Conditional Score-based Filter (CSF)

• CSF is a family of methods that represents filtering densities via learned score functions, bypassing finite ensemble limitations in high-dimensional, nonlinear filtering.
• It decomposes the posterior score into prior and likelihood terms, enabling robust reverse-time diffusion to accurately recover filtering posteriors from sequential observations.
• Variants of CSF include a set-transformer encoder for conditional modeling and a measurement-aware approach that integrates analytic likelihood guidance within the diffusion process.

Conditional Score-based Filter (CSF) denotes a class of score-based filtering methods that use diffusion-model score estimation and reverse-time sampling to approximate filtering posteriors in nonlinear state-space models. In the recent literature, the label has been applied to several closely related constructions: an early score-based nonlinear filter that embeds score modeling into the predict–update recursion (Bao et al., 2023), an efficient conditional diffusion filter that uses a set-transformer encoder and separates offline prior modeling from online posterior sampling (Zeng et al., 24 Sep 2025), and a measurement-aware score-based filter, also described as a Conditional (Measurement-Aware) Score-Based Filter (CSF/MASF), that builds the forward perturbation directly from the measurement equation so that the likelihood score is analytically tractable (Yoon et al., 3 Apr 2026). This suggests that CSF is best understood as a methodological family for Bayesian filtering and data assimilation rather than a single fixed algorithm.

1. Problem class and filtering objective

CSF methods are formulated for hidden-Markov or state-space models in which the latent state evolves dynamically and observations arrive sequentially. A common discrete-time specification is

$$x_t\in\mathbb{R}^d,\quad x_t \sim p(x_t\mid x_{t-1}),\qquad y_t\in\mathbb{R}^m,\quad y_t \sim p(y_t\mid x_t),$$

with Gaussian examples

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$

and filtering target

$p(x_t\mid y_{1:t}).$

The corresponding Bayes–Chapman–Kolmogorov recursion alternates prediction,

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$

and update,

$$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$

(Zeng et al., 24 Sep 2025, Bao et al., 2023).

The motivating problem is high-dimensional nonlinear filtering and data assimilation, where classical filters often struggle with accuracy or computational feasibility in high dimensions (Yoon et al., 3 Apr 2026). Score-based methods address this by representing the filtering density through a learned score function rather than only through a finite ensemble. In the formulation of the score-based nonlinear filter, “in stead of storing the information of the filtering density in finite number of Monte Carlo samples, in the score-based filter we store the information of the filtering density in the score model,” and reverse-time diffusion then allows one to “generate unlimited samples to characterize the filtering density” (Bao et al., 2023). A plausible implication is that CSF shifts the computational bottleneck from weight management and resampling toward score estimation and reverse-time integration.

2. Score decomposition and reverse-time sampling

A central identity underlying CSF is the posterior-score decomposition

$$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$

or, in time-dependent diffusion form,

$$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$

(Yoon et al., 3 Apr 2026, Zhang et al., 23 Oct 2025).

In early score-based filtering, one introduces an auxiliary pseudo-time $\tau\in[0,1]$ and a forward SDE whose marginal at $\tau=0$ is the target distribution and at $\tau=1$ is a standard Gaussian,

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$0

with score

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$1

If the exact score were known, the time-reversed dynamics become

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$2

which transports $$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$3 back to the target law (Bao et al., 2023).

The likelihood can be injected into the reverse-time dynamics in several ways. In the score-based nonlinear filter, the update is implemented by augmenting the prior score with an analytic likelihood gradient,

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$4

where $$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$5 satisfies $$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$6 and $$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$7 (Bao et al., 2023). In the efficient conditional CSF, the score network is trained conditionally,

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$8

and the reverse-diffusion Euler–Maruyama step is

$$p(x_t\mid x_{t-1})=\mathcal{N}(x_t;\,f(x_{t-1}),Q),\qquad p(y_t\mid x_t)=\mathcal{N}(y_t;\,h(x_t),R),$$9

(Zeng et al., 24 Sep 2025).

These constructions differ primarily in how the conditional information is encoded: directly in the drift through a likelihood gradient, in the score network through a conditioning code, or in the forward process itself. That distinction is central to later developments.

3. Architectural forms of CSF

The 2025 efficient CSF is built around a set-transformer prior encoder and a conditional diffusion model. The encoder consumes a Monte Carlo ensemble $p(x_t\mid y_{1:t}).$0 approximating the prior $p(x_t\mid y_{1:t}).$1, augments each particle with empirical mean and variance,

$p(x_t\mid y_{1:t}).$2

and maps the set to a conditional code $p(x_t\mid y_{1:t}).$3 through stacked self-attention blocks and pooling by multi-head attention. By construction, $p(x_t\mid y_{1:t}).$4 is “permutation-invariant and independent of $p(x_t\mid y_{1:t}).$5” (Zeng et al., 24 Sep 2025). The conditional denoising score-matching loss is

$p(x_t\mid y_{1:t}).$6

with $p(x_t\mid y_{1:t}).$7 (Zeng et al., 24 Sep 2025).

The 2026 measurement-aware score-based filter rethinks the forward process itself. Starting from the linear measurement model

$p(x_t\mid y_{1:t}).$8

it defines schedules $p(x_t\mid y_{1:t}).$9 and $$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$0 and then sets

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$1

The associated linear SDE has marginal

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$2

so that the measurement structure is folded into the forward perturbation itself (Yoon et al., 3 Apr 2026). This is the decisive difference from score-based filters whose forward processes are specified independently of the data assimilation problem.

A broader conditional score perspective also appears outside finite-dimensional filtering. In infinite-dimensional Bayesian linear inverse problems, the conditional score is defined by

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$3

and the analysis emphasizes that the conditional score, unlike the unconditional score, can blow up for small times (Baldassari et al., 2023). This suggests that conditioning fundamentally changes both the analysis and the numerical stability of score-based samplers.

4. Measurement-aware CSF and exact likelihood guidance

The measurement-aware construction yields an exact likelihood score at every diffusion time. If

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$4

then

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$5

and therefore

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$6

In the limit $$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$7, this reduces to the familiar linear-Gaussian likelihood score

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$8

(Yoon et al., 3 Apr 2026).

The prior score is learned by applying the measurement-aware forward map

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})\,p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$9

and training a neural score model $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$0 to approximate

$$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$1

through the denoising objective

$$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$2

The posterior score is then formed by Bayes’ rule: $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$3 with the learned prior term replaced by $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$4 and the likelihood term kept analytic (Yoon et al., 3 Apr 2026).

This eliminates one of the main weaknesses of earlier score-based data-assimilation filters. In the formulation of the measurement-aware paper, existing score-based filters often specify the forward process independently of the data assimilation, so “the measurement-update step depends on heuristic approximations of the likelihood score, which can accumulate errors and degrade performance over time.” By contrast, “no further approximations to the likelihood are required” in MASF; only the standard “Euler-Maruyama” freeze is used within each small reverse step (Yoon et al., 3 Apr 2026). A plausible implication is that the principal approximation burden shifts from likelihood modeling to score learning and reverse-time discretization.

5. Algorithmic workflow and reported performance

Across the literature, CSF methods alternate a forecast step with a score-based measurement update. The details differ by variant. The efficient CSF performs particle propagation, computes a conditioning code from the propagated ensemble, and then runs conditional reverse diffusion online without retraining; its reported per-step complexity is

$$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$5

where the set-transformer encoder costs $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$6 and posterior sampling costs $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$7 reverse-diffusion steps times $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$8 particles times the score-network evaluation cost $$p(x_t\mid y_{1:t})\propto p(x_t\mid y_{1:t-1})\,p(y_t\mid x_t)$$9 (Zeng et al., 24 Sep 2025). The measurement-aware MASF performs, at each measurement step, forward-process sampling, score-network training or fine-tuning on the current ensemble, and reverse-time sampling with

$$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$0

an Euler–Maruyama-style discretization of the reverse-time SDE (Yoon et al., 3 Apr 2026).

Setting	Reported comparison	Source
Double-well (1D)	CSF 0.038, SF 0.040, EnKF 0.474, PF 1.117 RMSE	(Zeng et al., 24 Sep 2025)
20D cubic sensor (correlated)	CSF 0.598, SF 0.661, EnKF 1.178, PF 1.937 RMSE	(Zeng et al., 24 Sep 2025)
Wall-clock per-step time	CSF $$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$1–$$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$2s vs. SF $$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$3–$$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$4s, a $$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$5 speedup	(Zeng et al., 24 Sep 2025)
Lorenz-63 ($$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$6, gap=100)	MASF RMSE $$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$7 vs EnKF $$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$8, SF $$\nabla_{x_t}\log p(x_t\mid y)=\nabla_{x_t}\log p(x_t)+\nabla_{x_t}\log p(y\mid x_t),$$9, SSLS $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$0	(Yoon et al., 3 Apr 2026)
Lorenz-96 ($$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$1)	MASF consistently lowest error across dimensions, strongest chaoticity, and sparsest measurements	(Yoon et al., 3 Apr 2026)
2D Kolmogorov flow (gap $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$2)	MASF retains SSIM $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$3, RMSE low, whereas SF collapses and SSLS blurs	(Yoon et al., 3 Apr 2026)

The earlier score-based nonlinear filter reports qualitatively similar behavior on classical filtering benchmarks. In the double-well model it tracked abrupt well-switches without lag, whereas APF and EnKF showed delayed updates under reduced process noise; in bearing-only tracking, CSF and APF performed comparably while EnKF diverged; and in Lorenz-96 with $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$4, APF even with $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$5 particles collapsed, while CSF with a two-layer net of approximately 400 hidden units maintained accurate tracking with log-RMSE $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$6 (Bao et al., 2023).

Taken together, these experiments support a recurrent pattern: score-based filters are most competitive when the posterior is non-Gaussian, the dimension is large, or the observation process is sparse or strongly nonlinear. Within that family, the literature distinguishes two concrete routes to better performance: eliminating repeated retraining through conditional amortization (Zeng et al., 24 Sep 2025), and eliminating heuristic likelihood guidance through a measurement-aware forward process (Yoon et al., 3 Apr 2026).

6. Limitations, ambiguities, and adjacent developments

The acronym CSF is not fully standardized. In filtering and data assimilation it refers to score-based posterior samplers driven by conditional or conditionable score models (Bao et al., 2023, Zeng et al., 24 Sep 2025, Yoon et al., 3 Apr 2026). In adjacent work it also names a source-separation method based on an $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$7-posterior and multi-level Gaussian smoothing, with objective

$$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$8

and stochastic-gradient updates derived from pre-trained denoisers (Jayashankar et al., 2023). A plausible implication is that “CSF” is now a broader label for conditional score-guided Bayesian inference procedures, not only sequential filters in the narrow state-estimation sense.

Several limitations recur across the filtering literature. The efficient CSF “requires an offline collection of prior-ensemble data via a particle filter or other sampler,” the “quality of meta-distribution $$S_{Z_t\mid Y}(z,t)=\nabla_z\log p_{Z_t\mid Y}(z\mid y)$$9 limits generalization,” posterior sampling “uses an approximate likelihood-gradient (Tweedie formula),” reverse diffusion is “still relatively slow for real-time constraints,” and there is “no rigorous convergence theory for conditional diffusion in filtering” (Zeng et al., 24 Sep 2025). The infinite-dimensional theory adds a different caution: the conditional score can blow up as $\tau\in[0,1]$0 when some observed modes have zero noise, so extending unconditional score models to the conditional setting requires “careful consideration” (Baldassari et al., 2023).

Related score-based assimilation methods further clarify what CSF variants are trying to fix. The original Ensemble Score Filter (EnSF) uses a heuristic weighted sum

$\tau\in[0,1]$1

which “matches the true $\tau\in[0,1]$2 only at $\tau\in[0,1]$3 and $\tau\in[0,1]$4, but is biased for intermediate $\tau\in[0,1]$5.” The Iterative Ensemble Score Filter (IEnSF) wraps an outer iteration around the reverse-time SDE solver to reduce this structural error, particularly when the state dynamics or the observation operator is nonlinear (Zhang et al., 23 Oct 2025). In that context, the measurement-aware CSF/MASF can be read as a more radical response: rather than iteratively correcting a heuristic likelihood term, it changes the forward process so that the likelihood score becomes exact for linear measurements (Yoon et al., 3 Apr 2026).

The current literature therefore presents CSF as a rapidly differentiating family of methods. One branch prioritizes amortization and encoder-based conditioning; another prioritizes analytic likelihood guidance through measurement-aware diffusion; a third investigates conditional score modeling in function spaces. What unifies them is the replacement of classical finite-ensemble update rules by score estimation and reverse-time stochastic transport, with the filtering posterior represented through a learned or analytically decomposed score field.