Title: PRESCRIBE: Predicting Single-Cell Responses with Bayesian Estimation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Single-cell biology

bioinformatics

gene expression

transcriptomics

uncertainty estimation
Abstract: In single-cell perturbation prediction, a central task is to forecast the effects of perturbing a gene unseen in the training data. The efficacy of such predictions depends on two factors: (1) the similarity of the target gene to those covered in the training data, which informs model (epistemic) uncertainty, and (2) the quality of the corresponding training data, which reflects data (aleatoric) uncertainty. Both factors are critical for determining the reliability of a prediction, particularly as gene perturbation is an inherently stochastic biochemical process. In this paper, we propose PRESCRIBE (PREdicting Single-Cell Response wIth Bayesian Estimation), a multivariate deep evidential regression framework designed to measure both sources of uncertainty jointly. Our analysis demonstrates that PRESCRIBE effectively estimates a confidence score for each prediction, which strongly correlates with its empirical accuracy. This capability enables the filtering of untrustworthy results, and in our experiments, it achieves steady accuracy improvements of over 3% compared to comparable baselines.

PRESCRIBE: Predicting Single-Cell Responses with Bayesian Estimation

Abstract

In single-cell perturbation prediction, a central task is to forecast the effects of perturbing a gene unseen in the training data. The efficacy of such predictions depends on two factors: (1) the similarity of the target gene to those covered in the training data, which informs model (epistemic) uncertainty, and (2) the quality of the corresponding training data, which reflects data (aleatoric) uncertainty. Both factors are critical for determining the reliability of a prediction, particularly as gene perturbation is an inherently stochastic biochemical process. In this paper, we propose PRESCRIBE ( PRE dicting S ingleC ell R esponse w I th B ayesian E stimation), a multivariate deep evidential regression framework designed to measure both sources of uncertainty jointly. Our analysis demonstrates that PRESCRIBE effectively estimates a confidence score for each prediction, which strongly correlates with its empirical accuracy. This capability enables the filtering of untrustworthy results, and in our experiments, it achieves steady accuracy improvements of over 3% compared to comparable baselines. Code is available at https://github.com/Bunnybeibei/PRESCRIBE .

1 Introduction

Predicting the effects of perturbations is crucial for advancing biological understanding and the development of targeted genetic therapies. Recent years have seen significant progress in machine learning models [1-3], data generation [4, 5], and benchmarking [6, 7] for this task. However, a critical challenge remains less explored: quantifying prediction uncertainty for individual predictions, especially for perturbations of genes that are not seen during training and are functionally distant to any of the genes in the training set. Fig. 1(a) illustrates this issue, showing that even models with high average accuracy can make substantial errors on specific predictions.

The first step towards estimating prediction uncertainty is to understand its sources. Prediction uncertainty arises from the interaction between two primary sources. First, data (aleatoric) uncertainty arises from the inherent stochasticity of biological systems, where perturbing a single gene can yield a diverse spectrum of cellular outcomes. Second, model (epistemic) uncertainty reflects the model's unfamiliarity with a given input, which is particularly high for out-of-distribution perturbations. A practical framework needs to account for both. For example, a prediction is the least reliable when the model's output is far from a highly consistent biological outcome (high model uncertainty, low data uncertainty). Equally, a prediction is intrinsically uncertain when the biological outcome itself is highly variable (high data uncertainty).

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Here, we propose a unified, data-driven metric to capture both data and model uncertainty, inspired by the energy distance (E-distance) [8]. E-distance quantifies the similarity between two cell populations by balancing the distance between them against the dispersion within them. In the context of perturbation prediction, E-distance reconciles both the model uncertainty and the data uncertainty (Fig. 1(b)). If the ground truth of a perturbation is known a priori, then the E-distance between the predicted post-perturbation population and the ground truth, composed of an inter-group distance compensated by a negative intra-group distance term, reflects the accuracy of the prediction (Fig. 1(c)). Apparently, the ground truth post-perturbation cell fate distribution of an unseen gene cannot be obtained. Therefore, E-distance, in its original form, is not directly applicable to the perturbation prediction uncertainty estimation task. In this work, we adapt the idea of E-distance by jointly modeling the prediction error and the intrinsic post-perturbation dispersion. We name our perturbation uncertainty estimation metric pseudo E-distance.

To estimate the pseudo E-distance for unseen perturbations, we introduce PRESCRIBE ( PRE dicting S ingleC ell R esponse w I th B ayesian E stimation). PRESCRIBE is a multivariate extension of a deep evidential learning framework (Natural Posterior Network [9]) that, instead of outputting a single expression profile, simultaneously predicts the post-perturbation state and estimates the prediction's confidence. PRESCRIBE has two key elements: a posterior distribution over the transcriptomic landscape and an evidence score derived from a learned latent density of the perturbation space. The pseudo E-distance (Fig. 2) is defined by combining these terms: the spread of the predicted distribution (measured by its entropy) quantifies data uncertainty, while the evidence score quantifies model uncertainty. Intuitively, the evidence score measures the distribution density of the training data in the latent perturbation space within a close vicinity of an unseen target gene or gene combination. A high evidence score indicates that a prediction is grounded by multiple functionally related gene perturbation instances in the training data, and a low evidence score indicates otherwise. We model the post-perturbation transcriptomic spread using a Normal-Wishart conjugate Bayesian framework, allowing the variance to be inferred from the perturbation embedding via a trained decoder. For perturbations far from the training data, the predicted distribution defaults to a null state (typically the unperturbed control cell population). This fallback ensures a safe output for unreliable predictions.

Our main contributions can be summarized as follows:

We propose PRESCRIBE, a novel framework that uses a predicted pseudo E-distance as a unified surrogate for both data and model uncertainty in single-cell perturbation prediction.

Weintroduce a multivariate extension of the Natural Posterior Network, utilizing an Inverse-Wishart prior to effectively model predictive distributions over multi-dimensional gene expression states.

We demonstrate through comprehensive experiments that PRESCRIBE generates well-calibrated uncertainty scores that improve predictive accuracy by enabling the filtering of unreliable results.

2 Related Work

2.1 Predicting Single-Cell Responses

In-silico prediction of single-cell responses offers an efficient alternative to costly single-cell perturbations in the wet lab. Current methods generally fall into two main categories: direct matching or disentanglement [7]. Direct matching methods, such as GEARS [1] and scGPT [10], map control cell gene expressions to perturbed expressions to predict responses to new perturbations. Disentanglement methods, like CPA [11], isolate perturbation effects from cellular features (e.g., cell type, dosage) to enable predictions under diverse conditions. Our work aligns with the direct matching strategy, focusing on the challenge of predicting responses to novel perturbations.

A key assumption in matching methods is the ignorability condition. It posits that, conditional on an adequate set of observed covariates, no unmeasured confounding factors would bias the comparison between control and treated cell populations. Within this framework, given a data set D = { X , y , c } , representing M types of perturbation and G genes, where X = { x 1 , x 2 , ..., x M } denotes the set of perturbations, y ∈ R G is the post-perturbation gene expression profile, and c is the pre-perturbation (control) gene expression profile, the predicted post-perturbation expression ˆ y x i can be modeled as:

where f ( x i ) is the learned effect under the perturbation x i .

Examples of direct matching models include CellOracle [12], which uses scRNA-seq and scATAC-seq data to infer gene networks for simulating linear perturbation effects. GEARS pioneered predictions for unseen perturbations using gene embeddings and Gene Ontology (GO) based perturbation embeddings. GraphVCI [13] employs counterfactual concepts from causal inference to enhance gene regulatory network learning. sams-VAE [14], uses a sparse additive mechanism shift variational autoencoder to disentangle specific perturbation effects. More recently, single-cell foundation models like scGPT and scFoundation [15] also perform single-cell response prediction as a downstream task.

Despite these advances, robust uncertainty is rarely reported. Some methods, such as GEARS (which uses Monte Carlo dropout [16]), do not fully account for the uncertainty from pair-wise distance to controls. Moreover, the variance of GEARS estimates is not well-calibrated, as it is inconsistent with generalization difficulty and actual prediction accuracy. These limitations underscore the need for more practical and specifically tailored uncertainty quantification frameworks for this task.

2.2 Natural Posterior Network for Uncertainty Quantification

The Natural Posterior Network (NatPN) [9] is an evidential deep learning method (Appx. § E.3) for quantifying uncertainty. Within a single forward process, it estimates epistemic uncertainty (model's confidence) through the distance between the learned posterior and the prior in the latent space, while aleatoric uncertainty (data randomness) is measured via predictive entropy.

NatPN updates beliefs using the Bayesian posterior theory (Appx. § E.2) for exponential family distributions. Briefly, for a likelihood P ( y | ω ) and its conjugate prior Q ( ω | χ prior , n prior ) , observing Mdata points { y j } M j =1 leads to a posterior Q ( ω | χ post , n post ) with parameters are updated as:

NatPN measures confidence with n post -H [ P ( y | ω )] , where n post (total evidence) captures epistemic uncertainty and entropy represents aleatoric uncertainty. The design of its density estimation architecture has been proven to drive the estimated evidence towards zero under out-of-distribution (OOD) conditions (e.g., far from the training set) (See Appx. § F.6 for details).

This confidence score shares conceptual similarities with E-distance calculations (Fig. 2(IV)). However, standard NatPN cannot be directly applied to single-cell perturbation prediction. It lacks mechanisms to incorporate pair-wise distances and does not offer a readily available multivariate extension suitable for complex gene expression data. Our work tries to resolve these limitations.

Ⅱ

. Perturbation Density Space

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3 Methods

Our core contribution is the pseudo E-distance, a metric from the model's outputs that unifies epistemic and aleatoric uncertainty. In this section, we first describe the model's probabilistic foundation, then define the pseudo E-distance and its calculation. Finally, we outline the network architecture and the optimization objective employed during training. For reference, detailed notation and hyperparameters are provided in Appx. Tabs. 5 and 6, and the core algorithm in Appx. Algo. 1.

3.1 Probabilistic Modeling of Gene Expression

To model both the distribution of gene expression and predictive uncertainty, we adopt a Bayesian approach. For a given perturbation category x i , we assume the gene expression vector y i follows a multivariate Gaussian distribution. Following related work, we place a Normal-Wishart conjugate prior on the parameters of this Gaussian, which enables analytical Bayesian updates. Thus, a posterior distribution is defined by four parameters ω i = { µ 0 x i ∈ R N , κ i ∈ R + , ν i ≥ N, L i } :

where Σ i = Λ -1 i is the covariance matrix and Λ i is its corresponding precision matrix. The scale matrix Ψ i is defined using a lower triangular matrix L i such that Ψ -1 i = ν i L i L T i . Here, N , W , and W -1 denote the Normal, Wishart, and Inverse Wishart distributions, respectively. N is the rank of the gene-gene interaction matrix, estimated via Principal Component Analysis (PCA).

The decoder outputs the effect of a perturbation x i as a set of sufficient statistics and evidence counters:

The network's outputs (denoted by the 'out' superscript). Since the evidence-related parameters are proportionally linked ( n i = κ i = 2 ν i ), we use ν to refer to them collectively for simplicity.

3.2 Pseudo E-distance as a Unified Uncertainty Surrogate

Definition. To capture both sources of uncertainty, we define pseudo E-distance (Fig. 2, Panel V) as:

Here, ˜ · denotes normalization. This metric comprises two key terms. The first, ˜ ν post i , represents the posterior evidence , which quantifies the model's epistemic uncertainty. High evidence indicates the prediction is well-supported by training data, while low evidence suggests an out-of-distribution input. The second term, -˜ H [ · ] , is the negative normalized entropy of the predictive distribution, which reflects aleatoric uncertainty or inherent output variability. As shown in Appendix F.5, ˜ E provably preserves the rank-ordering of the true E-distance under fixed prior conditions.

Calculation. We initialize a base prior using the control cell profile c and fixed hyperparameters ( κ prior i = 1 , ν prior i = 0 . 5 ). The posterior parameters are then obtained by combining this base prior with the model's outputs through the Bayesian update (Eq. 2). From this posterior, the evidence ν post i and the predictive entropy H [ · ] are normalized to the range [ N, 2 N ] . This operation serves two purposes: it places both components on a comparable scale and it ensures the degrees of freedom of the resulting Student's t -distribution remain in a regime that preserves its heavy-tailed properties, which is crucial for capturing sparse regions (low evidence) of the perturbation space.

Remark 1. Our model can distinguish low-confidence, out-of-distribution predictions from highconfidence predictions of perturbation with little to no effect. Although both may predict an outcome similar to the control state, they are separated by their significantly different evidence scores.

3.3 Model Architecture

As illustrated in Fig. 2, our model comprises three core modules: an encoder ( f α ; Panel I) that generates latent representations, a normalizing flow ( f ψ ; Panel II) that estimates evidence, and a decoder ( f β ; Panels III-IV) that produces the sufficient statistics for the predictive distribution.

Encoder f α . The encoder processes a perturbation x and the cell's basal state c into a latent embedding z ∈ R D that captures functional similarities. Assuming additive effects [17], the encoder comprises two components: f α 1 for individual perturbation effects and f α 2 for non-linear interactions. For a single perturbation x i , the embedding is calculated as follows:

For a set of perturbations x , the model aggregates their individual effects via summation. This operation ensures the resulting embedding is invariant to the order of the perturbations:

Here, f α 11 is a linear layer that uses pre-trained gene embeddings, while f α 12 and f α 2 are multilayer perceptrons (MLPs) with LeakyReLU activation. While the encoder design is agnostic to the choice of gene embeddings, we use those from scGPT [10] by default.

Normalizing Flow f ψ . The normalizing flow [18] module estimates the density of the training data in the latent space, which directly informs epistemic uncertainty. It takes the latent embedding z as input and outputs the evidence ν . High-density regions (familiar inputs) yield high evidence, while low-density regions (novel inputs) yield low evidence, forcing the prediction to revert toward the pre-defined null state. Specifically, the evidence ν i is calculated from the latent embedding z , and the posterior evidence ˜ ν post i is then updated and normalized as follows:

Here, N H is the total certainty budget, which we set to N . The update operation for ˜ ν post i scales the posterior evidence to the range [ N, 2 N ] and balances the influence of the base prior (when evidence ν i is low) against the data-driven prediction (when evidence ν i is high).

Decoder f β . The decoder is one linear layer that maps the latent embedding z i to the sufficient statistics χ out i . The outputs from the decoder χ out i and the normalizing flow ν out i are then combined with the prior through Bayesian update to form the final posterior distribution and evidence.

3.4 Optimization Objective

PRESCRIBE is trained by minimizing a composite loss function L designed to encourage both accurate predictions and meaningful perturbation density estimates:

Specifically, L consists of a primary objective for prediction accuracy ( L 1 ) and three auxiliary terms, weighted by hyperparameters λ 1 , λ 2 , λ 3 (see § 4.9 for grid search details):

Expected Log-Likelihood ( L 1 ). This term maximizes the likelihood of the observed data under the posterior predictive distribution, driving accurate predictions. The details can be found in Appx. F.1.

Entropy Regularization ( L 2 ). This term, weighted by prediction error, acts as a prior that favors uninformative distributions with high entropy. The details can be found in Appx. F.2.

E-distance Ranking Loss ( L 3 ). To supervise the model's uncertainty estimates, we introduce a ranking loss based on ListMLE [19]. This loss enforces consistency between the predicted and reference rankings within each training batch. Specifically, the predicted pseudo E-distances, ˜ E , are reordered according to the descending order of their corresponding reference E-distances, E :

The term ˜ E ( sorted ,x i ) refers to the i -th element of this sorted list. This objective encourages the model's predicted ranking of uncertainties to match the reference ranking.

Uncertainty Regularization Loss ( L 4 ). This term addresses the problem of vanishing gradients in low-evidence regions. The details can be found in Appx. F.3.

4 Results

This section details PRESCRIBE's empirical evaluation. The experiments were designed to: (i) assess the quality of its uncertainty ( § 4.3- 4.6)). (ii) demonstrate its utility in prediction accuracy ( § 4.7), and (iii) explore contributions of its core components and initialization strategies ( § 4.8- 4.9).

4.1 Experimental Setup

Datasets. We evaluated PRESCRIBE on three widely recognized benchmark datasets: Norman [4], Replogle2022_Rep1, and Replogle2022_K562 [5]. (Details are provided in Appx. § G.)

Comparison Baselines. To evaluate PRESCRIBE, we conducted a comprehensive benchmark against several existing methods. The baseline methods were divided into two main groups: recent methods without uncertainty estimation ( AverageKnown , Linear , Linear scGPT [20], CellOracle [12], samsVAE [14], GraphVCI [13], scFoundation [15], and scGPT [10]) and those with variance-based uncertainty estimation ( GEARS [1] and GEARS-Drop ). The evaluation also encompassed several adaptations and ablations of our approach and related models: GEARS-ens , an ensemble of five GEARS-Drop models; Ours-MLPs , a variant that employed the our model's encoder/decoder architecture but specifically used Multi-Layer Perceptrons (MLPs) to regress Edistance from the latent embedding; Ours-Null , representing our model's untrained state; Ours-Ens ,

an ensemble of five models using the our model's encoder and an MC dropout [16] decoder; and Ours-NOINFO , a variant that removed prior information by using zero vectors for the prior mean and covariance. The complete settings of these baselines can be found in Appx. §H.

4.2 Evaluation Metrics

Our evaluation employed two categories of metrics, each serving a different role.

Prediction Accuracy. These metrics are assessed by measuring the Pearson Correlation Coefficient ( r pred,truth) and directional accuracy (ACCpred,truth) between predicted and true gene expression. We also calculated these metrics on the top 20 differentially expressed genes (DEGs), denoted as r DEG pred,truth and ACC DEG pred,truth . All accuracy evaluations were performed on log-fold change values.

Calibration Quality. These metrics assess if a model's confidence scores are meaningful indicators of its predictive performance. We measure this using several metrics: (1) the Pearson ( r perf,conf) and Spearman's ( r s perf,conf ) correlation between confidence scores and actual performance (as measured by r pred,truth ); (2) a percentile-based classification accuracy (ACCperf,conf) to determine if low-confidence predictions are less accurate; and (3) the Expected Calibration Error (ECE) [21], denoted as ϵ perf,conf .

4.3 Estimation of E-distance

Our model is designed to efficiently estimate E-distance without requiring post-perturbation profiles. We therefore validated the practical effectiveness of our estimated 'pseudo E-distance' ( ˆ E ) in two dimensions. First, across all datasets, ˆ E exhibited a consistent positive correlation with a reference E-distance ( E ), which was computed using post-perturbation profiles (Tab. 1(a)). Second, this positive correlation strengthened significantly as the number of samples ( N ) used for computing the reference E-distance ( E ) increased (Tab. 1(b)). Given that E calculated with a larger N more accurately reflects the true underlying E-distance, this observed trend suggests that ˆ E can serve as an effective, and potentially asymptotic, approximation of the true E-distance's ranking.

4.4 Confidence Scaling with Generalization Difficulty

We utilized the Norman combination perturbation dataset to assess our model's performance under varying levels of generalization difficulty. We simulated increasing difficulty by varying the number of unseen perturbations in combinations of two (0, 1, or 2 unseen). To remove the confounding variables from E-distance, we controlled for the average E-distance across these scenarios, approximately 55 . As illustrated in Fig. 3(a), our model significantly reduced its confidence scores as the generalization difficulty increased. In contrast, others show a minorly decrease or inverse trend. This phenomenon demonstrates our model's ability to be more aware of epistemic uncertainty than other baselines.

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FOXA1+HOXB9

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4.5 Qualitative Visualization

We visualized the Normal Inverse Wishart latent space learned by our model using t-SNE [22] dimensionality reduction on the Norman dataset (Fig. 3(b)). This visualization represents control (prior), training, and test perturbations by red, gray, and blue, respectively. The intensity of the blue color corresponds to the r pred,truth prediction accuracy, with darker shades indicating higher quality predictions. The visualization aligns with expectations: training data points and large E values are generally located further from the control prior; the model assigns them higher confidence, reflecting their dominance over the prior, and indeed, most of them exhibit high prediction accuracy r pred,truth .

4.6 Accuracy-Based Calibration Analysis

We examined accuracy-based calibration curves for a more granular understanding of our model's practical utility. Furthermore, we computed the Spearman correlation ( r s perf,conf ) between prediction accuracy and the confidence scores produced by our model and baseline methods (for GEARS-Drop and GEARS-Ens, inverse values were used as their confidence scores represent variance). From Fig. 4, our model achieved the highest r s perf,conf across all datasets, and the visualizations of the calibration curves demonstrated that our model exhibits a consistent trend where confidence increased monotonically with accuracy, indicating its confidence score is well-calibrated overall. The expected calibration error ( ϵ perf,conf) results are presented in the supplementary materials (Appx. I).

4.7 Prediction Accuracy with Uncertainty-Guided Filtering

A practical use of reliable uncertainty estimation is identifying and potentially excluding lowconfidence predictions, thereby improving overall performance. We investigated this by comparing our model's accuracy after filtering out the 5% and 10% least confident predictions (Ours 5% and Ours 10%) against baseline models (Tab. 2). Ours-10% significantly outperformed all baselines, while Ours-5% also showed strong results. It is important to note that modeling distributions (as our model

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does) is inherently more complex than direct value prediction. Thus, models like GEARS-Drop and Ours, which learn an additional function for confidence evaluation, did not exhibit standout accuracy initially. However, only our model demonstrated stable, significant, consistent improvements across metrics after different degrees of filtering unreliable predictions.

4.8 Ablation Studies on Filtering and Key Model Components

We conducted several ablation studies further to understand the sources of our model's performance.

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pred,truth

43.25

55.61

22.41

ACCpred,truth

57.80

63.61

54.73

s

perf,conf

36.65

27.74

29.83

Impact of Filtering. We performed random filtering ( 10 repetitions) to assess whether the observed performance gains were merely an artifact of data removal. As shown in Tab. 3, randomly filtering 5% of predictions significantly decreased accuracy. Filtering 10% randomly brought accuracy back to approximately the initial unfiltered level, but did not yield significant improvements. This confirms that the performance gains of our model are not attributable to the filtering operation.

r

↑

r

↑

(a)

(b)

80

70

70

60

Metric Value (%)

60

Metric Value (%)

50

40

40

30

30

20

rpred,truth

20

pred,truth

ACCpred,truth

10

perf, conf

ACCpred,truth

10

perf,conf

0

0.0001

0.001

10000

100000

入2

Impact of Key Components. As demonstrated in Tab. 4, evaluating specific architectural and loss components revealed their distinct contributions. The Ours-NOINFO variant, despite comparable prediction performance to our model, showed significantly lower r s perf,conf values, underscoring the importance of informative prior settings for well-calibrated uncertainty. The loss term L 3 , which supervises learning from E-distance, proved critical; its removal led to a slight ACCpred,truth increase, potentially due to the adversarial nature of optimizing accuracy versus evidential uncertainty, but at the cost of substantially reduced uncertainty calibration. Finally, ablating the loss term L 4 resulted in a suboptimal model due to zero gradients, as mentioned in § 3.4.

4.9 Hyperparameter Search Strategy

We developed an efficient hyperparameter tuning protocol for applying our model to new datasets.

Selection Pipeline. The selection protocol can be summarized in the following steps:

Phase 1 λ 2 Optimization: Conduct a grid or random search for λ 2 (the weight for L 3 ).

Phase 2 λ 1 /λ 3 Ratio Tuning: Fix λ 3 (the weight for L 4 ) to 10 -5 and perform a focused search for λ 1 (the weight for L 2 ). This effectively tunes the ratio λ 1 /λ 3 .

Rationale and Example. Our efficient two-phase tuning pipeline reduces trials from 216 ( 6 × 6 × 6 ) to 42 ( 6 × 6 + 6 ) by fixing λ 3 in the second phase. This is justified by the adversarial relationship between L 2 (weighted by λ 1 , promoting prior adherence) and L 4 (weighted by λ 3 , encouraging evidence accumulation). Thus, we fix λ 3 = 10 -5 and tune λ 1 . For example, on the Norman dataset, an initial random search determined λ 2 = 0 . 1 (Fig. 5(a)). Tuning the λ 1 /λ 3 ratio (Fig. 5(b)) then revealed an inverse relationship between confidence calibration ( r s perf,conf ) and prediction performance ( r pred,truth, ACCpred,truth), leading us to select λ 1 /λ 3 = 0 . 01 (so λ 1 = 10 -7 for λ 3 = 10 -5 ).

5 Conclusion and Discussion

Conclusion. We introduce PRESCRIBE, a novel uncertainty-aware framework for single-cell response prediction. By employing a multivariate Natural Posterior Network to estimate the pseudo Edistance for each perturbation category, our model delivers well-calibrated, instance-level uncertainty. This metric acts as a unified surrogate for both aleatoric (data) and epistemic (model) uncertainty, and its application demonstrates practical utility by improving overall predictive accuracy.

Future Work. Recent research [23] highlights the critical impact of the reference state (often the unperturbed cell state) in this task. Therefore, future work will leverage PRESCRIBE's configurable 'null' state to explore how the choice of this reference influences predicted perturbation responses. The goal is to guide the optimal selection of a reference state, enabling the model to learn the specific effects of perturbations more easily along the pronounced direction of change.

6 Acknowledgment

This work is supported by STI2030-Major Projects 2022ZD0212400, the National Natural Science Foundation of China Project (No. 623B2086), TeleAI, STCSM grant 24510714300 and 20DZ2254400, GuangDong Basic and Applied Basic Research Foundation grant 2023B1515120006, and SJTU Science-Medicine interdisciplinary grant 24X010301456.

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Appendix Overall

A Notation Table

B Default Settings

The PRESCRIBE framework was trained on a single NVIDIA RTX 4090 GPU. We used the Adam [24] optimizer with an initial learning rate of 10 -4 and set the weight decay value as 10 -5 . The training was carried out for a maximum of 50 epochs using batch sizes of 4096 samples, with early stopping triggered if L 1 of the validation set did not show improvement for three consecutive epochs. A dynamic learning rate scheduler 'plateau' reduced the learning rate by 1% when the relative L 1 validation improvement remained below 0.01% for two consecutive epochs. The implementation leverages PYTORCH LIGHTNING [25] 2.4.0's modular architecture.

C Algorithm

The optimization process of PRESCRIBE can be summarized in the following algorithm 1

Algorithm 1 Training Process of PRESCRIBE 1: Input: 2: Perturbation types: X ; 3: Condition information: c , χ prior ; 4: Prior evidence: ν prior ; 5: Post-perturbed transcriptomics expressions: y i ; 6: E-distance of training samples: E i ; 7: Initialize: 8: θ = { θ α , θ ψ , θ β } ← initialize network parameters; 9: repeat 10: x i ← random mini-batch from X ; 11: z i ← f α ( x i , c ) ; // Encoder; 12: n i ← f ϕ ( z i ) ; // Flow; 13: χ i = { χ 1 , χ 2 } ← f β ( z i ) ; // Decoder; 14: // Bayesian Posterior Update; 15: χ i ← compute through Eq. 4; 16: ν post i , ν i ← compute through Eq. 8; 17: κ post i ← 2 · ν post i ; 18: χ post i ← ( ν prior χ prior + ν i χ i ) ν i + ν prior ; 19: µ post 0 x i ← χ post 1 ; 20: L post ← Cholesky(( χ post 2 -( χ post 1 ) 2 ) × ( ν post i ) 2 ) × ν post i ; 21: L ← compute through Eq. 9; 22: // Update parameters according to gradients; 23: θ ← θ -∇ θ L ; 24: until deadline reached

D Computational Complexity Analysis

The computational complexity of our model is characterized by three primary components: the Encoder, the Flow, and the Decoder. The complexities for each component are detailed below:

Encoder : The overall computational complexity for the Encoder is O ( d · ( S + G + d + D )) . This is derived from the complexities of its constituent functions:

f 11 : O ( S · d + d )

f 12 : O ( G · d + d )

f 13 : O ( d · d + d )

f 2 : O ( d · d + d )

f o : O ( d · D + D )

Flow : The complexity associated with the Flow component is O ( D ) .

Decoder : For the Decoder, which takes an input of dimension D and produces an output of dimension ( N · ( N +1) / 2 + N ) , the computational complexity is O ( D · N 2 ) .

E Preliminaries

E.1 E-Distance

E-distance [26], a statistical metric for assessing high-dimensional distributions, offers measures of the signal-to-noise ratio within a dataset. It is popularly referred to as energy distance, originally derived from the concept of gravitational energy in physics. In essence, E-distance determines

whether two distributions are distinct by comparing the distance between them - pairwise distance- to the variability within each distribution-self-distance'. The illustration can be referred to in Fig.2(IV). Applying E-distance in single-cell perturbation prediction task was introduced by scPerturb [8], which utilized E-distance to differentiate between strong and weak perturbations. For two given single-cell statuses, represented as X and Y , the E-distance calculation is defined as follows:

where σ X and δ XY are the self-distance within and pairwise distance between distributions:

Typically, scPerturb sets the control status as X , aligning with most deep learning models' assumptions. Our observation (Fig. 1(b)) suggests an apparent positive correlation between E-distance and predictive quality. However, since E-distance needs to access data, it poses challenges for predicting unseen perturbations, which is out-of-sample data.

E.2 Bayesian Posterior Update

In the context of distributions from the exponential family, we can express the conjugate prior and the posterior distribution after observing M target observations as follows:

The parameters ω , χ , and n correspond to the target distribution, prior distribution, and evidence, respectively. The posterior parameters can be updated as follows:

The method nature posterior network [9] has leveraged this framework to predict Bayesian uncertainty. Moreover, it employs normalizing flows [18] to develop a structural latent space, inferring that regions distant from the training data should exhibit higher uncertainty. However, this approach has yet to be expanded to the multivariate Gaussian distribution, which needs to be theoretically complemented in corresponding derivation to predict single-cell perturbation response better.

E.3 Deep Evidential Regression.

Evidential learning provides a promising solution to the challenge of overconfidence in deep learning models. This approach is more efficient than ensemble methods [27] or Monte Carlo approximations, as it learns high-order prior distributions by gathering evidence from training samples in a single forward pass. Currently, the research on multivariate regression within this framework is limited, with only one study [28] focusing on the parameters of a Normal Inverse-Wishart (NIW) distribution, which serves as a conjugate prior. In this setting, the target y ∈ R N is considered to be independently and identically distributed (i.i.d.) from a multivariate Gaussian distribution with an unknown mean µ and covariance matrix Σ . The model aims to predict not only the mean E [ µ ] , but also the aleatoric uncertainty E [ Σ ] and epistemic uncertainty var[ µ ] . These predictions can be expressed as follows:

To generate these predictions, this method optimizes the analytical solution of the likelihood function:

However, it lacks mechanisms to ensure a structured and continuous latent space, which is important to leverage genetic structural similarity. Additionally, some studies [29, 30] have noted that the model takes the risk of halting the optimization process in highly uncertain areas, denoted as 'zero-evidence'.

F Derivation and Proof

This section includes theoretical derivations and proofs of PRESCRIBE. First, § F.1 provides the derivation for the expected log-likelihood ( L 1 ) of a multivariate Gaussian distribution under a NormalInverse-Wishart prior. The derivation for the differential entropy of the Normal-Wishart distribution ( L 2 ) is presented in § F.2. § F.3 offers a justification for how Eq. 3.4 mitigates the issue of zero gradients in regions of high uncertainty. Furthermore, a multivariate approximation for the Gamma and Digamma functions is derived in § F.4. Moreover, this section also presents a proof demonstrating that the pseudo E-distance preserves the ranking order of the true E-distance in § F.5. Finally, we review the mechanism that NatPN uses to capture generalization difficulty. § F.6

F.1 Derivation of L 1

Expected Log-Likelihood L 1 . L 1 optimizes expected log-likelihood under predicted posterior ω , improving accuracy and reducing uncertainty. It also affects other data points' evidence due to flow normalization. Its closed form is as follows:

For stability, ψ N ( x ) is approximated by ∑ N n =1 ln( x -n +1 2 ) (Appx. F.4).

The Normal-Inverse-Wishart (NIW) distribution µ , Σ ∼ W -1 ( µ 0 , κ, ν, Ψ ) is the conjugate prior of the normal distribution y ∼ N ( µ , Σ ) . Note that both parameters κ and ν can be viewed as pseudo counts. However, PRESCRIBE enforces a single pseudo-count n corresponding to κ = 2 ν .

Target Distribution. The density and the entropy of the Multivariate Normal distribution are:

Conjugate Prior Distribution. The density of the NIW distribution is:

The full version of the probability density function is:

where Γ N is the multivariate gamma function and Tr( · ) is the Trace of the given matrix.

Expected Log-Likelihood. The expected likelihood of the Multivariate Normal distribution N ( µ , Σ ) under the NIW distribution W -1 ( µ 0 , κ, ν, Ψ ) is:

Here ψ N ( · ) denotes the multivariate digamma function. In PRESCRIBE's formulation, we can obtain Ψ -1 = ν LL T and the moment of the NIW distribution E [ µ Σ -1 ] = µ 0 · ν LL T , E [ Σ -1 ] = ν LL T , E [ µ T µ Σ -1 ] = µ 0 · ν LL T + 1 κ I , and the moment of the Inverse Wishart distribution is

F.2 Derivation of L 2

Entropy Regularization L 2 . L 2 is an entropy prior favoring high-entropy posterior distributions W post i . Its closed form:

Approximations for stability (Appx. F.4): xψ ( x ) ≈ x ln( x ) -1 2 and ln Γ N ( x ) ≈ N ( N -1) 4 ln(2 π ) + 1 2 ∑ N n =1 ( ln 2 π -( x +1 -n ) + ( x -n ) ln( x +1 -n 2 ) ) . In this section, F.1 gives the derivation of the expected log-likelihood of a multivariate Gaussian distribution under Normal Inverse Wishart prior Eq.20. F.2 presents the derivation of the differential entropy of the normal Wishart distribution Eq.33. In F.3, we justify that Eq.3.4 can avoid zero gradient in highly uncertain areas. Lastly, we derive an approximation of Gamma and Digamma in a multivariate version in F.4. Before deriving Eq.33, we first introduce two propositions as follows:

Proposition 1. For Λ ∼ W ( Ψ , ν ) and any positive definite matrix A ∈ R N × N ,

Proof. E [tr( Λ A )] = tr( E [ Λ ] A ) = ν tr( Ψ A )

Proposition 2. For Σ ∼ W -1 ( Ψ , ν ) and any positive definite matrix A ∈ R N × N ,

Proof. By definition, Σ -1 ∼ W ( Ψ -1 , ν ) , so according to Proposition 1, we can yield: E [tr( Σ -1 A )] = ν tr( Ψ -1 A ) .

The Differential Entropy of the Inverse Wishart Distribution. Using the Inverse Wishart density given in Eq.24, the Inverse Wishart differential entropy is:

where ( a ) uses Proposition 2 and Eq.32, and in ( b ) used tr( Ψ -1 Ψ ) = tr( I ) = N and simplified.

F.3 Derivation of L 4

Zero Gradient Regions. Research has shown that when the evidence approaches zero, the gradient will also become zero, resulting in stopping optimisation.

Proposition 3. The model cannot learn from samples in high uncertainty areas by optimizing L 1 .

Proof. Given the parameters p ν of the flow module, we can obtain ∂ν post as Eq.8. Therefore, the gradient of the loss function L 1 with respect to p ν is given by:

In a high uncertainty areas, p ν →-∞⇒ ( a ) → 0 , so it leads to zero gradient with respect to p ν . The model cannot learn from samples in these regions.

Then, we prove that the effectiveness of the proposed L 4 can be supervised not only by pairwise distance but also by avoidance of the zero gradient scenario.

Proposition 4. L 4 can help the model learn from samples within zero-gradient regions.

Proof. Through Eq.8, we can obtain ν post = ln( p ν -N 2 N -p ν · ν prior ) . Then we can derive the gradient of L 4 with respect to p ν is given by:

̸

The gradient remains non-zero when p ν →-∞ , which avoids zero gradients in these areas.

F.4 Approximation of Multivariate Gamma and Digamma Function.

The computation of a distribution's entropy often requires subtracting large numbers from each other. Although these numbers tend to be very close together, this introduces numerical challenges. For large parameter values, we approximate the entropy by substituting numerically unstable terms and simplifying the resulting formula. For this procedure, we use the following equivalences [31]:

According to the definitions of the multivariate Gamma functions and the multivariate Digamma Function:

we can obtain multivariate approximation versions as follows:

F.5 Proof of Ranking Order Preservation

Theorem 1. Under fixed prior conditions, ˆ E preserves the ranking order of E .

Proof. Under min-max normalization with fixed control states, we derive: E = 2 δ XY -Y , where δ XY , Y ∈ [ N, 2 N ] . Through model design constraints: distance from prior ν ∝ δ XY and target entropy H ∝ Y . Let ν = aδ XY and H = bY with a, b > 0 . Then:

Let C = -1 . Then ˆ E = 1 N E + C . Since N is a positive constant (as δ XY , Y ∈ [ N, 2 N ] implies N > 0 ), 1 N is a positive constant. This linear relationship, ˆ E = kE + C where k = 1 N > 0 , preserves ordinal rankings between ˆ E and E . If E 1 < E 2 , then kE 1 < kE 2 , and kE 1 + C < kE 2 + C , which implies ˆ E 1 < ˆ E 2 . Thus, the ranking order is preserved.

F.6 NatPN

The authors of NatPN [9] have proved that PRESCRIBE can estimate evidence that asymptotically approaches zero under OOD conditions (e.g., far from the training set) as follows:

Lemma 1. Let { Q l } R l be the set of linear regions associated to the piecewise ReLU network f ϕ ( x ) . For any x ∈ R D , there exists δ ∗ ∈ R + and l ∗ ∈ { 1 , . . . , R } such that δx ∈ Q l ∗ for all δ > δ ∗ [32] .

Theorem 2. Let a NatPN model parametrized with a (deep) encoder f ϕ with piecewise ReLU activations, a decoder g ψ and the density P ( z | ω ) . Let f ϕ ( x ) = V ( l ) x + a ( l ) be the piecewise affine

representation of the ReLU network f ϕ on the finite number of affine regions Q ( l ) [32]. Suppose that V ( l ) have independent rows and the density function P ( z | ω ) has bounded derivatives, then for almost any x we have P ( f ϕ ( δ · x ) | ω ) → δ →∞ 0 . i.e the evidence becomes small far from training data.

̸

Proof. Let x ∈ R D be a non-zero input and f ϕ be a ReLU network. Lem. 1 implies that there exists δ ∗ ∈ R + and l ∈ { 1 , . . . , R } such that δ · x ∈ Q ( l ) for all δ > δ ∗ . Thus, z δ = f ϕ ( δ · x ) = δ · ( V ( l ) x ) + a ( l ) for all δ > δ ∗ . Note that for δ ∈ [ δ ∗ , + ∞ ] , z δ follows an affine half line S x = { z | z = δ · ( V ( l ) x ) + a ( l ) , δ > δ ∗ } in the latent space. Further, note that V ( l ) x = 0 and ∥ z δ ∥ → δ →∞ + ∞ since x = 0 and V ( l ) has independent rows.

̸

Our model adaptation did not conflict with its architectural prerequisite.

G Dataset Details

Our datasets are downloaded from scPerturb [8] and perform the following preprocessing steps:

Quality control. To ensure consistent and homogeneous quality throughout the different data sets, we filter cells with fewer than 1 , 000 and genes with fewer than 50 cells per data set.

Feature Selection. We identified the top 2 , 000 highly variable genes (HVGs) from each dataset using standard procedures and included all perturbed genes in the dataset's feature list.

Normalization. The raw count data were normalized using the lognormalization method, implemented through the standard preprocessing workflow of Scanpy [33].

Identifying Differentially Expressed Genes (DEGs). We identified DEGs by comparing perturbed cells to control cells for each perturbation. Genes were ranked based on adjusted p-values computed using the Wilcoxon test in Scanpy, with those p-values < 0 . 05 designated as DEGs.

PCA. Considering that too large a dimension will cause the Student's t-distribution to not be distinguishable from the Gaussian distribution, we choose the PCA components as 10 . The PCA was trained on the training set and performed on the validation set and test set. Additionally, we save the PCA model for reconstructing all gene' values. The results are saved in adata.obsm[ ′ y _ pca ′ ].

E-distance Calculation. The E-distance for each perturbation was calculated to quantify the effect of the perturbation. E distance is a statistical measure of distance between two distributions, which is utilized to indicate the degree of perturbation effect in scPerturb. We utilized the scPerturb [8] Python library to compute this metric between the control and perturbed cells. For each perturbation, we subsampled all perturbations to their greatest common divisors to perform the calculation. The distance e, pairwise distance, and self-distance are saved in adata.obsm[ ' n ′′ ], adata.obsm[ ′ d ′ ] and adata.obsm[ ′ s ′ ], respectively. For normalization, we perform the max-min normalisation on all the training sets' E distances onto [ N, 2 N ] .

Spilt Dataset. The filtered perturbations were divided into training, testing, and validation sets according to the default operation in GEARS [1]. The training set was used to fit the models, the validation set was used to select the best model, and the testing set was used to evaluate the final model performance. And for more details, see Tab.7.

H Baseline Setting Details

To evaluate PRESCRIBE's effectiveness, we benchmarked it against seven baseline models and five PRESCRIBE variants. Unless otherwise specified, all baseline model parameters were configured according to this benchmark [6] research to ensure a fair comparison.

H.1 AverageKnown

AverageKnown is a fundamental approach that averages gene expression across all cells within known perturbations to predict unseen perturbations.

H.2 Linear

The linear method is derived from a benchmark study [20]. The optimization objective is given by

where Y train is the gene expression matrix of perturbed cells. The rows represent the gene feature set, and the columns represent the perturbed genes in the training data. G is the gene embedding matrix obtained from the top K principal components derived from principal component analysis (PCA) on Y train . P is a subset of G that includes the perturbed genes in the training data. The fitted matrix is expressed as b = 1 N ∑ N i =1 Y train : i , where b is the vector of average gene expressions from Y train . The matrix W is solved using the normal equations:

Here, we set k to 512 and λ to 0 . 1 . The fitted W is then used for prediction:

where Y represents the predicted gene expression, and P denotes the subset of G that corresponds to the embeddings of perturbed genes in the testing data.

H.3 Linear-GPT

Linear-scGPT [20] employs the same modeling approach as the linear method, with the key distinction that the gene embedding matrix G is derived from the scGPT model rather than the PCA components of the training data. We followed the instructions from the GitHub repository of the linear method to obtain the scGPT gene embeddings.

H.4 scGPT

The scGPT [10] model is versatile across various scenarios due to its ability to accept any length of gene input. In the unseen perturbation transfer and unseen cell type transfer scenarios, we adhered to the guidance in the scGPT GitHub repository to reformat data and train models. Default parameter settings were employed, with the model trained using all genes in the feature set. Similar to GEARS, we trained the scGPT model for 20 epochs and selected the best-performing model based on validation results for testing. For zero-shot transfer and cell state transition scenarios, the scGPT model was trained on a combined dataset comprising a total of 17 datasets, ensuring broader generalization across different biological contexts.

H.5 scFoundation

scFoundation [15] also employs transformer-based architectures, making it suitable for in silico perturbation tasks, similar to the training paradigm of scGPT. Specifically, we leveraged the encoder module of the scFoundation model to train the perturbation model. The training and testing procedures mirrored those in scGPT. Since scFoundation incorporates more transformer layers than scGPT, we set the learning rate to 1 × 10 -5 .

H.6 CellOracle

We used the code provided on the CellOracle [12] GitHub repository. Since our control group data contained significantly fewer cells compared to the tutorial dataset, we reduced 'n_iters' from 100 , 000 to 1 , 000 to improve computational efficiency.

H.7 GEARS

Standard. We followed the tutorials provided in the GEARS [1] GitHub repository. GEARS is designed for scenarios involving unseen perturbations. We first reformatted our data to align with GEARS' input requirements. The model was then trained using its default parameter settings, with the number of epochs set to 20 . The best-performing model, selected based on its performance on the validation dataset, was used to evaluate the test dataset.

GEARS-Drop. GEARS provides an additional version that predicts the variance of each gene under specified perturbations, utilizing a loss function based on negative log likelihood as follows:

where ˆ g u is the predicted post-perturbation scalar and ˆ σ 2 u is the variance, s u = log ˆ σ 2 u = w unc u h post-pert u + b unc u .

GEARS-Ens. We ensemble GEARS-Drop by employing different training/validation splits and evaluating on the same test set. The results are averaged for both the predicted values and the logarithmic variance. We set the ensemble number to 5 .

H.8 Variants Of PRESCRIBE

Ours-NOINFO. We did not use the control status's mean and covariance to initialize the prior distribution. Instead, we adopt zeros as the prior mean and a zero matrix as the prior covariance.

Ours-Null. Ours-Null represents our model's untrained state.

Ours-Drop. The Ours-Drop variant utilizes the same encoder architecture as PRESCRIBE but replaces the decoder with Multi-Layer Perceptrons (MLPs). It employs the same loss function formulation as GEARS-Drop.

Ours-Ens. We ensemble Ours-Drop by employing different training/validation splits and evaluating on the same test set. The results are averaged for both the predicted values and the logarithmic variance. We set the ensemble number to 5 .

Ours-MLPs. The Ours-MLPs variant is based on Ours-Drop but incorporates an additional mean squared error loss term. It uses two linear layers with a ReLU activation function to regress the E-distance from the latent embeddings.

I Expected Calibration Error Results

We evaluate the uncertainty calibration of our model using the Expected Calibration Error (ECE). As shown in Tab. 8, PRESCRIBE consistently achieves lower ECE scores across all three datasets compared to the baselines, indicating better-calibrated uncertainty estimates.

J Limitations

Input Embeddings. The model's performance relies on the quality of input embeddings from foundation models. Applying it to complex chemical perturbations, which are more intricate than genetic ones, requires developing correspondingly richer embeddings.

Continuous Data Assumption. We assume a continuous Gaussian space, while single-cell data is often represented as discrete counts. A potential solution is to use an autoencoder (e.g., scVI [34]) to project count data into a continuous latent space, apply our method, and then reconstruct the counts.

Representation of Epistemic Uncertainty. Our framework approximates epistemic uncertainty with latent space density. While this is a practical approach, it is an indirect measurement, as genuine epistemic uncertainty is difficult to quantify and interpret [35]. Developing more direct and robust methods for representing epistemic uncertainty is a crucial direction for future research.

K Broader Impacts

Our framework has broader applicability beyond its current focus on predicting gene perturbations. Many disciplines within the AI for Science landscape, such as materials science or drug discovery, face two common challenges: models are usually required to make predictions on novel, out-ofdistribution inputs, and the training data is often derived from experimental observations subject to significant inherent uncertainty. By providing a unified approach to quantifying both epistemic (out-of-distribution) and aleatoric (data) uncertainty, our method can enhance the reliability and trustworthiness of machine learning models in these critical scientific applications.

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a framework called PRESCRIBE for predicting responses to unseen gene perturbations in single-cell RNA sequencing (scRNA-seq) data using Bayesian estimation. The PRESCRIBE framework employs the previously proposed Natural Posterior Network (NatPN) to provide uncertainty estimates via a normal-inverse-Wishart distribution and incorporates scGPT for gene embeddings. Additionally, the paper introduces a pseudo E-distance as a confidence score derived from the multivariate NatPN, which can be computed for unseen perturbations. Experimental results on three-gene knockout datasets demonstrate that i) the pseudo E-distance score is correlated with the empirical E-distance, and ii) PRESCRIBE achieves performance competitive with baseline methods in terms of correlation-based evaluation metrics for inferring unseen gene perturbation transcriptional profiles.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The PRESCRIBE framework enables uncertainty estimation of model predictions.\n- The paper provides source code, which enhances reproducibility.\n\n**Weaknesses**\n\n- *Originality/Significance* - The proposed approach appears to be a straightforward combination of previously proposed methods, namely NatPN, scGPT, and E-distances, with minimal modifications and without justification compared to potential alternative approaches: \n1) What is the impact of using scGPT for gene embeddings compared to alternative gene embedding approaches, e.g., ESM-2, and GNN specified according to the GEARS formulation?\n2) What's the impact of using the normal-inverse-Wishart distribution versus the Gamma-Poisson distribution for the Negative Binomial used in scVI?\n\n- *Quality/Clarity* - The model formulation (section 3.3) should be expanded to provide complete modeling details:\n1) The use of scGPT for the gene perturbation embeddings should be highlighted; only two lines are provided to describe such an important modeling decision. \n2) It's unclear why the sum aggregation function is used for combined perturbations (Eq 7) instead of alternative set aggregation approaches, e.g., set transformers.\n3) It's unclear how the control profiles are encoded via $f(c)$ in Eqn 7. Also, how is $c$ selected for each $y$ given that only one is provided for a specific cell?\n4) The paper should provide complete parameterization of the decoder and the normalizing flow functions.\n\n- *Weak experiments* -  Given that the technical contributions are limited, the experimental setup does not seem to support the claims: \n1) While the paper provides an ablation study for some of the loss components in Eqn 9 (Table 4), they don’t seem comprehensive across all datasets, and the default NatPN objective (L1 and L2) should be provided.\n2) The paper reports correlation metrics as percentages, which is unconventional. These should be given in the standard range [-1, 1], ideally with tests of statistical significance.\n3) Figure 4: While PRESCRIBE seems to have better calibration than GEAR, it does not appear to be well-calibrated. Additionally, the calibration data points are few, with PRESCRIBE resulting in larger error bars. The paper should also provide alternative metrics for measuring uncertainty, including the coefficient of variation.\n4) Table 1: The paper claims that pseudo E-distance is correlated with E-distance; however, the correlation coefficient seems too small for it to be an effective measure of E-distance.\n5) It's unclear why non-correlation-based metrics, such as RMSE, are not considered.\n\nQUESTIONS:\n- What is the impact of using scGPT for gene embeddings compared to alternative gene embedding approaches, e.g., ESM-2, and GNN specified according to the GEARS formulation?\n- What's the impact of using the normal-inverse-Wishart distribution versus the Gamma-Poisson distribution for the Negative Binomial used in scVI?\n- Why is the sum aggregation function is used for combined perturbations (Eq 7) instead of alternative set aggregation approaches, e.g., set transformers?\n- Could you provide comprehensive non-correlation metrics, such as RMSE?\n\nLIMITATIONS:\n- The paper discusses some limitations regarding the use of scGPT for gene embeddings; however, it should also address the potential biases introduced by the preprocessing approaches, such as high variable gene selection and log-normalizing the counts data.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents **PRESCRIBE**, a model for predicting single-cell gene expression responses to genetic perturbations while estimating uncertainty. It extends the Natural Posterior Network to a multivariate setting and introduces a pseudo E-distance metric that approximates confidence without requiring post-perturbation data. The model is evaluated on standard benchmarks, demonstrating well-calibrated uncertainty estimates and improved prediction accuracy, especially when low-confidence outputs are filtered.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- **Practical relevance**: The authors address a key limitation in current single-cell perturbation methods; the lack of uncertainty estimation. \n- **Theoretical framework**: The derivation of the multivariate NatPN and its use of the Normal-Inverse Wishart distribution is a non-trivial extension of evidential learning approaches and seems appropriate for the single-cell perturbation task. The mathematical derivations are furthermore clearly explained.\n- **Comprehensive empirical evaluation**: The authors conduct thorough experimentation on three benchmarks, comparing to both methods without uncertainty estimation, and with. Moreover, the methods are evaluated on a range of relevant metrics.\n\n## Weaknesses\n- **Accessibility**: The paper is technically very dense, with advanced Bayesian machinery. This may limit accessibility to a broad ML and computational biology audience.\n- **Limited real-world validation**: The authors do not explore biological insights or interpretation of the uncertainty scores (*e.g.*, which perturbations are hard to predict and why). This is a missed opportunity to explore the model's utility beyond benchmarks.\n- **Lack of standard calibration metrics**: While Spearman correlation is used between prediction accuracy and confidence, no standard calibration metrics are utilized (*e.g.*, Brier score or ECE).\n- **Scalability**: The multivariate extension of NatPN with flows and Bayesian updates may introduce significant computational complexity, especially as the number of genes (dimensions) and perturbations increases. This isn’t deeply analyzed.\n\nQUESTIONS:\n**Q1**: Given the use of multivariate NatPN with normalizing flows, how does PRESCRIBE scale with increasing gene dimension or perturbation complexity? What are the memory/runtime bottlenecks?\n\n**Q2**: All benchmark datasets seem to share a common structure and origin. Did you test generalization on held-out cell types or across studies to evaluate transfer performance?\n\n**Q3**: In the filtering-based accuracy gains,what are the characteristics of the discarded predictions?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the PRESCRIBE method, which uses a natural posterior network, to provide uncertainty estimates for the effects of genetic perturbations on gene expression. This method crucially does not require ensembling or multiple forward passes. The authors justify this approach for uncertainty estimation by showing that it resembles an E-value, which has been used to establish statistical significance of perturbations in experimental data where there are sets of control and perturbed cells for each perturbation.\n\nSTRENGTHS AND WEAKNESSES:\n- The primary weakness of this paper is its clarity. (1) Multiple terms are used before they are defined (e.g. $\\chi^{\\text{prior}}$ and $n^{\\text{prior}}$). (2) Multiple variables are used with the same name (e.g. $n$, $N$, and $N_{H}$). (3) The methods section, instead of building up intuition, inundates the reader in equations. To be fair, I am not familiar with natural posterior networks and normalizing flows, so I might be the wrong audience, but even core ideas such as how their pseudo E-distance corresponds to the real E distance are not well explained (Figure 2V is more confusing than it is helpful). In its current incarnation, I cannot imagine how this paper would be particularly legible to people who are not already both experts on natural posterior networks and perturbational models.\n\n- It does seem to be the case that uncertainty estimates from their method are more calibrated than for GEARS, as they show in Fig. 4. But it's not the case that they are calibrated particularly well either, since they are quite far away from the y = x line. \n\n- It is good to see though that there predictive performance, as indicated in Table 2, is quite strong. In fact, the paper seems to undersell how strong the predictive performance is.\n\n- Since GEARS uncertainty estimates are so bad, I think two simple additional baselines should be included to prove the utility of this method: (a) The E-value between the predictions of an existing model like GEARS and the control cells. Specifically, I am wondering if the predicted magnitudes are more reflective of the model's certainty than its variance estimates. Obviously this is not helpful for distinguishing a perturbation with actually zero effect compared to a perturbation where the model doesn't know and therefore says zero. However, for perturbations with strong effects, the model's predicted effect magnitude might be a measure of its certainty.\n(b) Some measure of distance between the test set perturbations and the training set perturbations in some gene embedding space (like from scGPT).\n\nQUESTIONS:\n1. How is the PCC calculated in Fig. 1? Is it across gene LFCs for a given perturbation? \n2. Why should perturbation embeddings be additive in Equation 7? it seems like $f_{\\alpha_{2}}$ should be a set transformer or something similar.\n3. What is the intuition for setting $N_{H} = N$. \n4. It is unclear what you are showing in Table 1(b). Is that the correlation with the real E-value for perturbations where that are at least N cells measured post-perturbation?\n5. How does an untrained model (PRESCRIBE-Null) have such good predictive performance as is shown in Table 2?\n6. Is Ours-MLP trained with the E-distance ranking loss? If not, it seems like it should be for a fair ablation.\n\nLIMITATIONS:\nLimitations are not discussed in the conclusion at this point.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn order to address the issue of uncertainty estimation in modeling multidimensional perturbation datasets like those containing  transcriptomic readouts, the authors propose PRESCRIBE, an extension of the Natural Posterior Network (NatPN) accommodating multivariate normally distributed data. The authors additionally interpret the NatPN confidence score as \"pseudo\" energy distance (or E-distance), add a loss term to ensure that this interpretation holds, and attempt to show that the pseudo E-distance are well-calibrated with predictive performance.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Valid extension of NatPN to multivariate data and concepts of energy distance\n1. Extensive set of comparisons and ablations intending to establish the claims of predictive performance and well-calibrated uncertainty estimation\n1. The method accommodates combinatorial perturbations\n\nWeaknesses:\n1. The introduction of energy distance is poorly motivated\n1. There is application to only 3 datasets of one data and perturbation type, when this could have been fruitfully applied to many related datasets with different perturbation types, and even a toy datasets to illustrate the versatility of PRESCRIBE\n1. Following on the previous point, claims of PRESCRIBE being well-calibrated are limited to the narrow set of empirical results included in the paper. Results like Figure 3b seem cherry-picked.\n1. Model choices that could have been insightfully explored through ablation but were not: scGPT embeddings for gene perturbation, encoder architecture, PCA dimensions\n1. There is some confusion about the $\\lambda$s in equation 9 and the derivations in Appendix F. I think Section 3.4 should include a $\\lambda_i$ in front of each $\\mathcal{L}_i$, and the notation in Section 4.3 updated accordingly.\n\nQUESTIONS:\n1. Why did you focus on extending NatPN with energy distances? What motivated this? The paper suggests that it is because of an apparent similarity between E-distance and the NatPN confidence, but that is not obvious.\n1. The results of Table 3 are confusing - why does random filtering of 10% outperform that of 5% and match no filtering?\n1. Why is it important to keep the evidence in the range [N, 2N]? Preventing the t-distribution from resembling a Gaussian is mentioned, but I don't understand the negative implication that resembling a Gaussian would have.\n1. Why the form of the encoder mentioned in section 3.3? Is it appropriate to assume additive effects? The authors cite Gaudelet et a.l. (2024) for this result, but that paper is clear that the approach is more or less appropriate depending on the interaction. An analysis of the E-distance by interaction type would be an insightful extension of the analysis of that paper.\n\nLIMITATIONS:\n1. The recent paper [1] explores the performance issues of many of the baseline models and datasets used in this paper, and recommends new metrics that better demonstrate predictive performance. This paper should at least acknowledge this as a limitation, and should very likely redo some of the analyses with the suggested metrics.\n\n[1] Mejia, Gabriel M., et al. \"Diversity by Design: Addressing Mode Collapse Improves scRNA-seq Perturbation Modeling on Well-Calibrated Metrics.\" arXiv preprint arXiv:2506.22641 (2025).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a method called PRESCRIBE that predicts how single cells respond to perturbations. For each prediction, it also provides an uncertainty score, which it calls a \"pseudo E-distance\".\n\nThe model builds on the Natural Posterior Network (NatPN) but adapts it for this problem in a few key ways. It extends the network to a multivariate case using the Normal-Inverse-Wishart distribution. It also adds two new loss terms: one to make its uncertainty score mimic the ranking of true E-distances from the training data, and another to help with training by preventing zero gradients.\n\nThe paper tests PRESCRIBE's prediction accuracy and the quality of its uncertainty scores on three benchmark datasets and includes ablation studies to back up its design choices.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality:** Overall, I believe this is a high-quality paper. It addresses the critical need for reliable uncertainty quantification in single-cell response prediction. The authors present a convincing framework, PRESCRIBE, that is technically solid. The quality of the work is supported by extensive experiments on three benchmark datasets, as well as thorough ablation studies that justify the model's core components.\n\n**Clarity:** The paper is clearly written and well-structured, with helpful figures illustrating the method. However, the clarity could be improved by providing more explicit justification for certain design choices (e.g. see Questions).\n\n**Significance + Originality:** While my expertise is not in single-cell perturbation modeling, I’ve assessed the work based on the context provided in the paper. Within that frame, the work seems to be an original and significant contribution. The authors propose a novel method that balances prediction accuracy with uncertainty quantification, supported by ample theoretical motivation. They introduce well-motivated notions of uncertainty under the Normal-Inverse-Wishart model – which has been used in previous prediction models in this setting. The work's significance is clear as it has the potential to make in-silico screening more reliable by allowing researchers to know when to trust a specific prediction—a critical need for applications like drug discovery and the development of genetic therapies.\n\nQUESTIONS:\n- The author(s) initially present experimental results for all the datasets and then focus the rest of the exploration on the Norman dataset. What is the reason for focusing on the Norman dataset for the later analyses? Are the results robust across all three datasets? Similar experimental results for the other two datasets (e.g. in the Appendix) would be helpful here and could improve the clarity and significance of this work. If this is not feasible, an explanation for why this analysis is specific to the Norman dataset is necessary for transparency.\n\n- In the presentation of the method in 3.1, the author(s) fix the proportion between $n$, $\\kappa$ and $\\nu$ (Eq. 4). Could the authors provide the theoretical or empirical justification for this specific modeling choice? A brief ablation study showing that the model is robust to this choice, or that this choice is optimal, would strengthen the clarity of the model design choices here.\n\n- How robust the model is to misspecification of its Normal-Inverse-Wishart distribution? It may be nice to know how strong the regularization of the $\\mathcal{L}_3$ term is in handling any model misspecification here since it encourages the pseudo E-distance derived from the Normal-Inverse-Wishart prior to match the training data E-distance. A discussion of its robustness or a simple experiment on synthetic data with a known non-Gaussian ground truth (but still reasonable for single cell settings) to demonstrate that not only the predictions but also the uncertainty scores are robust would clarify on the limitations and strengths of this paper.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-23"}