913 and an Average Precision (AP) 0.4430, outperforming both classical models (e.g., logistic regression) and deep learning baselines such as U-Net. COMVLSTM, and UTAE. These results demonstrate the framework's 's ability to capture com- plex spatiotemporal dependencies in wildfire dynamics. Full quantitative results provided in Appendix[[ [ These results across diverse disaster scenarios validate our method's 's capability to han- dle multi-resolltion observations while maintaining interpretable causal structure, addressing key challenges in real-world disaster modeling applications.
CONCLUSION
We propose SVGDM, novel framework that integrates score-based diffusion with causal graphical models for inference on latent systems under multi-resolution observations. We leverage varia- tional inference approximate posterior distributions over latent variables while respecting known causal dependencies through our causal score decomposition approach. Theoretically, we establish sufficient conditions for structural identifiability and prove convergence guarantees under our error cascade analysis framework. Empirically, we demonstrate that SVGDM consistently outperforms baseline methods across both synthetic and real-world datasets, achieving superior performance in disaster estimation tasks with AUROC scores exceeding 0.93 across multiple earthquake events and wildfire scenarios. There are several limitations that warrant future investigation. First, our method assumes known causal structure, limiting applicability when graph topology must be inferred. Sec- ond, the locally Gaussian approximation may deteriorate under strong nonlinearities that violate our theoretical conditions. Third, computational complexity scales with the number of causal de- pendencies, potentially limiting scalability to very large systems. Future work could explore joint structure discovery, more flexible approximation families, and extensions to time- varying causal relationships. We leave these challenges for future work.
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ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation under Grant Nos. 2442712, 2332145, 2536601, and 2242590. Xuechun Li was supported part by Gordon and Betty Moore Foundation Postdoctoral Fellowship.
ETHICS STATEMENT
Our study uses openly available remote sensing datasets (InSAR, and optical imagery) and geophys- ical data curated by established scientific organizations (e-g.. USGS, NASA, StBER). No personally identifiable information, sensitive personal data, or human subjects are involved. Potential societal impacts of our work include both benefits (faster hazard assessment, improved disaster response) and risks (misuse or misinterpretation of automated hazard assessments in policy or emergency set- tings). To mitigate risks, we emphasize transparent reporting. open discussion of model assumptions and limitations, and reproducibility through shared code and data preprocessing steps. We declare no conflicts interest, financial or otherwise.
REPRODUCIBILITY STATEMENT
We are committed to ensuring reproducibility of our results. All experiments were run on publicly available datasets (SAR, InSAR, optical, and geophysical features) with clearly described prepro- cessing pipelines. To facilitate replication, we provide the full implementation of our framework, including model code, training scripts, and evaluation pipelines, at https: / com/ PaperSubmi ss issiioninnniiiiaII resolution satellite imagery), provide instructions for obtaining the data and scripts reproduce preprocessing. Together, these resources ensure that all reported results can be replicated.
For datasets requiring restricted access (e.g., high-
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A USE OF LARGE LANGUAGE MODELS (LLMS)
We used large language model as writing assistant improve grammar and readability. The model was not used for research ideation, technical derivations, experiments, or analysis. All scien- tific content, methodology, results, and conclusions were solely developed by the authors.
B SENSITIVITY TO CAUSAL GRAPH MISSPECIFICATION
In many physical systems, the qualitative causal directionality is known from scientific understand- ing, but some edges may be uncertain. To quantify the robustness of SVGDM under such uncer- tainty, we systematically perturb the causal DAG in the synthetic benchmark by: (1) dropping parent edges, (2) reversing edges, and (3) adding spurious edges. Table reports performance degradation relative to the correct DAG.
Table 2: Sensitivity of SVGDM causal graph misspecification.
Graph variant
NRMSE
MAPE (%)
ANRMSE
Correct DAG
0.098 + 0.006
7.6 0.5
Drop parent
0.112 0.008
8.7 0.6
+14%
Reverse edge
0.118 + 0.010
9.7
+20%
Two wrong edges
0.162 0.015
12.5
+65%
Add spurious edge
0.104 + 0.007
8.0 0.5
+6%
The results indicate that SVGDM robust to moderate structural errors, such as missing or extra edges, which introduce only small degradations. Severe incorrect directionality, which disrupts true information flow, naturally produces larger errors. This behavior is expected and supports the applicability of SVGDM even when the causal structure only partially known.
C
PROOFS
C.I PROOF OF THEOREM 2
Proof. We establish that the causal blanket relationship is preserved under diffusion through continuity argument.
Step 1: Initial Condition. At = 0, by the causal Markov property (Equation 6 (Anderson 1982
Rozet & Louppe 2023
(0) log Po (= (0)) = =0) (0),
This continuity- based argument follows the intuition in (Rozet & Louppe 2023 where blanket structures are preserved approximately under diffusion.
Step 2: Continuity. The score function 2.) log (z).N is continuous in t under Assumption
3 Similarly, 22.(1) po):((), (p)) is continuous.
Step 3: Bounded Approximation Error. For any t E [0, 1]. the approximation error satisfies:
Va,(1) log (z) V (o, (P() s Co(1)²
where depends on the strength of connections between 2; and non non-parent nodes.
Step 4: Noise-Dependent Convergence. As the noise level (tt) increases, the mutual information between (t) and non-parents decreases, making the causal blanket approximation increasingly accurate.
Therefore, the causal blanket relationship is preserved up to a controlled approximation eror that vanishes 0 and remains bounded for all i. [0, 1).
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C.2 PROOF OF PROPOSITION
Proof. From the definition of conditional probability density:
(z(z,, (7())) = pt ((72)))()7(()))))
(21)
Dividing both sides of Equation 21 by pr (((2)()) > 0 yields:
(((2))2(()) ((lz))p((2P()))))
(22)
Pt (=((())
Taking logs of Equation 22
Po (z) (7))) = log (2:) + log po (o(2)l27(0)) log (((2(())
(23)
Differentiating w.r.t 2j and noting that (1(()()) is independent of
2x logp(()/2p(a)) v., (o,) + v., (ogg(
(24)
C.3 PROOF OF PROPOSITION 2
Proof. Let Y = V (+) (og))(2(())2 and X = (t).. The DSM risk can be written as:
[[)(t)] sp. (X, t) Y
(25)
By the orthogonality principle of L minimization, the minimizer satisfies:
$/. (x, t) = []
(26)
The score marginalization identity Hyvärinen & Dayan 2005) states that:
E log (0)) = =. log Pz
(27)
Substitution Equation 27 into Equation 26 gives s (2;, t) = v., log (2:). which proves the claim.
This decomposition consistent with prior results graphical models (Wainwright et al. 20088 and posterior score decompositions in diffusion models (Song et al. 2020b).
D COMPUTATIONAL SCALABILITY ANALYSIS
While general DAGs can exhibit complex long-range dependencies that challenge computational scalability, our approach maintains tractability through several key strategies:
Local Computation Principle:
Each node requires computation only over its causal parents P(i), limiting complexity to the local neighborhood size rather than the full graph. For most real- world causal systems, IP(i) < N, ensuring that computational complexity scales as O(N d)
where d is the average parent set size.
Parallel Node Updates: Since each node's score computation depends only on its causal blan- ket, nodes with non-overlapping blankets can be updated in parallel, enabling efficient distributed computation across the graph structure.
Sparse Dependency Exploitation: Real-world causal graphs often exhibit sparsity, with most nodes having few parents. Our method naturally exploits this sparsity through the causal blanket decom- avoiding unnecessary computations for non-adjacent nodes.
position in Theorem 2
Complexity Comparison: Traditional approaches to inference in graphical models can require O(N*) complexity where w is the treewidth of graph. Our local score decomposition reduces this to O(N -max, (P())), providing significant computational advantages for sparse causal graphs.
D.1 SYNTHETIC EXPERIMENTS ON SCALABILITY
Table D.I reports results for systems with 10, 12, 15 latent variables under sparse and dense causal graphs. Accuracy degrades smoothly with reduced observability (VFO-+LFO-LLPO, and runtime scales approximately linearly with EEE, confirming that SVGDM remains computationally
feasible for larger causal systems.
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Table 3: Synthetic experiments for scalability analysis. Mean + s.d. aggregated over all latent
variables. Runtime is measured relative to the 3-variable VFO baseline.
Setting (N, graph)
NRMSE
MAPE (%)
Runtime (x baseline)
VFO (10, sparse)
0.11 0.02
9.3 1.4
3.3x
LFO (10, sparse)
0.14 + 0.02
1.5
3.6x
LPO (10, sparse)
0.05
14.7 1.9
3.9x
VFO (10, dense)
0.02
1.6
5,6x
VFO (12, sparse)
0.02
9.5
4.1x
LFO (12, sparse)
0.15
11.
4.4x
LPO (12, sparse)
0.20 1:0+:.00 0.04
1.1
4.6x
VFO (12, dense)
0.13 0.02
1.6
6.3x
VFO (15, sparse)
0.13
1.6
5.0%
LFO (15, sparse)
0.16
12.6
5
LPO (15, sparse)
0.21 0.04
2.2
5.7x
VFO (15, dense)
4.04
11.0
9.8x
E ROBUSTNESS TO NON-GAUSSIAN AND HAN-LOG-CONCAVE NOISE
The Gaussian and local log-concavity assumptions used in analysis (e.g., Theorem 3 dard regularity conditions that ensure convergence and bounded score approximation error. They are required by the implementation: the score model fully data-driven and capable of capturing heavy-tailed, heteroscedastic, mildly multimodal noise.
To verify robustness, we replace Gaussian noise with Laplace (heavy-tailed), Student-t, and skewed log-normal component. Across all conditions, SVGDM exhibits only modest degradation ($5-8% NRMSE on average), demonstrating that while the assumptions are helpful for theoretical analysis, the practical method remains stable under significant deviations.
Table 4: Robustness to non- an-Gaussian non-log-concave noise (VFO, N 10, sparse).
Noise type
NRMSE
MAPE (%)
Gaussian (baseline)
0.1158 + 0.0182
9.32 + 1.41
Laplace (same Var)
0.1167 = 0.0194
9.73 + 1.56
Student-t (v = 5)
0.1184 + 0.0205
9.96 + 1.63
Skewed log-normal (10% mix)
0.1199 + 0.0212
10.1 + 1.72
F IMPLEMENTATION ALGORITHMS
F.1 CONSISTENT CAUSAL SCORE SCORE TRAINING
Training Objective: We train the causal model parameters {Po, No} via maximum likelihood on forward samples, ensuring training-inference consistency:
(Oa) = Er,e(0) [ log (og ((Pp)(()))))))))))))))
(28)
where 0. (Me, Eej and (1) is computed using the current estimate.
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Algorithm Unified Training Algorithm
1: Input: Training data (=(0)(m) (-0))) causal graph G
2: Initialize: Score networks {80, J. causal networks {Mes Ee}
3: for epoch = to Nepachs do
4: for each node E V do
5: Sample =(0) po(2), (0, =N(,),
6: Generate (t)) via forward SDE (Equation 1)
7: Train marginal score: Update Sp, using L (Equation
8: Compute Tweedie reconstruction: E,(t) (1) + (1) Sy. (1) (/a(())
9: Train causal model: Update {Pe. ke} using Leausal (Equation 28
10: end for
11:
end for
12: Output: Trained score networks and causal models
F.2 CAUSAL SCORE COMPUTATION
During inference, the causal score is computed using only observable quantities:
Algorithm 2 Causal Score Computation
1: Input: Current state (t), parent states (P) (t), time
2: Compute marginal score: Smarginal = 8v, (t),
Compute Tweedie reconstruction:
2t (t)² marginaal (t(()
4: Compute causal mean: t4. =){(2;)
5: Compute chain rule components:
6: V He log (Ec) (Ec)- (2p(s) (2 (t) Mc)
7: Va,Mee Jacobian of [e at 2
8: V = 1 (t) Sor
9: Return: =., log Pi (1p()))) = Vp log Valee 22,
F.3 REVERSE SDE SAMPLING
To generate latent trajectories during inference, we simulate the reverse SDE (Equation 5 using predictor corrector mrdiccor-coorrector scheme. We use a predictor- corrector sampler with a fast -intentiiliinteeattoo step (Zhang & Chen (022) and DDIM-style updates (Song et al.] [2020a] Ho et al.) 2020 Song jet al., (020b)). This provides stable and efficient numerical integration across noise levels while maintaining high fidelity in reconstructions.
F.4 COMPUTATIONAL COMPLEXITY
The unified training algorithm has complexity consisting of score network training at O(N |V] T (coore) per epoch, causal model training
LEN--18I-Tcane per epoch, and forward SDE simulation at O(N T D) per sample, where N is batch size, | and [|] are graph sizes, T is time steps, and D is state dimension. The overall complexity per training epoch is dominated by O(N max([V) T Tecoee 181 - Temsstl T D)), which scales linearly with the number of nodes and edges in sparse causal graphs.
F.5 ENTROPY ESTIMATION IMPLEMENTATION
The flow-based entropy estimation can be implemented
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Algorithm 3 Flow-Based Entropy Estimation
1: Input: Samples {2(*)] Jn from 9v
2: Initialize entropy:
0
3: for each reverse SDE step k do
4: Sample prohe vectors: ((n) ~ 10,
5: Compute VJP: v(m) = =. ( )
6: Estimate log-det-Jacobian: log det Jkl Rs N (c(n)))2
7: Update entropy: H+ = log det J/l
8: end for
9: Return: H + H|]()r)]
G TRAINING CONSISTENCY AND CONVERGENCE
The apparent circular dependency between Tweedie reconstruction (t) depending on scores) and causal model training (causal parameters depending on & (t)) is resolved through our iterative train- ing procedure:
Algorithm 4 Consistent Joint Training
1: Input: Training data (=(m) (0)) causal graph G, tolerance т, max iterations Kmax
2: Initialize: Score networks {8pt J. causal networks {Me, De)
3: for epoch
N Napochs do
4: for each node E V do
5: Update marginal score su, via DSM loss (Eq. 9
6:
Compute current Tweedie estimate: 2,(() +- 2, (,,)))).(=(())))))...)))) (z(1).)/p:()
7: Update causal model (Je, Σej using current (t)
8:
end for
9: Compute Aconsistency = 1 (1+))) (t) 2(x) (2)
if' Agomsssency
epoch K Kmax then
11: break
end if
13:
end for
14: Output: Trained score networks and causal models
Convergence Guarantees: This iterative procedure converges because the DSM objective ensures score estimates improve with each update, leading to accurate Tweedie reconstructions. As reconstruction quality increases, causal model parameters stabilize around their optimal values. The joint objective (Eq. (7) is jointly convex in the score and causal parameters under the regularity conditions established in Section 4, ensuring that the alternating optimization procedure converges stationary point under standard convexity assumptions (Boyd & Vandenberghe 2004 Neal &
Hinton 1998
Stability Monitoring: We track the change in Tweedie reconstructions between iterations as Acomsistncyy = [[]|| [[]]=((+++))])) (() =(*) (1)]l2] Training terminates when Agomsistnncy < for a tolerance threshold r. indicating that the circular dependency has been resolved and the system has reached
consistent state where both score estimates and causal parameters are mutually compatible.
H REVERSE-TIVE TIME SAMPLER
For completeness, we summarize the sampler used to approximate the reverse-time diffusion dynam- ics in SVGDM. While the reverse SDE admits associated deterministic probability-fee ODE (the "normalizing flow" interpretation in Section 3.3), our implementation follows the stochastic reverse-SDE predictor-correcto sampler, which is standard in score-based based diffusion models (Song et al. 2020b).
At each discretized time step /. + L At, we apply:
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1. Predictor step (reverse SDE). Euler- Maruuyaa update of the reverse-time SDE:
At 2t + fa(z, t) At + g(t) se (Zr, 1) At + g(t)vALe, e~ N(0,
2. Corrector step (Langevin update). A few steps of:
z 2 a (z( V2 7~ N~0,0,
improving sample fidelity.
3. DDIM-style discretization. We use DDIM-style linear discretization to stabilize the tra- jectory; this does not change the underlying continuous-tine dynamics.
This stochastic sampler targets the same marginal distributions the reverse SDE and is widely used for numerical stability in diffusion models.
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I RESULTS
In this appendix, we provide supplementary visualizations and quantitative results that complement the main experiments.
I.1 SYNTHETIC DATA RECONSTRUCTIONS.
Figure 3 shows qualitative comparisons of reconstructed latent variables for the synthetic experi- ment. Our model's reconstructions closely match the ground truth (panel a), while baseline methods exhibit significant deviations, validating the quantitative improvements reported in the main text.
Ground Truth Real.:
Results: Our Model
Results: BBVI
Results: SIVI
Resutts: (MCMC)
Approximete
Approcimate
Approximate z;
Approsimate
Approximetes
Real Az
Approximate
Approxmate X;
Approximate
Approximate 2;
Approximate
volue
0.0
Real 2s
Approximato z;
Approximete 2g
Approximate Fg
Approximate 23
Approximato
Figure 3: Qualitative comparison synthetic data. (a) Ground truth latent trajectories 2;. (b)-(f) Reconstructions obtained by our model and baseline methods. Our model recovers sharper and more accurate latent trajectories compared to basclines.
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1.2 REAL-WORLD EARTHQUAKE BENCHMARKS.
Table reports AUROC results for seismic hazard estimation across three major earthquake events. Our method consistently outperforms both domain-specific methods (VBCI, DisasterNet) and generic variational inference baselines. In cases with limited missing ground truth (NGA), our model still demonstrates robust and generalizable performance.
Table 5: This table presents the comparison results (AUROC) using real-world data with baseline models for three unobserved variables that are causally related. NGA means no ground truth label is available
Earthquake
Model
Fts
21.
FBD
Model
0,9331
0.9317
0,9512
VCBI (Xu el al. 2022b
0,9012
0,9034
0,9123
DisusterNet (L
2023b
0.9293
0,9284
0.9413
2020 Puerto Rico EQ.
Prior Model (7hu et al. 2013 Nowicki Jessee et al. 2018
0.8712
0.8913
BBVT Kanganalh el al 2014
0.7912
0.7731
0.7423
SIVI Yim & Zhou 2018
0.7619
0.7846
0.7662
ADVI [Kucukelbir al. 2017
0.7763
0.6846
0.7492
NUTS (MCNIC) ([Tl)mm man
2014
0.7907
0.7183
0.7549
Our Model
0,9550
NGA
0,9587
VCBI (Xu et al. 2022b
0.9123
NGA
0,9123
DisasterNer (Li ct al 2023b
0.9421
NGA
0.9410
2021 Maiti EQ.
Prior Model (Zhu et al., 2013 Nowicki Jessee et al. 2018
0.8712 NGA
BBVI (Raneanath
2014
0.7729 NGA
0.8125
SIVI Y & Zhou 2018
0.7964 NGA
0.8123
ADVI (Kucukelbir el 2017
0.8155
NGA
0.8222
NUTS (MCMC)
(fool (1o)))mmmmm
al. 2014
0.7612
NGA
0.7747
Our Model
NGA
NGA
0,9488
VCBI (Xu el al. 20226
NGA
NGA
0.9025
DisasterNer [Li et al. 2023b
NGA
NGA
0,9315
2023 Turkey-Syria EQ
Prior Model Zhu al. 2013 NOWiEK Jesseee
2018
NGA
NGA
0,9391
oot
0,9300
BBVI (Ranganath et al. 2014
NGA
NGA
0,8233
SIVI & Zhou 2018
NGA
NGA
0.7947
ADVI (Kcckkebbr cl 2017
NGA
NGA
0.7315
NUTS (MCMC) (Hofmmn al. 2014
NGA
NGA
0,8013
J COMPARISON WITH MULTI-VIEW VAES
Our goal in SVGDM is to jointly infer multiple interacting latent physical processes { from multi- resolution observations, where (i) some observations inform only a single variable and (ii) others contain mixed signals of several variables, all defined at their native spatial resolutions. These latent variables correspond to different but causally linked physical processes (e.g., shaking, landslides, liquefaction, building damage), merely different "views" of single object.
This setting is fundamentally different from standard multi-view VAFs, which are designed to learn single shared latent representation the same object under different modalities views. In those models, each view is just a different measurement of one underlying entity, and all modalities assumed describe a common latent variable. Across different objects, they typically assume independence or, at best, impose non-causal graph regularizer (e.g., graph embedding or multi- view GCCA), so even a physical interaction graph exists, it is not encoded causal mechanism and does not drive information propagation during inference.
In contrast, SVGDM models multiple distinct latent physical processes 2; associated with differ- ent but causally linked objects, each with its own observation types defined at its native resolution, rather than treating them as views causal graphical model, so information is propagated according the specified causal dependencies among variables/objects. This allows SVGDM to fuse heterogeneous observations with differing spatial/teporra resolutions and noise characteristics without forcing them onto single likelihood family or common grid, preserving each modality's native sampling and uncertainty while still en- abling joint inference. Our objective is jointly infer these coupled latent processes from het-
single object. We embed score-based diffusion within a
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erogeneous, cepoluutoo--spcciiie observations, allowing information to flow directionally through the causal structure when some variables are sparsely or incompletely observed.
One could, in principle, treat the entire complex physical system single graph-level entity, c.g., as a graph variable as in Lin et al. 2023), and treat node-level observations as different "views" of shared latent graph representation. However, such graph-level multimodal VAEs do not model or approximate multi-scale noise diffussio, which is essential our setting because each observation modality undergoes its own resolution-dependent degradation process (e.g., radar interferometric phase noise vs. optical reflectance noise). Collapsing all processes and resolutions into one shared latent space ignores scale-dependent uncertainty and causal directionality, which leads to large per- formance degradation in our experiments.
To quantify this effect, we augment our synthetic benchmark with several representative multi-view VAE baselines: JMVAE (Suzuki et al. 2016), MMVAE (Shi et al. 2019), and MoPoE-VAn (Sutter et al. 202T). These models treat all observations different views of single latent representation and are trained using their standard multimodal ELBO objectives. Table 6 reports NRMSE and MAPE for all three latent variables the synthetic multi-resolution system.
Table Synthetic multi-resolution benchmark: comparison with multi-view VAEs. Mean over 5 seeds; lower is better. Multi-view VAEs collapse multi-resolution and graph structure into single shared latent, leading to substantially worse reconstruction of latent physical processes.
Model
Variable
NRMSE
MAPE (%)
31
0.0683
5.26
SVGDM (Ours)
0.1239
9.91
23
0.1031
7.63
21
0.0942
7.88
VBCI (Xu et al.
2022b
22
0.1463
10.97
23
0.1227
9.02
21
0.0998
8.32
DisasterNet (Li al.
2023b
72
0.1530
11.45
0.1281
9.37
21
0.2241
17.82
MoPoE-Van VAE Sutter et al.
2021
22
0.2665
20.91
23
0.2392
18.47
21 0.2479
18.94
MMVAE Shi et al.
2019
0,2858
21.57
23
0.2672
20.10
21
0.2313
18.05
JMVAE Suzuki et al.
2016
22
0.2789
21.14
23
0.2451
19.68
41
1,6981
68.9
BBVI (Ranganath et
2014
22
1,0814
103.6
1.3404
113.5
21
1.7215
66.9
SIVI Yin Zhou 2018
22
1,0832
99.8
23
1.3448
115.2
21
1.7248
69.6
ADVI Kucukelbir et al. 2017)
22
1 1.0839
103.8
23
1.3462
115.3
21
1.5242
69.7
NUTS Hooffman al. 2014
22
1.0435
103.9
23
1.3262
115.1
SVGDM consistently achieves the lowest eπrr across all variables. Multi-view VAEs perform markedly worse, even compared to graph deep models (VBCI, DisasterNet). This confirms
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that collapsing multi-resolution, causally linked processes into single shared latent representation fails to capture the structured, directional information flow needed for accurate multi-resolution in- ference. In other words, while multi-view methods can handle heterogeneous data, they assume conditional independence across variables given shared latent and cannot exploit causal structure for directional information iioegaaton-precciseee what is required when some variables com- pletely or partially unobserved at certain scales.
K ABLATION STUDY ON Loss COMPONENTS
To quantify the contribution of the proposed loss componenss--peeifccllly the local diffu- sion score-matching (DSM) term, the causal-blanket score term, and the observvtion-consiitenny term we conducted an ablation study on the 3-node synthetic system (VFO setting). Each variant removes one loss component while keeping the architecture, training process, and data identical.
Table 7: Ablation study on the 3-node synthetic system (VFO setting).
Model Variant
Included Loss Terms
NRMSE
A vs. Full Model
Full model (ours)
DSM Causal + Obs
0.103 + 0.005
No observation term
DSM + Causal
0.117 + 0.006
+14%
No causal term
DSM + Obs
0.129 + 0.007
+25%
No DSM term
Causal + Obs
0,142 + 0.009
+38%
DSM only
DSM
0.154 + 0.011
+50%
These results show that all three components contribute meaningfully and complement each other: (1) Removing the causal-blanker term prevents information from propagating along the causal graph. (2) Removing the DSM term eliminates the local score-matching objective and produces the largest degradation. (3) Removing the observation-coosiiseenyy term also yields worse accuracy.
The full model achieves the best performance by combining local diffusion learning, causal infor- mation propagation, and observation guidance.
L WILDFIRE SPREAD PREDICTION RESULTS
Table 8: Comparison of model performance wildfire spread prediction. AP stands for Aver-
age Precision.
Model
Fl Score
AP
SVGDM
0.5913
0.4430
Logistic Regression
0.432
0.279
U-Net
0.341
0.341
ConvLSTM (Mono-temporal)
0.310
0.292
ConVLSTM (Multi-temporal)
0.310
0.306
UTAE
0.350
0.372
25


Reviews:
[{"review_id": "P14dM4Z1Su", "reviewer": "Reviewer_SQUc", "summary": "This paper proposed the Score-based Variational Graphical Diffusion Model (SVGD), to address inference challenges in complex systems under heterogeneous and incomplete observations. By embedding score-based diffusion processes within causal graphical models, SVGD leverages a causal score decomposition mechanism to facilitate efficient information propagation across causally interdependent variables while maintaining the fidelity of the original observational data. Theoretical analysis and empirical evaluations on synthetic and real-world datasets demonstrate its superior performance compared to state-of-the-art baseline approaches.", "questions": "See the weaknesses above.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper proposes the Score-based Variational Graphical Diffusion Model (SVGD), which addresses the gap in inference under heterogeneous and incomplete observations in existing approaches.\n2. The authors validated the reliability and effectiveness of the model through theoretical analysis.", "weaknesses": "1. The problem addressed in the paper is based on a key assumption: \"the causal DAG is known,\" and the task is \"to perform inference based on the known causal structure,\" rather than \"learning the causal structure.\" However, the title only mentions \"Causal Inference,\" without explicitly stating \"based on a known causal structure,\" which may lead readers to mistakenly believe that the research problem is about \"learning causal structures from data and performing inference.\"\n2. The model assumes that the causal DAG is completely known, but in most real-world scenarios, the causal structure often needs to be inferred first. If the causal structure is unknown, the applicability of the model may be limited, as it relies on a predefined causal DAG. \n3. Although the authors provide theoretical analysis, it relies on assumptions such as \"log-concavity of conditional distributions\" and \"additive Gaussian observational noise.\" If these assumptions are not satisfied, the proposed method in the paper would not be applicable.\n4. Some notations in the paper lack definitions. For example, in Eq. (1), what does $W_i$ mean?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PaoQlhrJyW", "reviewer": "Reviewer_vHa7", "summary": "This paper proposes SVGD, a framework that combines diffusion process and causal graphical models. Techinically, this paper (i) formulates the node-wise forward and reverse process with causal relations, (ii) introduce a causal score decomposition based on \"causal blanket\", (iii) estimate the marginal distribution with continuous‑time DSM and causal term via a locally Gaussian conditional model. Experiments on synthetic and real-world data report consistent gains over baseline models.", "questions": "In Sec. 3.3, the author claims \"reverse SDE defines a normalizing flow from noise to data\", which introduce a deterministic flow. However, in the appendix, the author uses \"reverse SDE with predictor–corrector\" and \"DDIM-style update\", which is not determinstic. Could the author explain this confliction?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The motivation is clear and well-demonstrated. The idea to model the diffusion process with modularity is simple and intuitive. The combination of graphical‑model locality with score‑based diffusion through a causal score decomposition is, to my knowledge, novel and significant.\n2. The loss functions are closely related to the theorem. The learning recipe is concrete. Besides, a cascade error bound and plus convergence under error control are provided.\n3. The exeperiments on both synthertic and real-world applications achieves a better performance.\n4. Codes are provided.", "weaknesses": "1. Some ablation studies and component analysis can better attribute gains, i.e., if some of the loss terms are ignored.\n2. Appenix C provides a \"COMPUTATIONAL SCALABILITY ANALYSIS\" with an O(Nd) complexity. Please provide measured runtimes vs. N and d , and GPU memory scaling, to substantiate these claims.\n3. The model requires a known causal graph. In the real world experiments, where does the prior come from? Moreover, what will happen if the causal graph is partially wrong? Can you conduct an experiment to analyse the sensitivity?\n4. Theoretical validity (Theorem 3) requires local log‑concavity and bounded Tweedie error. In strongly non‑Gaussian regimes (heavy tails/multimodality), how robust is it?\n5. The abbreviation is the same as Stein Variational Gradient Descent [1].\n\n[1] Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "O7UZLHjInQ", "reviewer": "Reviewer_TJQU", "summary": "The paper introduces the Score-based Variational Graphical Diffusion Model (SVGD), a novel framework for causal inference on latent systems observed through heterogeneous and multi-resolution observations. The authors address key limitations of existing methods by exploiting the causal structure and multi-resolution data. SVGD integrates score-based diffusion processes with a known causal DAG. The core technical innovation is the causal score decomposition, which leverages the Markov blanket property to enable joint inference and information propagation across causally-connected variables. The framework provides theoretical analysis, including the existence of node-wise causal SDEs and an error cascade analysis for the locally Gaussian approximation used in the causal consistency term. Empirical results on synthetic systems and real-world disaster estimation tasks (earthquakes and wildfires) demonstrate the performance.", "questions": "1. Table 2 \"SVGDM\" should be \"SVGD\"? Plus, it looks like $\\phi$ in the problem formulation part (line 135)  has different meaning with $\\phi$ in eq 2? If yes please clarify.\n\n2. Given that the assumption of a known causal structure is the primary weakness, could the authors propose a concrete future direction for jointly learning the causal DAG G alongside the diffusion model parameters? \n\n3. The observation model assumes additive Gaussian noise. How does the method perform empirically when this assumption is violated? Some empirical results would help.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Clarity: The paper is generally well-written with a clear structure, though there are some parts not clear to me (please see my questions). \n\nOriginality: The SVGD framework, built around the causal score decomposition, is a original approach to handle systems where data quality is heterogeneous (e.g. varying resolution). The problem itself is highly significant for critical applications like climate modeling and disaster assessment.\n\nTheoretical Rigor: The paper includes a substantial theoretical backbone, proving the existence of the SDE system, showing the approximate preservation of the causal blanket under diffusion, and providing an explicit error analysis.", "weaknesses": "Assumption of Known Causal Structure: The most significant limitation is the reliance on a known causal DAG. In many of the complex systems discussed (e.g., Earth systems), the structure especially involving latent variables is unknown and itself a challenging problem.\nAs a comparison, some causal representation learning methods can learn the structure among latent variables. This drastically limits the significance of the proposed method.\n\nLimited Synthetic Experiment Complexity: The synthetic experiments, while demonstrating the proof-of-concept, are conducted on a very limited setting (i.e., only three latent variables). The paper also notes that computational complexity scales with the number of causal dependencies. More complex synthetic experiments (e.g., more than 10 latent variables with different sparse and dense graphs) are needed to convincingly validate the proposed method and the computational scalability. Given that the method already assumes the structure is given, a synthetic experiment with only three latent variables looks far from sufficient.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8Uxx5ScE5j", "reviewer": "Reviewer_TiC8", "summary": "This paper proposes the use of causally structured diffusion priors to integrate multi-resolution data. They call the method SVGD, which involves a graphical factorization in latent space, augmenting the drift term of the SDE to encourage multi-resolution integration, and variational inference for inference. An error analysis and experimental results are provided on real and synthetic datasets.", "questions": "- Is the end goal of the methods proposed in this paper to infer the shared latents under integrated multi resolution observations? If so why are there no comparisons to other multi-modal methods, e.g., multi-view VAEs?\n\n- The actual model structure is unclear. Are the $z_{p_l}$ variables in Section 3 referring to $z$ at $t=1$? Presumably $t=0$ indicates pure noise. But how can $p_0(z)$ factorise as the causal graph in Theorem 1? Please clarify. \n\n- Do we need to know the graphical structure a priori? How is this determined for the latent scale?\n\n- Do we need to infer the forward and backward maps $\\phi$? Do we run into identifiability issues in doing so?\n\n- In Theorem 1, I'm not sure an SDE with drift only influenced by parents preserves causal structure at all time points. Consider an ODE (i.e., degenerate SDE) and the graph $z_1 \\to z_2 \\to z_3$ with independent initial condition $p_0(z)$. Then, as we propogate dynamics, even though $z_1$ is not a parent of $z_3$, the indirect effect gives a dependence and the factorisation will be $p_t(z) = p_t(z_1)p_t(z_2 | z_1) p_t(z_3| z_2, z_1)$ which is not the original graph structure. This can be shown with change-of-variables, i.e., the Jacobian of the implied flow has a non-zero term $J_{1,3}$. Could the authors double check this? \n\n- The authors write that (l99) PGMs fail to model variables with different resolutions or noise levels. I don't see why this has to be the case? PGMs only give a formalism for factorising the joint distribution, it doesn't specify the actual model.", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "- Use of latent space for multi-resolution observations is sound and seems natural. The way that the multiple frequencies encourage similar latent trajectories seems novel.\n- Embedding a graphical model within the diffusion model is an interesting idea.", "weaknesses": "- The motivation behind the methods proposed in this paper are unclear to me. Why are latent graphical models useful for data integration? Why is diffusion the right choice?\n\n- With respect to graphical structure, the authors repeatedly list benefits such as \"respecting graphical structure\", but there is no mention of how the graphical structure arises. \n\n- Scalability is given a few times as a benefit of score-based diffusion, but given that the diffusion model is over latent space it doesn't seem like it would be a big issue. \n\n- The use of PGMs is not inherently causal and thus I find the title and abstract (e.g., relating to causal inference) to be misleading. \n\n- The paper is very difficult to follow. At many times it is unclear what the overall goal is and what problems are being solved precisely.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)