Flow-Based Density Ratio Estimation for Intractable Distributions with Applications in Genomics

Egor Antipov; Alessandro Palma; Lorenzo Consoli; Stephan Günnemann; Andrea Dittadi; Fabian J. Theis

cs.LGFeb 27, 2026

Flow-Based Density Ratio Estimation for Intractable Distributions with Applications in Genomics

Authors: Egor Antipov, Alessandro Palma, Lorenzo Consoli, Stephan Günnemann, Andrea Dittadi, Fabian J. Theis

Organizations: Institute of Computational Biology, Helmholtz Munich · School of Computation Information and Technology, Technical University of Munich · Munich Center for Machine Learning (MCML) · Munich Data Science Institute (MDSI) ·

Abstract

Estimating density ratios between pairs of intractable data distributions is a core problem in probabilistic modeling, enabling principled comparisons of sample likelihoods under different data-generating processes across conditions. While exact-likelihood models such as normalizing flows offer a promising approach to density ratio estimation, naive evaluations are computationally expensive and prone to discretization errors because they require simulating each distribution's likelihood independently. In this work, we leverage condition-aware flow matching to derive a single dynamical formulation for tracking density ratios along generative trajectories. We demonstrate competitive performance on simulated benchmarks for closed-form ratio estimation, and show that our method supports versatile tasks in single-cell genomics data analysis, where likelihood-based comparisons of cellular states across experimental conditions enable treatment effect estimation and batch correction evaluation.

Figures & tables

Figure 1: Naively approximating the likelihood ratio

q(\bm{x}_{1}\mid\bm{y})/q(\bm{x}_{1}\mid\bm{y}^{\prime})

requires estimating the likelihood of

\bm{x}_{1}

separately under two conditional generative models

p_{t}^{\theta}(\cdot\mid\bm{y})

and

p_{t}^{\theta}(\cdot\mid\bm{y}^{\prime})

, incurring two independent integral evaluations with compounding discretization errors. scRatio instead estimates the log-ratio directly as the solution of a single ODE tracked along trajectories simulated from data to noise under a vector field

b_{t}

. Here, we illustrate the case where

b_{t}:=u_{t}^{\theta}(\cdot\mid\bm{y})

, i.e., trajectories are simulated under the numerator field.

Figure 2: a. The average MSE across 5 training runs between true and estimated likelihood ratios across models and simulation parameters. SB paths stand for Schrödinger Bridge probability paths. b. Runtime in seconds for likelihood ratio estimation comparing scRatio with the naive approach across simulation parameters. Dynamical ratios are simulated with the adaptive dopri5 solver.

Model	$d$
	20	40	80	160	320
Classifier baseline	0.27 ± 0.04	9.99 ± 0.00	19.51 ± 0.18	0.83 ± 0.11	26.68 ± 3.70
TSM with SB paths	0.03 ± 0.01	0.38 ± 0.02	0.25 ± 0.10	0.89 ± 0.11	3.55 ± 0.12
CTSM with SB paths	0.06 ± 0.01	0.09 ± 0.04	0.23 ± 0.03	0.87 ± 0.10	2.15 ± 0.08
scRatio $\sigma_{\mathrm{min}}=0,\lambda=0$	0.05 ± 0.01	0.05 ± 0.02	0.05 ± 0.03	0.16 ± 0.05	21.18 ± 0.14
scRatio $\sigma_{\mathrm{min}}=0.1,\lambda=0$	0.04 ± 0.02	0.03 ± 0.02	0.02 ± 0.01	0.16 ± 0.07	21.00 ± 0.26

Table 1: MAE (

\downarrow

) between true and predicted MI across data dimensions. Results are averaged over three training runs and reported as mean ± standard error. The best and second-best results per dimension are highlighted in bold and underlined, respectively.

Model	$\rho(\textrm{AUC},a)$ ( $\uparrow$ )	AUC ( $a\geq 0.3$ ) ( $\uparrow$ )	$\rho(\textrm{NAR},a)$ ( $\uparrow$ )	NAR ( $a\geq 0.3$ ) ( $\uparrow$ )	$\rho(\textrm{CSP},a)$ ( $\uparrow$ )	CSP ( $a\geq 0.3$ ) ( $\uparrow$ )
MrVI	0.71 ± 0.09	0.89 ± 0.00	0.83 ± 0.01	7.33 ± 0.27	0.67 ± 0.06	0.97 ± 0.00
MELD	0.48 ± 0.00	0.93 ± 0.00	0.17 ± 0.00	6.45 ± 0.00	0.27 ± 0.00	0.99 ± 0.00
scRatio ( $\sigma_{\mathrm{min}}=0,\lambda=0$ )	1.00 ± 0.00	0 .95 ± 0.00	1.00 ± 0.00	9 .64 ± 0.45	0 .95 ± 0.01	0 .98 ± 0.00
scRatio ( $\sigma_{\mathrm{min}}=0.1,\lambda=0$ )	0 .99 ± 0.01	0.96 ± 0.00	1.00 ± 0.00	11.39 ± 0.30	0 .95 ± 0.03	0 .98 ± 0.00
scRatio ( $\sigma_{\min}=0,\lambda=0.01$ )	0 .99 ± 0.01	0 .95 ± 0.00	0 .98 ± 0.02	9.39 ± 0.19	0.98 ± 0.01	0 .98 ± 0.00

Table 2: Comparison of scRatio with three distinct scheduling settings and baseline methods across increasing levels of DA controlled by the parameter

a

(Eq. 14 ). We report (I) the Spearman correlation

\rho(\cdot,a)

between each metric and the ground-truth DA level

a

, and (II) the average metric value over high-DA regimes (

a\geq 0.3

). Results are averaged over three independent training runs and reported as mean

\pm

standard error. In bold and underlined are the best and second-best results per metric, respectively.

Figure 3: Ratio-based batch correction evaluation on the NeurIPS 2021 (panel a ) and C. Elegans (panel b ) datasets. Left: Distribution of the absolute estimated $\log$ -ratio before and after batch correction for each of the measured batch and cell type combinations in the two datasets, estimated by training scRatio alternatively on the original and batch-corrected representations. Right: UMAP plots calculated before and after batch correction with scVI. To compute the UMAP, we use 50-dimensional representations obtained by PCA and scVI for uncorrected and corrected data, respectively. Batch names are annotated below each plot.

Figure 4: On the x-axis, the mean absolute $\log$ -likelihood ratio $\log r^{\theta}(\bm{x}\mid(d_{1},d_{2}),d_{1})$ evaluated on cells perturbed with both $(d_{1},d_{2})$ and only $d_{1}$ . On the y-axis, the mean log-odds of a classifier trained to discriminate cells treated with $(d_{1},d_{2})$ from cells treated with $d_{1}$ only. Each point is a combination. UMAP plots show extreme cases of the combinatorial effect and lack thereof.

Figure 5: Patient-specific differential response to perturbation. ( a ) Depiction of the experimental setup. Following Oesinghaus et al. (2025) , we divide donors into two groups responding differently to distinct cytokine treatments. (b) Absolute

\log

-likelihood ratios between treated and control distributions evaluated on perturbed cells from different donor groups. A higher ratio indicates evidence of a strong response to a cytokine by a donor group, while

\log

-ratios around 0 refer to a lower shift from controls.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

$\alpha_{t}$	$\sigma_{t}$	$u_{t}(\bm{x}_{t}\mid\bm{x}_{1})$	$\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t}\mid\bm{x}_{1})$
$t$	$1-t$	$(\bm{x}_{1}-\bm{x}_{t})/(1-t)$	$(t\bm{x}_{1}-\bm{x}_{t})/(1-t)^{2}$
$t$	$1-(1-\sigma_{\mathrm{min}})t$	$(\bm{x}_{1}-(1-\sigma_{\min})\,\bm{x}_{t})/(1-(1-\sigma_{\min})\,t)$	$(t\bm{x}_{1}-\bm{x}_{t})/(1-t+t\sigma_{\mathrm{min}})^{2}$
$t$	$C=\left[(1-\lambda)t^{2}+(\lambda-2)t+1\right]^{\frac{1}{2}}$	$(((\lambda-2)t+2)\bm{x}_{1}+(\lambda+2(1-\lambda)t-2)\bm{x}_{t})/(2C^{2})$	$(t\bm{x}_{1}-\bm{x}_{t})/((1-\lambda)t^{2}+(\lambda-2)t+1)$

Appendix

Table 3: Closed-form conditional vector field

u_{t}(\bm{x}_{t}\mid\bm{x}_{1})

and score

\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t}\mid\bm{x}_{1})

for different conditional probability path schemes. The tuple

(\alpha_{t},\sigma_{t})

represents the schedule of the path between noise and data. A more detailed description of the schedules is in Sec. B.4 .

$\alpha_{t}$	$\sigma_{t}$	$\dot{\alpha}_{t}$	$\dot{\sigma}_{t}$	$\nabla_{\bm{x}_{t}}\log p^{\theta}_{t}(\bm{x}_{t})$	$u_{t}^{\theta}(\bm{x}_{t})$
$t$	$1-t$	$1$	$-1$	$(tu_{t}^{\theta}(\bm{x}_{t})-\bm{x}_{t})/(1-t)$	$(\bm{x}_{t}+(1-t)\nabla_{\bm{x}_{t}}\log p_{t}^{\theta}(\bm{x}_{t}))/t$
$t$	$1-(1-\sigma_{\mathrm{min}})t$	$1$	$\sigma_{\mathrm{min}}-1$	$(tu_{t}^{\theta}(\bm{x}_{t})-\bm{x}_{t})/(1-(1-\sigma_{\mathrm{min}})t)$	$(\bm{x}_{t}+(1-(1-\sigma_{\mathrm{min}})t)\nabla_{\bm{x}_{t}}\log p_{t}^{\theta}(\bm{x}_{t}))/t$
$t$	$C=\left[(1-\lambda)t^{2}+(\lambda-2)t+1\right]^{\frac{1}{2}}$	$1$	$(\lambda-2(\lambda-1)t-2)/2C$	$(2(tu_{t}^{\theta}(\bm{x}_{t})-\bm{x}_{t}))/((\lambda-2)t+2)$	$(2\bm{x}_{t}+((\lambda-2)t+2)\nabla_{\bm{x}_{t}}\log p_{t}^{\theta}(\bm{x}_{t}))/2t$

Appendix

Table 4: Closed-form relation between score and vector field for different noise schedules. The tuple

(\alpha_{t},\sigma_{t})

represents the schedule of the path between noise and data, while

(\dot{\alpha}_{t},\dot{\sigma}_{t})

is the respective time derivative. The relationship between score and vector field is obtained from Eq. 12 .

Hyperparameters	Values
Latent space size	32, 64, 128, 256, 512
$\lambda$	0, 0.1, 0.25, 0.5, 0.75, 1
$\sigma_{\mathrm{min}}$	0, 1e-4, 1e-3, 0.01, 0.1
Use sinusoidal time embedding	True, False
Sinusoidal time embedding size	32, 64, 128
Condition embedding size	32, 64, 128

Appendix

Table 5: The hyperparameter grid explored for the Gaussian simulation dataset. Each model with a different configuration of hyperparameters is compared based on the test-set ratio estimation performance measured as MSE.

$d$	MI
20	5
40	10
80	20
160	40
320	80

Appendix

Table 6: The real MI values for different data dimensions.

Hyperparameters	Values
$\lambda$	0, 0.1, 0.25, 0.5
$\sigma_{\mathrm{min}}$	0, 0.01, 0.1

Appendix

Table 7: The hyperparameter grid explored for the MI estimation experiment.

Hyperparameters	Grid
$\lambda$	0, 0.01, 0.1, 0.25, 0.5
$\sigma_{\mathrm{min}}$	0, 0.01, 0.1
Batch size	256, 512,1024
Top-k PCs	10, 20, 30, 40, 50

Appendix

Table 8: The hyperparameter grid explored for the DA estimation experiment on the PBMC68k dataset.

Figure 6: a. UMAP plot of the PBMC68K dataset colored by cluster labels. The clusters are assigned to each cell through the Leiden algorithm ( Traag et al., 2019 ) . b. The proportion of treated cells per cluster for different values of $a$ (see Eq. 14 for a description of $a$ ). c. UMAP plot colored by the treatment label sampled for each cluster using different values of $a$ .

Figure 7: Mean Squared Error (MSE) of different methods across varying data dimensionalities and values of $s$ (mean shifts from the origin). TSM and CTSM denote baseline models, and SB paths indicate Schrödinger Bridge probability paths.

Figure 8: a. Comparison between scRatio and a standard classifier baseline on the task of predicting likelihood ratios between Gaussians (see Sec. 5.1 ). Performance is evaluated for different numbers of features (x-axis) as the MSE between ground truth and predicted ratios. b. Overlap between 2D Gaussian distributions as a function of the distance between their means. In all plots, one of the distributions is fixed as a standard Gaussian.

Figure 9: scRatio’s MSE results for different levels of sample complexity on the Gaussian log-likelihood ratio estimation task. From Sec. 5.1 we use the harder setting with

s=2

and

d\in\{2,5,10,20,30\}

( Yu et al., 2025 ) . For the best hyperparameter configuration, we retrain the model with different numbers of training points and report the performance on the test set.

Figure 10: scRatio’s MSE results for neural parameterization and closed-form reparameterization of the scores on the Gaussian log-likelihood ratio estimation task. From Sec. 5.1 we use the harder setting with

s=2

and

d\in\{2,5,10,20,30\}

( Yu et al., 2025 ) . For the best hyperparameter configuration in Sec. 5.1 , we compare the performance of the model across dimensions between closed-form reparameterization and neural parameterization of the score and vector field in Prop. 4.1 .

Figure 11: Mean Squared Error (MSE) as a function of the absolute value of the log-ratio, shown for different locations and dimensionalities.

Figure 12: Test MSE as a function of dimensions and path variance for CTSM and scRatio across hyperparameter configurations. Boxplots represent the distribution of MSE (linear scale for CTSM, log-scale for scRatio) as a function of input dimensions. Boxes for both models are colored by the level of path stochasticity in the interpolants (Schrödinger Bridge and flow matching path variance, respectively).

Figure 13: Heatmap of Mean Squared Error per number of training steps for velocity $u_{t}^{\theta}$ and score $s_{t}^{\psi}$ fields. On the $x$ -axis the number of steps used to train the velocity field $u_{t}^{\theta}$ is presented, the $y$ -axes shows the number of steps for the score field $s_{t}^{\psi}$ , while the hue value encodes the MSE attained by the combination of the two.

Figure 15: Right: Four UMAP plots showing the gradual increase of combinatorial effects across several single perturbations and combinations thereof. Left: Corresponding box plots for each UMAP, displaying the absolute values of the estimated log-likelihood ratios.

$d$	$\lambda$	$\sigma_{\mathrm{min}}$	MSE ( $s=1.0$ )	MSE ( $s=2.0$ )
2	0.0	0.0	0.01	0.05
		0.0001	0.01	0.05
		0.001	0.01	0.13
		0.01	0.01	0.05
		0.1	0.02	0.17
	0.10	0.0	0.02	0.07

Appendix

Table 9: Sweep results on Gaussian simulation experiments for different noise schedules.

\lambda

controls the stochasticity along conditional probability paths from noise to data, while

\sigma_{\mathrm{min}}

is the variance of Gaussian noise around data points. The

\alpha_{t}

parameter is kept fixed to

\alpha_{t}=t

, with

t\in[0,1]

. MSE results for different shifts from the origin (

s

) are averaged across three training replicates. Best results are highlighted per dimensionality.

Figure 16: Performance as a function of number of features for different model hyperparameters (number of condition embedding dimensions, learning rate, time feature size, number of latent dimensions). Evaluations are conducted on Gaussian data from Sec. 5.1 with

s=2

. The performance quality is measured through the MSE between ground-truth and estimated likelihood ratios.

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Open paper AskPDF arXiv

Flow matching has emerged as a powerful generative family for simulation-based inference (SBI). However, sampling and density evaluation require numerical integration at inference time. This makes posterior sampling in neural posterior estimation costly. More importantly, it can make repeated likelihood evaluation prohibitively expensive and impractical for Markov chain Monte Carlo (MCMC) sampling, which may require thousands of repeated density evaluations. Here, we develop conditional mean flows for posterior and likelihood approximation, combining few-step sampling with tractable density evaluation. We show that the mean flow objective identifies conditional flow maps and that independent masking enables direct estimation of marginal likelihoods. We evaluate both posterior and likelihood estimation across several tasks and show that conditional mean flows retain the flexibility of flow-based density estimators while significantly reducing inference time. In particular, single-step conditional mean flow density evaluation yields estimation error comparable to 200-step flow matching while enabling about 200-fold faster downstream MCMC sampling.

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