PRESCRIBE: Predicting Single-Cell Responses with Bayesian Estimation
Authors: Jiabei Cheng
Changxi Chi
Jingbo Zhou
Hongyi Xin
Jun Xia
Venue: NeurIPS 2025 poster ()
Keywords: Single-cell biology
bioinformatics
gene expression
transcriptomics
uncertainty estimation
Primary area: machine_learning_for_sciences

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
Jiabei Cheng', Changxi Chi², Jingho Zhou?, Hongyi Xin Jun Xia
Shanghai Jiao Tong University, "Westlake University,
3 AIMS Lab, The Hong Kong University of Science and Technology (Guangzhou), Thhe Hong Kong University of Science and Technology {jiabei_ @jiabe_cchen,, hongyi xin]asjto jtt.uuu...cco ead.edd.co. junxia@hkust- edu. en
Abstract
In single-cell perturbation prediction, a central task is to forecast the effects perturbing 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 the corresponding training data, which reflects data (aleatoric) uncer- tainty. Both factors critical for determining the reliability of prediction, particularly as gene perturbation is an inherently stochastic biochemical process. In this paper. we propose PRESCRIBE (PREdicting Single- Response wIth Bayesian Estimation), a multivariate deep evidential regression framework de- signed 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 comparable baselines. Code available at https: //gi thub. httm://unnybbbbbb///SStH..
Introduction
Predicting the effects of perturbations is crucial for advancing biological understanding and the development targeted genetic therapies. Recent years have seen significant progress in machine learning models [133, data generation 4 [. and benchmarking 6 7 for this task. However, 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. ((a) illustrates this issue, showing that even models with high average accuracy can make substantial errors specific predictions.
The first step towards estimating prediction uncertainty is understand its sources. Prediction uncer- tainty arises from the interaction between two primary sources. First, data (aleatoric) uncertainty arises from the inherent stochasticity 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 given input, which particularly high for out-of-distribution perturbations. A practical framework needs to account for both. For example, prediction is the least reliable when the model's output far from highly consistent biological outcome (high model uncertainty, low data uncertainty). Equally, prediction is intrinsically uncertain when the biological outcome itself is highly variable (high data uncertainty).
Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Average POC;
E-010sann tescored normaltedt!
scGPt Corr 67.47%
21
20.03 0.00 20.0
Fold Change (5)
-1 -I Change
Change
ED
00
(b)
(c)
Figure 1: Preliminary experiments on the Norman dataset using three representative models: (a) High overall predictive accuracy does not ensure individual prediction reliability, (b) E-distance exhibits strong correlation with prediction accuracy across these models. (c) Calibration analysis demonstrates that E-distance percentiles effectively stratify prediction accuracy.
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. [(b)). the ground truth of a perturbation is known a priori, then the E-distance between the predicted post-ped cest-perturbation population and the ground truth, composed of an inter- group distance compensated by a negative intra- intra-group distance term, reflects the accuracy of the prediction (Fig. [(e). 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, 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 (PREdicting Single- Cell Response wIth Bayesian Estimation). PRESCRIBE is multivariate extension of deep evidential learning framework (Natural Posterior Network [9] that, instead of outputting a single expression profile, simultancously predicts the post-perturbation state and estimates the prediction's confidence. PRESCRIBE has two key elements: a posterior distribution over the transcriptomic landscape and evidence score derived from 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 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 low evidence score indicates otherwise. We model the post-perie pestuuerturbttio transcriptomic spread using a Normal-Wishart conjugate Bayesian framework, allowing the variance to be inferred from the perturbation embedding via 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 safe output for unreliable predictions.
Our main contributions can be summarized as follows:
We propose PRESCRIBE, novel framework that uses predicted pseudo E-distance as unified surrogate for both data and model uncertainty in single-cell perturbation prediction.
We introduce a multivariate extension of the Natural Posterior Network, utilizing an Inverse-Wishart prior 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.

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2 Related Work
2.1 Predicting Single-Cell Responses
In-silico prediction of single-cell responses offers an efficient alternative to costly single-cell pertur- bations in the wet lab. Current methods generally fall into two main categories: direct matching or disentanglement 77] Direct matching methods, such as GEARS [ and scGPT [10], map control cell gene expressions to perturbed expressions to predict responses new perturbations. Disentanglement methods, like CPA [ 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 data set D (X, y, c), representing M types of perturbation and G genes, where X {x), I2, 2 ( denotes the set of perturbations, y É R is the post-perturbation gene expression profile, and is the pre-perturbation (control) gene expression profile, the predicted post-perturbation expression Vz. can be modeled as:
yr, = c (axe).
where (xx) is the learned effect under the perturbation Wi.
Examples of direct matching models include CellOracle [12]. which uses scRNA-seq and scATAC-seg 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. Graph VCI [3] employs counterfactual concepts from causal inference to enhance gene regulatory network learning. sams-VAE (14), uses 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 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. Morcover, 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 evidential deep learning method (Appx.š E.33 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.5 F.2 for exponential family distributions. Briefly, for a likelihood P(y w) and its conjugate prior Quw M M data points { leads to posterior Q (w m poot with parameters updated as:
p priur, observing
x x
peir + M u(3)))
+
poos = pprior + M
NatPN measures confidence with post [H[ (3 w)], where possi (total evidence) captures epis- temic uncertainty and entropy represents aleatoric uncertainty. The design of its density estimation architecture has been proven to drive the estimated evidence towards zero under disti---itt (OOD) conditions (e.g., far from the training set) (See Appx. For details).
This confidence score shares conceptual similarities with E-distance calculations (Fig IVV). How- ever, standard NatPN cannot be directly applied to single-cell perturbation prediction. It lacks mechanisms to incorporate pair-wise distances and does not offer readily available multivariate extension suitable for complex gene expression data. Our work tries to resolve these limitations.
3

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Input Space
Perturbation Density Space
Normal Inverse Wishart S pace
Control Cels
Seen P arturbed Colls
Low Entropy [Low Data Uncortalnty
Unseen Perturbed Cell
13 (r, (or)
x1 (rr
Normalizing Flow P(nj(n)
V2 My ((ayloa) 2 Ta(as)
High (figh Data Unceriaintyl
High Dens (Low Macel Uncertainty)
X
fa
1g ((sl(o))))
post (g(ss)
Density (High N Uncertainty)
Encoder
M. Decoder f Optimization Process
Uncertainty Indicator
Normelicing Flow
x
prior
Control Celss
prior
2-1ove MiLP
Perturbations
Deep
Bayesian Update
pou sirliey
[MP((s)0))
Embedding
Linus
ppost
E
2y (PP ())
Perturbation Supersised Signels
Modal Uncartannty Data Uncertainty
Figure 2: Overview of PRESCRIBE. Each perturbation X₃ under condition first mapped onto a perturbation density space by the encoder fa. From 24, the decoder fa derives the parameter update X nut while normalizing flow fo yields the evidence update 1/!Tt. Posterior parameters w, are obtained from a weighted combination of prior " and updated parameters according to u!a..
3 Methods
Our core contribution is the pseudo E-distance, metric from the model's outputs that unifies epistemic and aleatoric uncertainty. 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. I
3.1 Probabilistic Modeling of Gene Expression
To model both the distribution of gene expression and predictive uncertainty, adopt Bayesian approach. For a given perturbation category Hi, we assume the gene expression vector y, follows a multivariate Gaussian distribution. Following related work, we place a Normal-Wishar conjugate prior on the parameters of this Gaussian, which enables analytical Bayesian updates. Thus, a posterior distribution is defined by four parameters w;
(Moz, E EN Kr E R+
vi N, L.}:
PP ( M4 Ad) ~ (H., AT), P(A,) = w (v, V. P(ET P(E.) W- (vi, W;);
P(N, A,) N (Hoz, (KG,,
where E; = A,- is the covariance matrix and A, is its corresponding precision matrix. The scale matrix V is defined using a lower triangular matrix L, such that - = L. Here, N, W, and W-1 denote the Normal, Wishart, and Inverse Wishart distributions, respectively. N the rank of the gene-gene interaction matrix, estimated via Principal Component Analysis (PCA).
The decoder outputs the effect of a perturbation 2y a of sufficient statisties and evidence counters:
X1
out
out Uur.
X:
X2
out
out Mi0e
(1
L out
T (utt
T(L
n2:
Ki
20 out

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The network's 's outputs (denoted by the "out" superseript). Since the evidence-related parameters are proportionally linked (n, K; 2n:), we use refer 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. Panel V)
as:
È 20 [y) w:)]-
(5)
Here, - denotes normalization. This metric comprises two key terms. The first, 8 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 out-of out-of-distributtin input. The second term, -H. is the negative normalized entropy of the predictive distribution, which reflects aleatoric uncertainty 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 base prior using the control cell profile and fixed hyperparameters (Ne = po = 0.5). The posterior parameters are then obtained by combining this base prior with the model's s outputs through the Bayesian update (Eq. 2 From this posterior, the evidence v and the predictive entropy Hi[-] are normalized the range [ 2.V]. This operation serves two purposes: it places both components
comparable scale and it ensures the degrees of freedom of the resulting Student's I-distribution remain in a regime that preserves its heavy-tailed properties, which is crucial for capturing sparse regions (low evidence) of the perturbation space.
Our model can distinguish low-confidence, confidence predictions of perturbation with little to no effect. Although both may predict an outcome similar the control state, they are separated by their significantly different evidence scores.
Remark 1.
aut-of-distributiion predictions from high-
3.3 Model Architecture
As illustrated in Fig. 2 our model comprises three core modules: an encoder (fa: Panel I) that generates latent representations, a normalizing flow (f& Panel II) that estimates evidence, and decoder (fs; Panels III-IV) that produces the sufficient statistics for the predictive distribution.
Encoder fa. The encoder processes a perturbation x and the cell's basal state e into a latent embedding E RD that captures functional similarities. Assuming additive effects [17), the encoder comprises two components: fa, for individual perturbation effects and fa, for non- non-linear interactions. For single perturbation 2;, the embedding is calculated follows:
2, = fa (2,) = fais (x²) + faiz(c))
(6)
For a set of perturbations æ. the model aggregates their individual effects via summation. This operation ensures the resulting embedding is invariant to the order of the perturbations:
z = fo Σ (a.( + foz ESan
Here, fa.: is linear layer that uses pre-trained gene embeddings, while fauz and fay are multilayer perceptrons (MLPs) with LeakyReLU activation. While the encoder design agnostic to the choice of gene embeddings, we use those from scGPT [10 by default.
Normalizing Flow fo- 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 v. 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 in is calculated from the latent embedding 2, and the posterior evidence i.Peea is then updated and normalized as follows:
NV;
14 exp(fx (z.) NH).
Post
+N [N,2N].
(8)
1: + pptiiee
Here, Nu is the total certainty budget, which we set to N. The update operation for ipoat scales the posterior evidence to the range [N, 2N and balances the influence of the base prior (when evidence i is low) against the data-driven prediction (when evidence is high).
5

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Decoder fa. The decoder is one linear layer that maps the latent embedding to the sufficient statistics X; The outputs from the decoder X; and the normalizing flow 1/tth 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 composite loss function designed encourage both accurate predictions and meaningful perturbation density estimates:
C
E winw 1ao [l (y: wi)l -Ai y. pasl
H.
W
post
L1
L2
B
(9)
22
In
-Aal
Moz,
1l
In
N
B
2
((ooneiaz))
2N-
1
Specifically, L consists of primary objective for prediction accuracy (C1) and three auxiliary terms, weighted by hyperparameters \1, X2, A (see 4.9 for grid search details):
Expected Log-Likelihood (L1). This term maximizes the likelihood of the observed data under the posterior predictive distribution, driving accurate predictions. The details can found in Appx. F.1
Entropy Regularization (C2). This term, weighted by prediction error, acts as prior that favors uninformative distributions with high entropy. The details can found in Appx. F2
E-distance Ranking Loss (Cz). To supervise the model's uncertainty estimates, we introduce ranking loss based on ListMLE (99] 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:
Extted
E Jargsort E, descending)].
(10)
The term 2s 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 (C1). This term addresses the problem vanishing gradients in low-evidence regions. The details can be found in Appx. F3
4 Results
This section details PRESCRIBE's empirical evaluation. The experiments were designed to: (i) assess the quality of its uncertainty 4.3 (.6)). (ii) demonstrate its utility in prediction accuracy
4.7
and (iii) explore contributions of its core components and initialization strategies (8 4.8
4.1 Experimental Setup
Datasets. We evaluated PRESCRIBE on three widely recognized benchmark datasets: Norman
4
Replogle2022_Reet Repl, and Replogle2022 K562 [5. (Details are provided in Appx. G
Comparison Baselines. To evaluate PRESCRIBE, we conducted comprehensive benchmark against several existing methods. The baseline methods divided into two main groups: recent methods without uncertainty estimation (AverageKnown, Linear, Linear scGPT [20] CellOr- acle [12] samsVAE [14, GraphVCI [3], scFoundation [15] and seGPT (1O) and those with variance-based uncertainty estimation (GEARS (I] and GEARS-Drop). The evaluation also en- compassed several adaptations and ablations of our approach and related models: GEARS-ens, an ensemble of five GEARS-Drop models; Ours-MLPPs. variant that employed the our model's encoder/decoder architecture but specifically used Multi-Layye Perceptrons (MLPs) to regress E- distance from the latent embedding; Ours-Null, representing our model's untrained state; Ours-Ens,

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an ensemble of five models using the our model's encoder and an MC dropout [6 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 different role.
Prediction Accuracy. These metrics are assessed by measuring the Pearson Correlation Coefficient (Fpred.truh) and directional accuracy (ACCpred.uut) between predicted and true gene expression. We also calculated these metrics on the top 20 differentially expressed genes (DEGs), denoted as DEG pred,truth All accuracy evaluations were performed on log-fold change values.
and ACC pred.tmuhh
Calibration Quality. These metrics assess a model's confidence scores are meaningful indicators of its predictive performance. We measure this using several metries: (1) the Pearson (fper(.cont) and Spearman's 's (r (fptt.coo))) correlation between confidence scores and actual performance (as measured by "pred.rrth): : (2) a percentile- based classification accuracy (ACCper..ont) to determine if low-confidence predictions are less accurate; and (3) the Expected Calibration Error (ECE) [21). denoted Epert.conf-
4.3 Estimation of E-distance
Our model is designed efficiently estimate E-distance without requiring post-perturbation profiles. We therefore validated the practical effectiveness of our estimated "pseudo E-distance" (É) in two dimensions. First, across all datasets, É exhibited consistent positive correlation with a reference E-distance (E), which was computed using post- cost-perturbation profiles (Tab, aa). Second, this positive correlation strengthened significantly as the number of samples (N) used for computing the reference E-distance (E) increased (Tab. [b)). Given that E calculated with a larger N more accurately reflects the true underlying E-distance, this observed trend suggests that can serve as an effective, and potentially asymptotic, approximation of the true E-distance's ranking.
Table 1:
Comparison of pseudo É with reference E-distance (E): (a) across various datasets; (b) across increasing sample size (N) used for reference E calculation on the Norman dataset.
Cross dataset performance
Norman.
Repl.
K562.
Method
"Ecoat
ACCa,cot
TEccot
E.out 1
ACC
"Econt
"Ecoot
ACC
"E,our
Ours-Null
0.76
16.13
-3.28
-3.61
19.10
-2.34
-9.58
17.65
-10.96
Ours-MLP
27.49
25.03
29.56
.45 64
22.81
-50.59
-3.61
18.89
-2.34
Ours
35.56
25.81
33.84
12.18
23.53
21.80
24.74
31.58
16.86
(b) Cross sample size N performance
Ours
Ours-MLPs
Ours-Null
N
ACC1.coa
E.out
ACCr.cof f
"ioood
ACCr.cof
54
35.56
25.81
33.84
29.03
27.49
29.56
16.13
0.76
-3.28
200
41.37
33.33
45.44
33.33
32.71
31.52
7.69
-14.68
-9.42
500
80.00
50.00
67.55
50.00
20.00
24.22
NaN
33.33
NaN
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 the generalization difficulty increased. In contrast, others show a minorly decrease or inverse trend. This phenomenon demonstrates model's ability be more aware of epistemic uncertainty than other basclines.
7

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88
Tain Data
Confidunce Score te-ccored nurmalzee)
Text
77:32
Cesnsrro
OurvOrop
10
Control State
Avaraged E-distencce
Companent
2AP HSLC382
141.2
BAKI-TM5B-X
E-55.15
55
E-
-10
CBL-UBBASHB
F 00.41
0.0
Uunteer of3
1 Lreeen of? Dificultyy
7 Unssen #12
Component
(a)
(b)
Figure 3: (a) The line plot depicts confidence scores from different models plotted against generaliza- tion difficulty. An accompanying bar plot displays the average E-distance for corresponding levels of generalization difficulty. (b) t-SNE visualization of the Normal-Inverse Wishart space.
Table Perturbation Prediction Performance Comparison (in %)
Nurmankh gestanck
Repl. Specooch
K563.
Modele
Tyed mac
Tpecauck
AvergrRaanen
Linenz
37.68
53.54
03.34
35.01
401.0
41.65
38.15
15.70
12.42
$2.54
52.36
Linear-u GPT
39.30
39.66
27.15
03.34
50.00
$5.55
43.09
31.31
33.85
42.23
$4.70
$4.31
CollOrde
3.40
12:48
17:07
3066
27.05 05
13.05
16.30
16-10
41:41
25.00
0.73
38.41
05 15
ecThendeeiee
46.34
030
66
19
GEARS
45:30
63.19
26.00
09.05
43 18
$3.59
$1.08
52.55
33.51
42.08
56.53
56.14
GEARS Doop
44.96
09:38
08.28
$1.05
$3.25
57.00
31.26
03.38
54
$7.67
GLARS Lhop-ItP IPP
49.22
05:25
31.57
75.15
48.38
36.35
GEARS-EEZ Eiaa
43:34
02.84
29.31
09:35
47.87
49.92
53.91
35.30
34.59
42-99
22
36.16
09.00
40-12
10
GEARS
50.91
FRESCRIBE Nall
14.20
43.54
$1.19
65.07
8.50
16.55
51.83
33.59
10.95
21.80
52.58
$6.30
PRESCRIBE-SS 5-5
70.04
PRESCRIBE 109
54.32
6961
66.33
75:93
00.38
06:13
$7.89
90.0
38.58
47.52
01.04
11.21
4.5
Qualitative Visualization
We visualized the Normal Inverse Wishart latent space learned by our model using t-SNE dimensionality reduction on the Norman dataset (Fig, ((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 Tpredauuh prediction accuracy, with darker shades indicating higher quality predictions. The visualization aligns with expectations: training data points and large 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 prrdd.rruhh
4.6 Accuracy-Batt Calibration Analysis
We examined accuracy-based calibration curves for a more granular understanding of our model's practical utility. Furthermore, computed the Spearman correlation (re- (rpaat.cont) between prediction accuracy and the confidence scores produced by our model and baseline methods (for GEARS-Drop and GEARS-ER -nnS, inverse values were used as their confidence scores represent variance). From Fig. 4 our model achieved the highest r perf.conf across all datasets, and the visualizations of the calibration curves demonstrated that our model exhibits consistent trend where confidence increased monotonically with accuracy, indicating its confidence score is well-calibrated overall. The expected calibration error (Eperf.oont) results are presented in the supplementary materials (Appx.
4.7 Prediction Accuracy with Uncertainty Guided Filtering
A practical use of reliable uncertainty estimation is identifying and potentially excluding low- confidence 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. 22). Ours-10% significantly outperformed all baselines, while Ours-5% also showed strong results. It important to note that modeling distributions (as model
8

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SPEETN in 97.39
SPCTINE 20.00
(800110 82.85
SPEONN 18.84
(a)
(d)
PCS Parcarsie (M)
PCG: Paccendle
PCC Faceeniia 131
PC5 Parerdia
(h)
SPECTS in 12-43
SPECTSI -6.74
SPOC/NG - 21.23
SPcOni 21.33
Racomtic
Figure Calibration curves relating model-estimated confidence scores to prediction accuracy (rrediiuh)) Each row corresponds to dataset, while columns and colors distinguish different models (in order: Ours, Ours-MI.Ps, GEARS-Drop. and GEARS-Ens). Text in each subplot indicates the Spearman Correlation Coefficient (ppent.oon)) between (peddtruth and confidence scores.
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.
Table Ablation Studies on Filtering (in %)
Table 4: Ablation Studies on Modules (in %)
Model
Fpredumnh
ACC ACCprod.mutt
Model
Tpeetinaa
ACCrrssmm
pertosa
Ours
58.38
63.24
NOI-NOONNFO
43.25
$7.80
36.65
Random Filtering-S5 5%
53.35 1,16 58.28:11.59
60.96.10.97
w/o La
55.61
63.61
27.74
Random Filtering- 10%%
57.49112.25
w/o C4
22.41
54.73
29.83
Impact of Filtering. We performed random filtering (10 repetitions) to assess whether the observed
performance gains were merely an artifact data removal. As shown in Tab. 3 randomly filtering 5% of predictions significantly decreased accuracy. Filtering 10% randomly brought accuracy back 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.

---

43
ACC
(a)
(b)
Figure 5: Hyperparameter Search on Norman Dataset. (a) Box plots illustrating the relationship
between X2 and the distributions of "peed.tuth. pred and r perfssont- (b) Line plots illustrating the
relationship between the ratio / λa and metrics ppred.muth ACCpred.ruth and pprt.ccon when 22 0.1.
Impact of Key Components. As demonstrated in Tab. 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 perf,comf values, underscoring the importance informative prior settings for well-calibrated uncertainty. The loss term La. which supervises learning from E-distance, proved critical; its removal led to slight tCCpred,tnuth 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 L4 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 be summarized in the following steps:
Phase 1 >22 Optimization: Conduct grid or random search for A2 (the weight for L3).
Phase /r Ratio Tuning: Fix aa (the weight for L4) to 10-5 and perform focused search for \₁ (the weight for L2). This effectively tunes the ratio \//
Rationale and Example. Our efficient two-phase tuning pipeline reduces trials from 216 (6 6 x 6) to 42 (6 x + 6) by fixing A3 the second phase. This is justified by the adversarial relationship between L2 (weighted by Å1, promoting prior adherence) and L1 (weighted by Ag, encouraging evidence accumulation). Thus, we fix Å3 = 10-5 and tune A₁ For example, on the Norman dataset, initial random search determined X2 0.1 (Fig. 5(a)). Tuning the À / ratio (Fig. 5(b)) then revealed an inverse relationship between confidence calibration (r pert.comt) and prediction performance ('predttuhh ACC tres.,tthh) leading us to select \ 0.01 (so \ 10-7 for aa 10-5).
Conclusion and Discussion
Conclusion. We introduce PRESCRIBE, a novel uncertainty-awant framework for single-cell response prediction. By employing a multivariate Natural Posterior Network estimate the pseudo E- distance for each perturbation category, our model delivers well-calibrated, instance-level uncertainty. This metric acts unified surrogate for both aleatoric (data) and epistemic (model) uncertainty, and its application demonstrates practical utility by improving overall predictive accuracy.
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 to guide the optimal selection of reference state, enabling the model to learn the specific effects of perturbations more casily along the pronounced direction of change.
Future Work. Recent research (3
10

---

6 Acknowledgment
This work is supported by ST12030-Major Projects 20222ID0112400, the National Natural Sci- ence Foundation of China Project (No. 623B2086), TeleAI, STCSM grant 24510714300 and 20DZ2254400, GuangDong Basic and Applied Basic Research Foundation grant 102311515120066 and SJTU Science-Medicine interdisciplinary grant 24X010301456.
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13

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Appendix Overall
Notation Table
15
Default Settings
16
Algorithm
17
Computational Complexity Analysis
17
Preliminaries
17
E-Distance
17
E.2
Bayesian Posterior Update
18
E,3
Deep Evidential Regression.
18
F
Derivation and Proof
19
Derivation L
19
F2
Derivation of Lg
20
Derivation of L
21
Approximation of Multivariate Gamma and Digamma Function.
22
F5
Proof of Ranking Order Preservation
22
F6
NatPN
22
Dataset Details
23
H Baseline Setting Details
23
AverageKnown
24
2 Linear
24
H.3 Linear-GPT
24
scGPT
24
scFoundation
24
H.6 CellOracie
25
GEARS
25
H8 Variants OF PRESCRIBE
25
Expected Calibration Error Results
25
J Limitations
25
Broader Impacts
26
NeurlPS Paper Checklist
27
14

---

A
Notation Table
Table Notation and Descriptions for Key Variables in PRESCRIBE
Notation
Description
Mean value
Σ
Covariance matrix
Number of perturbation types
X
Genetic perturbation
Post-perturbed transcriptomics expressions
Pre-perturbed transcriptomies expressions
P
Exponential family distribution
Q
Conjugate prior
N
Normal distribution
Variance
W
Wishart distribution
W-1
Inverse Wishart distribution
A
Precision matrix
N
Rank of gene-gene interaction matrix Hidden dim Latent dim
d
S
Token count in scGPT embedding
Ho, v, KS, L.
Parameters in NTW+; L: lower triangular matrix
V
NIW parameter: Simplified NIW parameters: {uo, Ki L}
3 -
n
Evidence (data observations)
X
Sufficient statisties for NIW
E
Pseudo E-distance
E
E-distance (energy distance)
B
Batch size
III
Entropy
Digamma function
r
Gamma function
0a
Encoder learnable parameters
8
Decoder Icarnable parameters
0
Flow model learnable parameters
L1
Expected log-likelihood
C2
Entropy regularization
Ca
E-distance supervised loss Uncertainty-regularizee loss
C4
X1-3
Weights for C2. La. L4
Tpredaruh
Pearson correlation (predicted vs. true log fold change)
ACC ACCpred.uuth
Directional accuracy (predicted
true log fold change)
Pearson correlation between E and *
Pperf,conf
ACC₂
Quintile classification accuracy (É x*)
Spearman correlation (E: vs. x²*)
perf.conf
NNWW Normal Inverse Wishart distribution; *x defaults PCC, unless specified.
15

---

B Default Settings
The PRESCRIBE framework was trained single NVIDIA RTX 4090 GPU. We used the Adam 24 optimizer with an initial learning rate of 10 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 L1 of the validation set did not show improvement for three consecutive epochs. dynamic learning rate scheduler "plateau" reduced the learning rate by 1% when the relative L validation improvement remained below 0.01 % for two consecutive epochs. The implementation leverages PYTORCH LIGHTNING 25 2.4.0's modular architecture.
Table 6: Experiment Configurations.
Hyperparameter
Value
Flow layers
10
Model Settings
Flow size
0.774231384
Flow hidden
2
Bound
30
D
64
N
10
Warm up epochs
5
Warm up learning rate
0.001
Optimizer
Adam
Scheduler
Plateau
Scheduler change step
2
Training Details
Scheduler reduce rate
0.99
Patience
3
Learning rate
le-4
Total epochs
50
Weight decay
le-5
B
4096
Gradient Accumulation
4
A1
le-7
Aa
0.1
A3
le-5
16

---

C
Algorithm
The optimization process of PRESCRIBE can be summarized in the following algorithm
Algorithm 1 Training Process of PRESCRIBE
1: Input:
2:
Perturbation types: X;
3: Condition information: c, x
4:
Prior evidence: yprior;
5: Post-perturbed transcriptomics expressions: y;
6: E-distance of training samples: E;;
7: Initialize:
8:
0 {0a. Ow 03} < initialize network parameters;
9: repeat
10:
< random mini-bated from X;
2 (a,, c): Encoder;
12:
< fo(=)); // Flow;
X. {x1, X2) < fa(z;); Decoder;
14:
// Bayesian Posterior Update;
15:
Xi < compute through Eq. 4
up post V + compute through Eq.
17:
post
+ 2 v.
( x +4xx),
X
M. Motee
19:
Mor.
post
X1
pant
post
Cholesky ((x post (xF pun) x (2 x viou
21:
L + compute through Eq.
22:
I/ Update parameters according to gradients;
23:
0 с 0 VoL;
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:
1. Encoder: The overall computational complexity for the Encoder is O(d (Sdd-S$-) G +++++++ D)). This derived from the complexities of its constituent functions:
fu: O(S d d)
f12: OGG d
f1:: O(d d
fe: O(d-dd+-
fo:
2. Flow: The complexity associated with the Flow component is O(D).
3. Decoder: For the Decoder, which takes input of dimension D and produces output of dimension (N (N + 1)/2 + N), the computational complexity is O(D. N2).
E
Preliminaries
F.1
E-Distance
E-distance [66 statistical metric for assessing high-dimensional distributions, offers measures of the signal-to-noise ratio within dataset. popularly referred as energy distance, originally derived from the concept of gravitational energy in physics. In essence, E-distance determines
17

---

whether two distributions are distinct by comparing the distance between them pairwise distance- to the variability within each ses-uttoonnssf--dsttneeee" The illustration can referred to in FV) Applying E-distance in single-cell perturbation prediction task was introduced by scPerturb [ 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:
E(X, == 26xx ox oY:
(11)
where ox and dxy are the self-distance within and pairwise distance between distributions:
N
ox N(N
cل - -x-xzl1
(12)
6XY
1 NM
Σ
| Byll.
(13)
Typically, scPerturb the control status X, aligning with most deep learning models' assump- tions. Our observation (Fig. ((b)) suggests an apparent positive correlation between E-distance and predictive quality. However, since E-distance needs 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:
Ply w) hly) exp (0 uly) A(w))
(14)
@ww x. n(x: n2) exp (nw" nA(w)).
(15)
Q (w x n m exp
(n non -n0
(16)
The parameters w. X. and n correspond to the target distribution, prior distribution, and evidence, respectively. The posterior parameters can be updated as follows:
pe prir S u(ys)
X X
(17)
Post
prior
M
The method nature posterior network 99] has leveraged this framework 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 be expanded to the multivariate Gaussian distribution, which needs to be theoretically complemented in corresponding derivation 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 Normal Inverse- -iishart (NIW) distribution, which serves as conjugate prior. In this setting, the target y E R is considered to be independently and identically distributed (i.i..)) from a multivariate Gaussian distribution with an unknown mean μ and covariance matrix Σ. The model aims to predict not only the mean B[ja], but also the aleatoric uncertainty E| and epistemic uncertainty var [H]). These predictions can be expressed as follows:
E[E]
Eip Mo-
E v - - LL", var(u]
(18)
1/2x
To generate these predictions, this method optimizes the analytical solution the likelihood function:
P(y. wx)= y Poss 2 WT Lts,, ,,,,,,
(19)
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".
18

---

Derivation and Proof
This section includes theoretical derivations and proofs of PRESCRIBE. First, F.1 provides the derivation for the expected log-likelihood (L1) of multivariate Gaussian distribution under Normal- Inverse- Wishart prior. The derivation for the differential entropy of the Normal- -iisart distribution offers justification for how Eq. 3.4 mitigates the issue of zero
(C2) is presented in F.2 F3 gradients in regions of high uncertainty. Furthermore, a multivariate approximation for the Gamma and Digamma functions derived $ F.4 Moreover, this section also presents 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
Expected Log-Likelihood L1. L1 optimizes expected log-likelihoode under predicted posterior w, improving accuracy and reducing uncertainty. It also affects other data points' evidence due to flow normalization. Its closed form is follows:
N
In(2
1 (In 12 L. VN
(1)) 2
(20)
2
2
1
(v, -Mor) (y, MON.) )² vi L.
2
2ve
For stability, (x) is approximated by Σn In( 1----1 (Appx. F.4
The Normal-Inverse- Wishart (NIW) distribution μt, Σ w- w- (Mo: K; v, M is the conjugate prior of the normal distribution y [J,, )). Note that both parameters k and can be viewed as pseudo counts. However, PRESCRIBE enforces a single pseudo-count n corresponding
2v.
Target Distribution.
The density and the entropy of the Multivariate Normal distribution are:
N(H, Σ) exp 1x T (x 14)
(21)
((2T) E
N 2 In(2 + In E
H Σ)]
(22)
Conjugate Prior Distribution.
The density of the NIW distribution is:
-
W w-1 (m, E Mo. R, 1 = =N
Mo:
w-1(>
w
1 V.)).
(23)
//2
w w v)
exp
1r
(Σ- wr)
(24)
2
The full version of the probability density function is:
K: N/2 TT 1/2 Σ
{ - T - (4- Mn) Σ (E Mn)
1
w-1 (H,E) Mgg K, V
exp
(2T) N/22 5
(25)
where Ny is the multivariate gamma function and Tr(-)) is the Trace of the given matrix.
19

---

Expected Log-Liken Log-Likeiihood. The expected likelihood of the Multivariate Normal distribution N(H, Σ) under the NIW distribution W W Ki, 24 4 is:
E),,E)-W- 1 20, v)
[In N(y
p.,
Σ)
(26)
N
1 ln(2 In | (y: - 11) E (y: - 14)
= E
(27)
2
2
E
[y:
11) Σー (y: (y:-1) m)] E In Σ N Nln(2m)
(28)
(y: 2²) (y: n) E |]] + E In |ー
E
Tr
N In(2T)
(29)
(-y y:E
[ + y E [ 1 + y;ll - - E [u μ 1 21)
2
+ 2 2 + d (5) Nln (2x)
1
(NNn2
(30)
2 (v: -00 (y: - 1o) v² LL T + 2v
1
I + In (2vLL" + thn (/) N Vln(2x)
(31)
Here un (-) denotes the multivariate digamma function. In PRESCRIBE's formulation, we can obtain
L vLL T and the moment of the NIW distribution E mnΣ 'I
Mo E [ [-- vLL".
p Po vL
2/2 I, and the moment of the Inverse Wishart distribution
In In | 1hN (5)
T
E In
E
Inn
(32)
2
2
F.2 Derivation of C2
Entropy Regularization Cz. L2 is an entropy prior favoring high-entropy posterior distributions
Whest Its closed form:
N
(5)
In
L.
+
+IIN
2
1/4 +
N
+
1//
D&N
UN
+ 2
(33)
2
Approximations for stability (Appx.
F.4 xp((x) ~ x In(x)) 1 and In x N(N-1) In in(2T)
- (In 2n - (x + 1 1 n) + (z n In( ) ). In this section, F.1 gives the derivation the expected ligelikelihood of a multivariate Gaussian distribution under Normal Inverse Wishart prior
20 F2 presents the derivation of the differential entropy of the normal Wishart distribution Eq 33 justify that Eq (3.4 can avoid zero gradient in highly uncertain areas. Lastly, we derive an approximation of Gamma and Digamma in multivariate version F44 Before deriving Eq 33 we first introduce two propositions as follows:
Eq
In
Proposition 1. For A W(V, v) and any positive definite matrix A RNN,
= =
(34)
Proof. [[(r(AA)] = (r(E[A])) v AV).
Proposition 2. For E ~ w- (LL v) and any positive definite matrix A E RNN,
El E-1
(ArEE-A)] A)] = v (r
(35)
Proof. By definition, Σ 1
W(K-1 so according to Proposition
we can yield:
Ett(EE-AA A)] = // /rr(( ( - A).
20

---

The Differential Entropy of the Inverse Wishart Distribution. given Eq 24 the Inverse Wishart differential entropy is:
Using the Inverse Wishart density
H
Σ
E
In W
Σ
L
(36)
V
E
E
1
vN
(2)
V N
In
+
In2
ININ (2)
+
E In V
(37)
2
2
2
2
V'
In
|
vtr
V
V
VN
In +
InIN
N +1
In
N In 2
NN
1/
2
2
2
2
(38)
(6)
N
+1
In
V
+
PN
In
++
N
(
V
(39)
2
where (a) uses Proposition
and Eq 32 and in (b) used tr y -1 v/) tr()) = NN and simplified.
F.3
Derivation of L4
Zero Gradient Regions. Research has shown that when the evidence approaches zero, the gradient will also become zero, resulting in stopping optimisation.
Proposition The model cannot learn from samples high uncertainty areas by optimizing L1.
Proof. Given the parameters of the flow module, we can obtain Ov as Eq Therefore, the gradient of the loss function L1 with respect Pv given by:
0C1
0L1
Bossss
Op,
Dupon
Opv
ILi
eP
N
(40)
Öypan
NN + uprior32
QLi N.
(41)
Sypan
ePrtNu 2yprior vpriore e (no+NH)
In a high uncertainty areas, + 000 (a) + 0, so it leads to zero gradient with respect to P The model cannot learn from samples in these regions.
Then, we prove that the effectiveness of the proposed L4 can be supervised not only by pairwise distance but also by avoidance of the 7ero gradient scenario.
Proposition 4. La can help the model learn from samples within zero- zero- gradient regions.
Proof. Through Eq we can obtain yoo
In( Pu-N
urtoor) Then we can derive the gradient of
2N -p
C4 with respect to pa is given by:
0L4
0Ls dv
Opr Jupost Opr
(pr N) (2N -py) N (p. - N) (2N Pv)
N
1/ Mozil
(42)
N2
yy:
Moz.
* 0.
(43)
The gradient remains non-zero when P. + -0, which avoids zero gradients in these areas.
21

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F.4 Approximation of Multivariate Gamma and Digamma F Function.
The computation of a distribution's '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 equívalences [31]
In (x)) ~
In 2T
x +
I
In J
(44)
2
2
(xx) =
+
~
In
(45)
2x
According to the definitions of the multivariate Gamma functions and the multivariate Digamma Function:
InIN(()= N(N
1)
N
I 11x+1-n1 )
In(2T)
Σ
In
(46)
4
N
1)
(N(
(47)
2
n=1
we can obtain multivariate approximation versions follows:
In [x)
N(N 1)
In(2
Σ
N
2n - (x + n) + (x
n) In( x = -) (48)
(+))-))
4
N
I 12
1b (x)
~
In
(49)
2
F.5 Proof of Ranking Order Preservation
Theorem 1. Under fixed prior conditions, È preserves the ranking order of E.
Proof. Under min-max normalization with fixed control states, derive: F. 25xx - -Y, where 5XY, Y E [N,2N] 2NJ. Through model design constraints: distance from prior v x 6xy and target entropy H x Y. Let adxy and H = bY with a, b > O. Then:
= 2 alōxx N) MY-N-
aN
bN
2 5xy N
Y
N
N
25xx Y N
N E-1
N
Let C' -1. Then É 3/2 E' + C. Since is a positive constant (as 8xy, E: [N,2N] implics
N > 0), N is a positive constant. This linear relationship. É = kE + C where k = N 0, preserves
ordinal rankings between E and E. If E₁ < Eg, then EEi < kEg, and KE, < &E2 + C, which implies En < Eg. Thus, the ranking order preserved.
F.6 NatPN
The authors of NatPN 19 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 {QI) be the set of linear regions associated to the piecewise ReLU network fa(x). For any D., there exists 8* E R and l" {1...., RR such that öx Qi- for all 8> 5" [2]
Theorem 2. Let NatPN model parametrized with (deep) encoder fo with piecewise ReLU activations, a decoder g6 and the density P(z]w). Let fa(x) ((0)z + !!)) be the piecewise affine
22

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representation of the ReLU network f on the finite number of affine regions Q (32]. Suppose that Vli have independent rows and the density function P[(]w) has bounded derivatives, then for almost any a we have ( x)(w) + O. Le the evidence becomes small far from training data.
64200
Proof. Let E EP be a non zero input and fo be ReLU network. Lem. I implies that there exists 8" E It+ and E {l,.... , R} such that 8- Qt)) for all > 8*. Thus, 2s = fo x) = 5 -(VV)) + a for all 8 8 Note that for E "*"+oo], 26 follows an affine half line S = {z|z = ô (V()) + a 8 > 8*} in the latent space. Further, note that V o and || + +oo since a + and (()) has independent rows.
50000
Our model adaptation did not conflict with its architectural prerequisite.
G
Dataset Details
Our datasets are downloaded from scPerturb 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, imple- mented through the standard preprocessing workflow of Scanpy [3
Identifying Differentially Expressed Genes (DEGs). We identified DEGs by comparing perturbed cells control cells for each perturbation. Genes ranked based adjusted p-values computed using the Wilcoxon test in Scanpy, with those v-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 pea'].
E-distance Calculation. The E-distance for each perturbation was calculated to quantify the effect of the perturbation. E distance is statistical measure of distance between two distributions, which utilized 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 their greatest common divisors to perform the calculation. The distance e, pairwise distance, and self-distance are saved in adata.obsm, (nnmm")"), adata.obsm| d'' and addata.bsmm s'), respectively. For normalization, we perform the max-min normalisation on all the training sets' E distances onto [N, 2N..
Spilt Dataset. The filtered perturbations were divided into training, testing, and validation sets according the default operation in GEARS [ The training set was used fit the models, the validation set 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.
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Table Summary Statistics of Different Datasets
Norman2019
Replogic2022, Repl
Replogke2022
E-distance
59.172741
135,737438
31,014967
Number train/vallted
8071 22475866009
121242/1 4910/57863
133412/159344114
Type of trainhvaliest
213/32/52
1036/115 15/364
754/82/272
Deseription
Only douille gene knock-out perturbation
Only single gene knock-oot pestorbation Onlyy single gone knock-out perturiation
UMI Count
Cell Type
KS62
RPEI
KS62
H.1 AverageKnown
AverageKnown is a fundamental approach that averages gene expression across all cells within known perturbations predict unseen perturbations.
H.2
Linear
The linear method is derived from a benchmark study (20). The optimization objective is given by
Y - -GGP( + +b) b) 1|
arg min W
(50)
where Yimii 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 Yumiin P is subset of G that includes the perturbed genes in the training data. The fitted matrix is expressed as b N Σi= y where 6 the vector of average gene expressions from Yimin The matrix W is solved using the normal equations:
W (G" AI)
GT (yttain - b) P
(51)
Here, we set k to 512 and À to 0.1. The fitted W is then used for prediction:
Y b GWPT.
(52)
where Y represents the predicted gene expression, and P denotes the subset of that corresponds to the embeddings of perturbed genes in the testing data.
H.3 Linear-GPT
Linear-seGPT (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 the feature set. Similar to GEARS, trained the scGPT model for 20 epochs and selected the performinnongggg model based validation results for testing. For zero-shot transfer and cell state transition scenarios, the scGPT model was trained combined dataset comprising total of 17 datasets, ensuring broader generalization across different biological contexts.
H.S scFoundation
scFoundation 15 also employs transformer-base architectures, making it suitable for in silico perturbation tasks, similar to the training paradigm scGPT. Specifically, we leveraged the encoder module of the scFoundation model to train the perturbation model. The training and testing procedures mirrored those scGPT. Since scuuddationn incorporates more transformer layers than scGPT, we set the learning rate to 5
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H.6 CellOracle
We used the code provided on the CellOracle GitHub repository. Since our control group data contained significantly fewer cells compared to the tutorial dataset, we reduced 'n jiter's' from 100, 000 1, 000 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 data to align with GEARS' input requirements. The model was then trained using its default parameter settings, with the number of epochs to 20. The best-performing model, selected based on performance on the validation dataset, was used to evaluate the test dataset.
GEARS-Drae 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:
Th
Lame
1 T
Σ k=1
exp (-8w) (g.. g..) )
(2+%)
(53)
where Su is the predicted post-perturbation scalar and & is the variance, 8a log o
W wil h pert-pert + bune
GEARS-EnS. We ensemble GEARS-Drop by employing different training/validation splits and evaluating 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 the control status's mean and covariance 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 model's untrained state.
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-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.
The Ours-MLI variant based on Ours-Drop but incorporates an additional mean squared error loss term. It uses two linear layers with ReLU activation function to regress the E-distance from the latent embeddings.
Ours-MLPs.
I Expected Calibration Error Results
We evaluate the uncertainty calibration of model using the Expected Calibration Error (ECE). As shown in Tab,8) PRESCRIBE consistently achieves lower ECE scores across all three datasets compared the baselines, indicating better-calibrated uncertainty estimates.
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.
25

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Model
Norman
Rep1
KS62
PRESCRIBE (Ours)
30.59
50.38
41.47
GEARS-unc
32.45
63.96
64.76
GEARS-Ens
38.63
53.85
43.07
Table 8: Expected Calibration Error (ECB), multiplied by 100. Lower values indicate better calibra- tion. Our model, PRESCRIBE, compared against two baselines.
Continuous Data Assumption. We assume continuous Gaussian space, while single-cell data is often represented as discrete counts. A potential solution is to use an autoencoder (e.g., Sc [34] to project count data into 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 practical approach, 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 crucial direction for future research.
K
Broader Impacts
Our framework has broader applicability beyond its current focus 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-of- distribution inputs, and the training data is often derived from experimental observations subject to significant inherent uncertainty. By providing unified approach to quantifying both epistemic cout-of-distriuuiin and alcatoric (data) uncertainty, our method can enhance the reliability and trustworthiness of machine learning models in these critical scientific applications.
26

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Please refer to our LLM policy (https://neurips /att:p///earrrppccc/ooooeereeee/ cc////ooeeeeeecs//22/////.. for what should or should not be described.
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Reviews:
[{"review_id": "XiQEYTE1lS", "reviewer": "Reviewer_YtU5", "summary": "This 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.", "questions": "- 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.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I already gave the paper an accept rating and the authors were able to clarify on my concerns.", "rebuttal": "Dear Reviewer YtU5,\n\nThank you for all your helpful comments. The following are our responses.\n\n### **Metrics and Their Descriptions**\n\n| Metric | Purpose | Description |\n| :--- | :--- | :--- |\n| **PCC1** | Performance | The Pearson Correlation Coefficient between predicted and true log fold changes for genes under a specific perturbation. |\n| **ACC1** | Performance | The accuracy of predicting the sign (i.e., up- or down-regulation) of the log fold change. |\n| **ACC2** | Calibration | The accuracy of using our uncertainty metric to classify predictions into quintiles of final quality (PCC1). |\n| **SPCC** | Calibration | The Spearman Rank Correlation between our proposed uncertainty metric and the final prediction quality (PCC1). |\n\n> **Q1. 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(1) We conducted experiments across all datasets, with the exception of the N-increasing analysis, generalization analysis, and ablation study. The following are the reasons and supplement experiments.\n\n(2) We supplement the N-increasing analysis for all datasets here. \n\n| Cell Count (Rpe1) | SPCC |\n| :--- | :--- |\n| 300 | 53.92 |\n| 400 | 61.90 |\n| 500 | 64.85 |\n\n| Cell Count (k562) | SPCC |\n| :--- | :--- |\n| 300 | 50.32 |\n| 400 | 65.85 |\n| 500 | 70.90 |\n\n(3) For the generalization analysis, we focused on the Norman dataset because it involves double knock-outs, which allows us to mimic different degrees of generalization. The other datasets involve single knock-outs, for which the degree of generalization is harder to define. Therefore, for those datasets, we provide the average training and test set confidence scores instead. \n\n| Model (Rpe1) | Seen | Unseen |\n| :--- | :--- | :--- |\n| GEARS (uncertainty)| 29.47 | 25.97 |\n| Ours  (confidence)| 15.70 | 8.76 |\n\n| Model (k562) | Seen | Unseen |\n| :--- | :--- | :--- |\n| GEARS (uncertainty)| 36.24 | 31.61 |\n| Ours (confidence)| 18.10 | 14.35 |\n\n(4) For the ablation study, constrained by time, we only provide an ablation on the loss function here. \n\n| Model (Rpe1) | PCC1 | ACC1 | SPCC |\n| :--- | :--- | :--- | :--- |\n| Ours | 59.18 | 67.36 | 28.69 |\n| w/o L2 | 58.86 | 63.72 | -1.49 |\n| w/o L3 | 8.50 | 51.83 | 5.75 |\n| w/o L4 | 8.50 | 51.83 | 5.75 |\n| NO-INFO | 58.89 | 68.72 | -5.00 |\n\n| Model (K562) | PCC1 | ACC1 | SPCC |\n| :--- | :--- | :--- | :--- |\n| Ours | 36.20 | 60.27 | 12.43 |\n| w/o L2 | 35.57 | 60.74 | -11.09 |\n| w/o L3 | 10.96 | 52.38 | -5.13 |\n| w/o L4 | 36.14 | 60.56 | 10.00 |\n| NO-INFO | 36.00 | 61.34 | -1.00 |\n\n(5) The hyperparameters are different from those for the Norman dataset, but the tuning pipeline remains the same.\n\n> **Q2/W1.2.In the presentation of the method in 3.1, the author(s) fix the proportion between, and (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(1) This modeling choice is based on previous research[1]. \n\n(2) The Problem (Ambiguity): The original loss function ($\\mathcal{L}_{1}$) does not depend on the individual parameters that control uncertainty ($\\kappa$ and $\\nu$). Instead, it depends on a *combination* of them. This means that infinitely many different combinations of these parameters can produce the exact same loss value, making it impossible for the model to learn the correct, unique uncertainties. This issue is called \"degeneration\".\n\n(3) The Solution (Fixing the Ratio): To solve this, the authors propose to link, or \"couple,\" the two parameters with a constant ratio, $r$, such that $\\nu_{i}=2\\kappa_{i}$. This removes the ambiguity and eliminates the need for a separate, flawed regularization term that was used in prior work.\n\n> **Q3.How robust is the model to misspecification of its Normal-Inverse-Wishart distribution? It may be nice to know how strong the regularization of the 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 the limitations and strengths of this paper.**\n\nThis is useful advice, and we conducted the following experiments in response. \n\nIn single-cell preprocessing, data is usually first log-normalized. Although raw count data is very sparse, it sometimes contains a few very large counts. Thus, we first normalize to increase differences, then take the logarithm to reduce the long tail. In this way, single-cell data can be treated as having a somewhat Gaussian distribution. However, as you suggested, in a non-Gaussian setting, we did not perform log-normalization. Instead, we used min-max linear normalization, which yields a very asymmetric distribution as our input. Other operations remained the same. We still use the Norman dataset as an example, and the following are the results.\n\nWe conducted two trials: the first uses a non-normal input and a non-normal E-distance, while the second uses a non-normal input and a normal E-distance.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG |\n| :--- | :--- | :--- | :--- | :--- |\n| Ours(no_normal) | 16.37 | 22.25 | 53.10 | 71.21 |\n| Ours(no_normal_signal_normal) | -47.71 | -15.32 | 27.01 | 36.55 |\n\n| Model | SPCC | ACC2 | \n| :--- | :--- | :--- | \n| Ours(no_normal) | -13.31 | 41.94 | \n| Ours(no_normal_signal_normal) | 10.98 | 41.94 |\n\nThe results show that our model cannot handle non-Gaussian distributions. We also note that using an E-distance derived from a normal distribution assumption deteriorates model performance in this setting. We think this may be because the E-distance is a distance calculation suitable for Gaussian distributions. This inspires the idea that the E-distance could be replaced with another distribution distance metric, but that is outside the scope of this paper. However, we sincerely thank you for your advice and will include this limitation regarding the distribution assumption in our paper.\n\n[1] Multivariate Deep Evidential Regression."}, {"review_id": "sRjhVe12Uu", "reviewer": "Reviewer_HczP", "summary": "In 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.", "questions": "1. 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.", "limitations": "1. 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).", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "Authors thoroughly addressed my concerns.", "rebuttal": "Dear Reviewer HczP,\n\nThank you for all your helpful comments. The following are our responses, following the order of questions, weaknesses, and limitations.\n\n> **Q1/W1.Why did you focus on extending NatPN with energy distances?**\n\nOur motivation was to construct trustworthy perturbation models, using NatPN as it is efficient, flexible, and can capture both model and data uncertainty. We used the E-distance because previous perturbation-related research[1] suggests that the perturbation E-distance can reflect data quality, and we further find that it can efficiently discover unsatisfactory results, so we try to combine them.\n\nWe admit that the link with pairwise distance with evidence causes some confusion, as we want to incorporate them together. Actually, for broad ML users, they don't need to consider E-distance and L3; they can just use our extended multivariate NatPN. For perturbation models, we prioritize the utility, so we introduce pairwise distance and E-distance.\n\n> **Q2.The results of Table 3 are confusing - why random..**\n\nThis table aims to ablate the impact of filtering. Performance can be significantly enhanced by filtering out the few worst predictions, such as those with negative PCCs in Fig. 1(a). Therefore, as the number of filtered items increases, there is a higher probability of removing the worst predictions, which in turn increases performance. Thus, the stable trend is more significant.\n\n> **Q3.Why is it important to keep the evidence in the range [N, 2N]?**\n\nWhen the t-distribution degenerates into a Gaussian, the model behaves like GEARS, which only mimics variance. This means it tends to interpolate and underestimate epistemic uncertainty in missing or sparse areas of the perturbation space, e.g., for new perturbation inputs.\n\n> **Q4.Why the form of the encoder mentioned in section 3.3? Is it appropriate to assume additive effects?**\n\n1.  Model Design: The encoder module actually is flexible, requiring only a deterministic mapping from inputs (c, x) to latent embeddings z (Appx.F. 6, Theorem 2), so we take a simple and efficient one.\n\n2. We do not find that the E-distance has a significant relationship with the interaction type. Here are the mean E-distances for several common interaction types analyzed on Norman.\n\n| Interaction | E-distance |\n| :--- | :--- |\n| Additive | 62.02 |\n| Epistasis | 66.65 |\n| Neomorph | 19.32 |\n| Redundant | 176.76 |\n| Suppressive | 43.86 |\n| Synergy | 80.42 |\n\n> **W2.There is an application to only three datasets of one data and perturbation type, when ..**\n\nActually, our baselines, like scGPT and GEARS, all use these three typical datasets, so we followed the same convention. As for transfer performance, we consider two scenarios. \n\nThe first is perturbation transfer, such as predicting the effects of new genetic or drug perturbations; the adaptation is simple, replacing $x_i$ with the new perturbation embeddings. \n\nThe second is cell type or basal state transfer, which is trickier since different cell lines have very different systems. We conducted a trial where we held out some perturbations in one cell type and used another known cell type to predict them. The design is that two cell types share the same encoder but have different flow and decoder modules. We used the dataset from Nadig et al. (2025), which contains CRISPR screens in Jurkat and HepG2 cells as an example.\n\n**Results & Interpretation:**\nAs the table below shows, PRESCRIBE's uncertainty metric remains well-calibrated (SPCC > 0.38) in this transfer task. And by filtering out the 5-10% of predictions with the highest uncertainty, the performance on the remaining predictions consistently improves.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG | SPCC | ACC2 |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: |\n| Ours (PRESCRIBE) | 30.77 | 33.16 | 58.88 | 71.32 | 38.45 | 26.32 |\n| Ours-5% Filtered | 31.94 | 34.10 | 59.23 | 71.45 | - | - |\n| Ours-10% Filtered | 33.12 | 34.68 | 59.50 | 71.53 | - | - |\n\n> **W3.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.**\n\n(1) We think that one limiting factor is the calculation of the ground-truth E-distance itself, which requires a large number of cells to be genuinely accurate, a number often unavailable in real datasets. However, the consistent trend of performance improvement upon filtering (as shown in Table 2) demonstrates that our uncertainty metric has already been put into practical use.\n\n(2) Additionally, we updated more standard calibration metrics using Expected Calibration Error (ECE), a standard metric for calibrated regression (Kuleshov et al., ICML 2018). Lower ECE values indicate better calibration.\n\nThe table below reports the ECE (multiplied by 100) for PRESCRIBE and baselines.\n\n| Model | Norman | Rep1 | K562 |\n| :--- | :---: | :---: | :---: |\n| Ours | 30.59 | 50.38 | 41.47 |\n| GEARS-unc | 32.45 | 63.96 | 64.76 |\n| GEARS-Ens | 38.63 | 53.85 | 43.07 |\n\nWe will add the formal definition of ECE and a detailed discussion of these results to the revised manuscript.\n\n> **W4.Model choices that could have been insightful..**\n\nWe provide ablation studies below.\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG |SPCC | ACC2 |\n| :--- | :--- | :--- | :--- | :--- |:---: |:---:  |\n| Ours (scFoundation) | 57.79 | 56.86 | 63.35 | 70.34 |37.02 | 22.58|\n| Ours (scFoundation)-5% | 59.68 | 58.36 | 63.75 | 70.56 |:---: |:--:|\n| Ours (scFoundation)-10% | 61.46 | 58.63 | 64.47 | 70.65 |:---: |:---:  |\n| Ours (GNN) | 14.31 | 44.47 | 50.90 | 66.94 |15.15 | 16.13 |\n| Ours (GNN)-5% | 14.58 | 45.29 | 50.87 | 66.90 |:---: |:---:  |\n| Ours (GNN)-10% | 14.76 | 45.62 | 51.03 | 67.04 |:---: |:---:|\n| Ours(Set) | 57.16 | 56.15 | 63.00 | 68.33 | 12.50 | 16.13 |\n| Ours(Set)-5% | 57.40 | 56.61 | 63.31 | 68.55 | :---: |:---:  |\n| Ours(Set)-10% | 57.67 | 57.79 | 63.35 | 68.62 | :---: |:---:  |\n| Ours | 58.38 | 64.44 | 63.24 | 74.68 |67.19|38.71|\n| Ours-5% | 61.58 | 66.36 | 64.08 | 75.69 |:---: |:---:|\n| Ours-10% | 64.32 | 68.61 | 64.73 | 75.93 |:---: |:---:|\n\n> **W5.There is some confusion about the $\\lambda$ s in equation 9 ...**\n\n(1) We do not add a $\\lambda$ coefficient for $L_1$ because it is the central optimization function, and we wanted to reduce the hyperparameter tuning burden; other parameters are all smaller than 1.\n\n(2) We think the coefficient does not affect this core design: when $\\partial L_1/\\partial p_v$ approaches zero, $\\lambda_3 \\partial L_4/\\partial p_v \\neq 0$ when $\\lambda_3 \\neq 0$.\n\n> **Limitation The recent paper [1] explores the performance issues of many of the baseline models and datasets used in this paper.**\n\nAfter reading this paper, we agree that the reference frame is essential in perturbation models. We think your concern may be that the E-distance is only correlated with the old metrics. Therefore, we conducted a quick preliminary study where we used $\\mu_{all}$ to recalculate the GEARS metric and obtained the following results:\n\n| Model | PCC(new) |\n| :--- | :--- |\n| unc | 23.42 |\n| E-distance (new)| 56.05 |\n| E-distance (old)| 55.57 |\n\nWe find that although the new E-distance is more correlated with the new PCC, it does not dominate the strong E-distance signal with PCC(new), as the old E-distance is still highly correlated with the new PCC.\n\nAnd since the paper was published very recently, we do not have time to update all our research, but would like to include it in the future.\n\n[1] Peidli, S., Green, T.D., Shen, C. et al. scPerturb: harmonized single-cell perturbation data. Nat Methods.(2024)"}, {"review_id": "BvvCDV0ITD", "reviewer": "Reviewer_UDmR", "summary": "This 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.", "questions": "1. 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.", "limitations": "Limitations are not discussed in the conclusion at this point.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "- 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).", "was_revised": true, "final_justification": "I have increased my rating from a 3 to a 4 due to the author's additional experiments comparing their method to two baselines to compute uncertainty: GEARS (E-distance) and scGPT gene embeddings. \n\nThe advance seems real and the method intuitively seems compelling, but it is concerning that GEARS (E-distance) performs almost as well as their method on their nonstandard calibration metrics. And when the authors move to a more standard calibration metric (ECE), a worse version of GEARS performs almost as well as their method based on their discussion with reviewer tFuR.\n\nIf the authors sufficiently improve the paper's clarity---and during the discussion period, they did make it clear that they would at least try to---then I do still feel that this method is methodologically interesting not only for this application but potentially for others. It may not be as big an advance as originally advertised, but it still seems appropriate for a venue like NeurIPS.", "rebuttal": "Dear Reviewer UDmR,\n\nThank you for your thoughtful review and constructive comments. We have addressed each of your points below.\n\n**Metrics and Their Descriptions**\n| Metric | Usage|Description|\n|------|------|------|\n| PCC1 | Performance Evaluation| The Pearson correlation between the predicted and true log fold change under specific perturbations.|\n| ACC1 | Performance Evaluation| The directional accuracy between the predictive and true log fold change under specific perturbations.|\n| ACC2 | Caliberation Evaluation|The quintile classification accuracy between certain metric and PCC1|\n| SPCC| Caliberation Evaluation| The Spearman correlation between certain metric and PCC1|\n\n***\n\n### **Responses to Questions**\n\n> **Q1. How is the PCC calculated in Fig. 1? Is it across gene LFCs for a given perturbation?**\n\n**Yes.** This is a standard evaluation metric in the field. We use log-fold changes (LFCs) for two reasons:\n1.  **Biological Relevance:** Accurately predicting the *change* induced by a perturbation is more useful since the downstream study on the perturbation effect is conducted by comparison.\n2.  **Scoring Robustness:** Evaluating PCC directly on post-perturbation expression can lead to artificially inflated scores due to the high sparsity in single-cell data.\n\n\n> **Q2. Why should perturbation embeddings be additive in Equation 7? It seems like it should be a set transformer or something similar.**\n\n1.  **Model Design:** Our encoder in Equation. 7 models both an additive effect (first term) and a non-linear interaction effect (second term). This architecture was adopted from prior work [1] and chosen for its effectiveness and simplicity. The encoder module is flexible, requiring only a deterministic mapping from inputs (c, x) to latent embeddings z (Appx.F. 6, Theorem 2).\n\n2.  **Experiment with a Set Transformer:** Following your suggestion, we implemented a version of our model using a Set Transformer as the encoder. The results are presented below.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG | SPCC | ACC2 |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: |\n| Ours(Set) | 57.16 | 56.15 | 63.00 | 68.33 | 12.50 | 16.13 |\n| Ours(Set)-5% | 57.40 | 56.61 | 63.31 | 68.55 | :---: |:---:  |\n| Ours(Set)-10% | 57.67 | 57.79 | 63.35 | 68.62 | :---: |:---:  |\n\nOur detailed adaptation is \n\n$\\sum f_{a1}(x_i) + f_{a2}(\\sum f_{a1}(x_i)) \\rightarrow f_{set}(CONCAT[f_{a1}(x_1),f_{a1}(x_2),...,f_{a12}(c)])$.\n\nWhile this is a promising direction, the Set Transformer did not outperform our original model on this dataset, potentially due to the large data requirements for training a representative transformer.\n\n> **Q3. What is the intuition for setting $N_H=N$?**\n\nOur choice of $N_H = N$ is based on an empirical evaluation of four heuristics proposed in the NatPN paper. We tested these strategies on the Norman dataset to select the optimal setting for our model.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG |SPCC | ACC2 |\n| :--- | :---: | :---: | :---: | :---: |:---: | :---: |\n| Ours(exp-half) | 14.31 | 44.47 | 50.90 | 66.94 |17.88 | 16.13 |\n| Ours(constant) | 14.31 | 44.47 | 50.90 | 66.94 |15.15 | 16.13 |\n| Ours(normal) | 50.69 | 58.02 | 61.32 | 68.32 |62.90 | 29.03 |\n| Ours(normal)-5%| 52.32 | 58.68 | 61.86 | 69.14 |:---: | :---: |\n| Ours(normal)-10%|  54.60 | 60.19 | 62.61 | 69.63 |:---: | :---: |\n\nAs shown, the 'exp-half' ($N_H = e^{N/2}$) and 'constant' ($N_H = 1$) settings led to training collapse. The 'normal' setting ($N_H=e^{log(\\sqrt{4\\pi})N}$) underperformed the 'exp' setting ($N_H = N$) (Please refer Table 2 in the paper). We will include these analyses in the appendix. \n\n> **Q4. It is unclear what you are showing in Table 1(b).**\n\nIn Table 1(b), $N$ represents the number of cells sampled from the experimental data to calculate the \"ground truth\" E-distance. The purpose of this table is to show that the correlation between our predicted E-distance and the ground truth improves as the ground truth becomes more reliable (i.e., as $N$ increases). This suggests that the performance reported in Table 1(a) may be underestimated, as the ground-truth E-distance calculation itself is limited by the number of available cells in the dataset.\n\n> **Q5. How does an untrained model (PRESCRIBE-Null) have such good predictive performance as is shown in Table 2?**\n\nThe PRESCRIBE-Null model, while its network weights are random, makes predictions by leveraging the prior distribution, which is conditioned on the control cell profile. However, its performance is substantially lower than a more meaningful baseline like 'AverageKnown', which highlights the value of our fully trained model.\n\n> **Q6. Is Ours-MLP trained with the E-distance ranking loss? If not, it seems like it should be for a fair ablation.**\n\nThank you for this valuable point. For a fair comparison, we re-trained the 'Ours-MLP' baseline using the same E-distance ranking loss as our whole model. The updated results are below.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG | SPCC | ACC2 |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: |\n| Ours-MLP | 55.60 | 55.86 | 62.40 | 69.84 | 28.71 | 19.35|\n\n***\n\n### **Responses to Weaknesses**\n\n> **W1. The primary weakness of this paper is its clarity... the methods section... inundates the reader in equations...**\n\nWe sincerely apologize for this and will revise the manuscript to improve clarity.\n\n(1)&(2). In the original paper, we have compiled a comprehensive notation table in Appendix A. We recognize that overloading the symbol $N$ created unnecessary confusion (Q4), and we apologize for this oversight. In the revision, we will implement the following changes: The symbol $N$ will be used exclusively to represent the latent dimension. A new, distinct symbol (e.g., $N_{cell}$) will be introduced for the number of sample cells used in the E-distance calculation.\n\n(3). We will reframe the method to clarify our method: **to create a unified confidence score** that handles multiple sources of uncertainty.\n * **For seen or familiar perturbations**, E-distance is a powerful metric because it considers both prediction variance (like GEARS) and the predicted effect magnitude. (Fig.1 & Table 1), So identifying this case is our primary goal.\n * **For unseen perturbations**, our NatPN-based approach is designed to map these cases to a low E-distance, which constructs a common feature for them (Fig.3(a)), which is better than arbitrary prediction.\n\n    To demonstrate this works, we supplemented an experiment that shows our model's advantage at identifying these cases (PCC<0.2 & E>0). Analysis conducted on 600+ test points across three datasets.\n\n    | % of Data Filtered by Metric | 5% | 10% | 15% | 20% |\n    | :--- | :---: | :---: | :---: | :---: |\n    | Ours Discovery Rate (%) | 17.1 | 22.9 | 25.7 | 31.4 |\n    | GEARS Discovery Rate (%)| 8.2 | 12.2 | 16.3 | 24.5 |\n\n> **W2. It does seem to be the case that uncertainty estimates from their method are more calibrated... But it's not the case that they are calibrated particularly well either...**\n\nWe think that one limiting factor is the calculation of the ground-truth E-distance itself, which requires a large number of cells to be genuinely accurate, a number often unavailable in real datasets. However, the consistent trend of performance improvement upon filtering (as shown in Table 2) demonstrates that our uncertainty metric has already been put into practical use.\n\n> **W3. It is good to see though that there predictive performance... is quite strong. In fact, the paper seems to undersell how strong the predictive performance is.**\n\nOur primary focus was on validating our uncertainty quantification framework, so we emphasized the consistent trend of performance improvement upon filtering.\n\n\n> **W4. Since GEARS uncertainty estimates are so bad, I think two simple additional baselines should be included...**\n\nThank you for suggesting these excellent baselines. We have run both experiments.\n\n**(a) GEARS Predicted Effect Magnitude as Uncertainty:** We calculated the E-distance between GEARS's predictions and the control state to use as an uncertainty score.\n\n| Uncertainty Metric | SPCC | ACC2 |\n| :--- | :---: | :---: |\n| GEARS (Original Unc.) | 37.32 | 28.12 |\n| GEARS (E-distance) | 59.82 | 46.07 |\n\nThe results show this is indeed a strong baseline. We think in perturbation models (different from just a regression task). E-distance is a more comprehensive metric than variance alone. It is not a parallel concept to variance; instead, it's an inclusive one, capturing both variance (via self-distance) and effect magnitude (via pairwise-distance). The following correlations between these components and GEARS's original variance score support this:\n\n| E-distance Component | Correlation (SPCC) w/ GEARS (Original Unc.) |\n| :--- | :---: |\n| Self-distance | 10.15 |\n| Pairwise-distance | -92.38 |\n\n**(b) Embedding Similarity as Uncertainty:** We also evaluated using the cosine similarity between gene embeddings of test and training perturbations as a confidence score. We used the Rpe1 and K562 datasets, as defining similarity for the double knockouts in the Norman dataset is ambiguous.\n\n| Model (Rpe1) | SPCC | ACC2 |\n| :--- | :---: | :---: |\n| mean | 15.50 | 24.14 |\n| max | 19.83 | 22.02 |\n| median | 13.56 | 25.46 |\n\n| Model (K562) | SPCC | ACC2 |\n| :--- | :---: | :---: | \n| mean | 1.58 | 18.52 |\n| max | 10.77 | 22.96 | \n| median | 0.24 | 20.37 |\n\nThis straightforward similarity-based approach does not yield a well-calibrated uncertainty metric, highlighting the need for a more sophisticated method.\n\n***\n\n### **Responses to Limitations**\n\n> **Limitations are not discussed in the conclusion at this point.**\n\nYou can find limitations in Appendix J, where we discuss the model's reliance on the quality of perturbation embeddings and the challenges that remain in fully disentangling epistemic uncertainty.\n\n[1] SEASON COMBINATORIAL INTERVENTION PREDICTIONS WITH SALT & PEPER. ICLR (2024)"}, {"review_id": "XbiOvERiOV", "reviewer": "Reviewer_tFuR", "summary": "This 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.", "questions": "**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?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "The rebuttal addressed my initial concerns and included additional experiments that further support the proposed method. Critical points raised by Reviewer UDmR, such as the lack of ablation studies, were addressed with concrete results. With these new results and clarifications, and the commitment to include them in the camera-ready version, the paper demonstrates a strong contribution to the literature. I therefore maintain my positive score.", "rebuttal": "Dear Reviewer tFuR,\n\nThank you for your insightful comments and valuable suggestions. Our point-by-point responses are detailed below.\n\n**Metrics and Their Descriptions**\n\n| Metric | Purpose | Description |\n| :--- | :--- | :--- |\n| PCC1 | Performance | The Pearson Correlation Coefficient between predicted and true log fold changes for genes under a specific perturbation. |\n| ACC1 | Performance | The accuracy of predicting the sign (i.e., up- or down-regulation) of the log fold change. |\n| ACC2 | Calibration | The accuracy of using our uncertainty metric to classify predictions into quintiles of final quality (PCC1). |\n| SPCC | Calibration | The Spearman Rank Correlation between our proposed uncertainty metric and the final prediction quality (PCC1). |\n| ECE | Calibration | Expected Calibration Error. Measures the discrepancy between the model's predictive confidence and its actual accuracy. Lower values indicate better calibration. |\n\n---\n\n> **Q1. Scalability: 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\nWe benchmarked PRESCRIBE's runtime while varying the model's latent dimension, $N$. The results below show that the average batch time scales in a near-linear fashion.\n\n| Latent Dim. | Avg. Time / Batch (s) | Max Epochs to Converge |\n| :--- | :---: | :---: |\n| N | 261.53 | 15 |\n| 2N | 268.86 | 28 |\n| 3N | 272.31 | 32 |\n| 4N | 275.16 | 42 |\n| 5N | 340.66 | 50 |\n\n---\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 To supplement our standard benchmarks, we performed a new cross-cell-type generalization experiment.\n\n**Experimental Design:** We used the dataset from Nadig et al. (2025), which contains CRISPR screens in Jurkat and HepG2 cells. We created a transfer task where the model was trained on all perturbations in HepG2 and a subset of perturbations in Jurkat. It was then tested on the held-out perturbations in the Jurkat cell line. The model uses a shared encoder with cell-type-specific flow and decoder modules.\n\n**Results & Interpretation:**\nAs the table below shows, PRESCRIBE's uncertainty metric remains well-calibrated (SPCC > 0.38) in this transfer task. And by filtering out the 5-10% of predictions with the highest uncertainty, the performance on the remaining predictions consistently improves.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG | SPCC | ACC2 |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: |\n| Ours (PRESCRIBE) | 30.77 | 33.16 | 58.88 | 71.32 | 38.45 | 26.32 |\n| Ours-5%| 31.94 | 34.10 | 59.23 | 71.45 | - | - |\n| Ours-10%| 33.12 | 34.68 | 59.50 | 71.53 | - | - |\n\n---\n\n> **Q3 / W2. In the filtering-based accuracy gains, what are the characteristics of the discarded predictions?**\n\nLow confidence metric our model predicted induced from two uncertainty sources: \n\nFor seen or familiar perturbations, low confidence means either high prediction variance (like GEARS) (learnt by entropy) or low effect magnitude (supervised by ranking loss). (Fig.1 & Table 1).\n\n For unseen perturbations,  we expected the flow module to generate low evidence or likelihood so that it constructs a common feature for those epistemic uncertainties (Fig.3(a)) rather than an arbitrary prediction.\n\n\n> **W1. The paper is technically very dense... This may limit accessibility...**\n\nThank you for this feedback. We will revise the introduction and methods to improve accessibility. We will clarify the core contributions for different audiences:\n\n* **For the Machine Learning Audience:** They can ignore E-distance and prior settings to directly use a multivariate nature posterior network, which mainly focuses on estimating variance and epistemic uncertainties.\n* **For the Computational Biology Audience:** We will highlight the practical utility, thus we consider the magnitude of the effect by using the E-distance concept.\n\n> **W3. While Spearman correlation is used... no standard calibration metrics are utilized.**\n\nWe agree and have now benchmarked our model using Expected Calibration Error (ECE), a standard metric for calibrated regression (Kuleshov et al., ICML 2018). Lower ECE values indicate better calibration.\n\nThe table below reports the ECE (multiplied by 100) for PRESCRIBE and baselines.\n\n| Model | Norman | Rep1 | K562 |\n| :--- | :---: | :---: | :---: |\n| Ours | 30.59 | 50.38 | 41.47 |\n| GEARS-unc | 32.45 | 63.96 | 64.76 |\n| GEARS-Ens | 38.63 | 53.85 | 43.07 |\n\nWe will add the formal definition of ECE and a detailed discussion of these results to the revised manuscript."}, {"review_id": "oMwIyaFOuH", "reviewer": "Reviewer_Z96M", "summary": "The 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.", "questions": "- 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?", "limitations": "- 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.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 2, "originality": 1, "strengths_and_weaknesses": "**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.", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer Z96M,\n\nThank you for your constructive feedback. We have addressed each of your points below.\n\n**Metrics and Their Descriptions**\n| Metric | Usage|Description|\n|------|------|------|\n| PCC1 | Performance Evaluation| The Pearson correlation between the predicted and true log fold change under specific perturbations.|\n| ACC1 | Performance Evaluation| The directional accuracy between the predictive and true log fold change under specific perturbations.|\n| RMSE | Performance Evaluation| Root mean square error between the predictive and true log fold change under specific perturbations.|\n| ACC2 | Caliberation Evaluation|The quintile classification accuracy between certain metrics and PCC1|\n| SPCC| Caliberation Evaluation| The Spearman correlation between certain metric and PCC1|\n\n> **Q1/W1.1/W2.1. Gene Embedding Ablation Study**\n\nThank you for the suggestion. To clarify, the ESM model operates on amino acid sequences to produce protein embeddings, not gene embeddings, so a direct comparison is not applicable.\n\nHowever, to address the core of your question about the impact of the gene encoder, we conducted an ablation study comparing our model with three alternatives:\n1. GEARS-GNN embeddings.\n2. scFoundation embeddings.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG |SPCC | ACC2 |\n| :--- | :--- | :--- | :--- | :--- |:---: |:---:  |\n| Ours (scFoundation) | 57.79 | 56.86 | 63.35 | 70.34 |37.02 | 22.58|\n| Ours (scFoundation)-5% | 59.68 | 58.36 | 63.75 | 70.56 |:---: |:--:|\n| Ours (scFoundation)-10% | 61.46 | 58.63 | 64.47 | 70.65 |:---: |:---:  |\n| Ours (GNN) | 14.31 | 44.47 | 50.90 | 66.94 |15.15 | 16.13 |\n| Ours (GNN)-5% | 14.58 | 45.29 | 50.87 | 66.90 |:---: |:---:  |\n| Ours (GNN)-10% | 14.76 | 45.62 | 51.03 | 67.04 |:---: |:---:|\n| Ours | 58.38 | 64.44 | 63.24 | 74.68 |67.19|38.71|\n| Ours-5% | 61.58 | 66.36 | 64.08 | 75.69 |:---: |:---:|\n| Ours-10% | 64.32 | 68.61 | 64.73 | 75.93 |:---: |:---:|\n\n> **Q2/W1.2: Choice of Distribution**\n\nWe chose the Normal-Inverse-Wishart (NIW) distribution because our model, like the GEARS baseline, operates on preprocessed, normalized expression data, which is continuous. The NIW is the conjugate prior for a multivariate Gaussian likelihood, making it a natural fit for this data type. \n\nIn contrast, the Negative Binomial distribution is designed for modeling discrete count data. While adapting our evidence-based framework to a discrete likelihood is a good proposal for future research, it would require a non-trivial theoretical derivation.\n\n> **Q3/W2.2/W2.3. Encoder Ablation Study**\n\n1.  **Model Design:** Our encoder in Equation. 7 models, both an additive effect (first term) and a non-linear interaction effect (second term). This architecture was adopted from prior work [2] and chosen for its effectiveness and simplicity. The encoder module is flexible, requiring only a deterministic mapping from inputs (c, x) to latent embeddings z (Appx.F. 6, Theorem 2).\n\n2.  **Experiment with a Set Transformer:** Following your suggestion, we implemented a version of our model using a Set Transformer as the encoder. The results are presented below.\n\n| Model | PCC1 | PCC1_DEG | ACC1 | ACC1_DEG | SPCC | ACC2 |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: |\n| Ours(Set) | 57.16 | 56.15 | 63.00 | 68.33 | 12.50 | 16.13 |\n| Ours(Set)-5% | 57.40 | 56.61 | 63.31 | 68.55 | :---: |:---:  |\n| Ours(Set)-10% | 57.67 | 57.79 | 63.35 | 68.62 | :---: |:---:  |\n\nOur detailed adaptation is \n\n$\\sum f_{a1}(x_i) + f_{a2}(\\sum f_{a1}(x_i)) \\rightarrow f_{set}(CONCAT[f_{a1}(x_1),f_{a1}(x_2),...,f_{a12}(c)])$.\n\nWhile this is a promising direction, the Set Transformer did not outperform our original model on this dataset, potentially due to the large data requirements for training a representative transformer.\n\n> **Q4/W3.5: Comprehensive Non-Correlation Metrics (RMSE)**\n\nAs requested, we now provide Root Mean Squared Error (RMSE) metrics (multiplied by 100).\n\n| Model | Norman-RMSE | Rep1-RMSE |K562-RMSE |\n| :--- | :--- | :--- | :--- | \n| GEARS | 10.52 | 17.85 | 15.42 | \n| Ours | 10.16 | 17.27 | 11.63 | \n| Ours-5% | 10.05 | 17.14 | 11.45 |\n| Ours-10% | 9.99 | 16.89 |11.40 | \n\n> **W1: Originality and Significance**\n\nThe three major contributions of our work are\n1. Motivation Originality. Our paper tries to solve overconfidence in single-cell response prediction tasks, a crucial yet often overlooked limitation in previous research. \n2. Theoretical Derivation Originality. The derivation and application of a multivariate evidence-based framework for single-cell perturbation prediction. This includes the novel multivariate loss function and the associated parameter update mechanisms.\n3. Model Originality. The newly designed model is expected to achieve uncertainty-aware single-cell responses prediction and is expected to benefit by increasing the accuracy of predictive models.\n\n> **W2.3: Control State Encoding**\n\nWe apologize for the lack of clarity. The control profile $c_{i,j}$ for a given perturbation is the mean expression profile of all control cells in that experiment. We will state this explicitly in the revised manuscript.\n\n> **W2.4: Decoder and Normalizing Flow Parameterization**\n\nThe overall architecture is described in Lines 150-158, and hyperparameters are listed in Appendix B.\n\n> **W3.1: Ablation Study on losses and other datasets**\n\nThe $L_1$ term serves as the main regression target (like Mean Absolute Error for GEARS), so we did not ablate it.\n\nThe effect of the $L_2$ regularization term is studied in the hyperparameter search (Figure 5b), where setting $\\lambda_1 = 0$ corresponds to its ablation.\n\nThe other ablations on the other two datasets are shown below:\n\n| Rpe1 | PCC | ACC | SPCC |\n| :--- | :--- | :--- | :--- |\n| Ours | 59.18 | 67.36 | 28.69 |\n| w/o L2 | 58.86 | 63.72 | -1.49 |\n| w/o L3 | 8.50 | 51.83 | 5.75 |\n| w/o L4 | 8.50 | 51.83 | 5.75 |\n| NO-INFO | 58.89 | 68.72 | -5.00 |\n\n| k562 | PCC | ACC | SPCC |\n| :--- | :--- | :--- | :--- |\n| Ours | 36.20 | 60.27 | 12.43 |\n| w/o L2 | 35.57 | 60.74 | -11.09 |\n| w/o L3 | 10.96 | 52.38 | -5.13 |\n| w/o L4 | 36.14 | 60.56 | 10.00 |\n| NO-INFO | 36.00 | 61.34 | -1.00 |\n\n\n> **W3.2: The paper reports correlation metrics as percentages, which is unconventional.**\n\nRegarding the calibration plot (Figure 5), the y-axis and x-axis show the percentile rather than PCC values, so it is not in the [-1, 1] interval.\n\n> **W3.3: 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.**\n\n(1) We think that one limiting factor is the calculation of the ground-truth E-distance itself, which requires a large number of cells to be genuinely accurate, a number often unavailable in real datasets. However, the consistent trend of performance improvement upon filtering (as shown in Table 2) demonstrates that our uncertainty metric has already been put into practical use.\n\n(2) Actually, a large vertical error bar is a sign of good calibration. To conveniently visualize the data, we discretize continuous confidence scores and PCC values by binning them into percentile intervals, but variance within intervals is expected. Therefore, a better visualization might be to show 45-degree slanted error bars, which we are considering adopting.\n\nWe updated more standard calibration metrics using Expected Calibration Error (ECE), a standard metric for calibrated regression (Kuleshov et al., ICML 2018). Lower ECE values indicate better calibration.\n\nThe table below reports the ECE (multiplied by 100) for PRESCRIBE and baselines.\n\n| Model | Norman | Rep1 | K562 |\n| :--- | :---: | :---: | :---: |\n| Ours | 30.59 | 50.38 | 41.47 |\n| GEARS-unc | 32.45 | 63.96 | 64.76 |\n| GEARS-Ens | 38.63 | 53.85 | 43.07 |\n\nWe will add the formal definition of ECE and a detailed discussion of these results to the revised manuscript.\n\n> **W3.4: Pseudo E-distance Correlation***\n\nTable 1(b) shows that the correlation between our predicted E-distance and the ground truth improves as the ground truth becomes more reliable (i.e., as $N$ increases). This suggests that the performance reported in Table 1(a) may be underestimated, as the number of available cells in the dataset limits the ground-truth E-distance calculation itself."}]

Author rebuttal:

Meta-review: **(a) Summary**  
This paper introduces PRESCRIBE, a framework for predicting single-cell transcriptional responses to genetic perturbations while providing calibrated uncertainty estimates. The method extends the Natural Posterior Network (NatPN) to the multivariate case with a Normal-Inverse-Wishart prior, introduces a pseudo E-distance metric to jointly capture epistemic and aleatoric uncertainty, and evaluates the approach on three benchmark single-cell perturbation datasets. The authors provide ablation studies, calibration metrics, and transfer experiments, positioning PRESCRIBE as a practically useful tool for uncertainty-aware perturbation modeling.

**(b) Strengths**  
- Addresses a critical gap in single-cell perturbation prediction by quantifying prediction uncertainty, an important step for practical applications in biology and drug discovery.  
- Provides a theoretically motivated extension of NatPN to the multivariate setting and carefully derives a pseudo E-distance surrogate for uncertainty.  
- Backed by extensive experiments across multiple datasets, including calibration analyses, ablations on model components and embeddings, and a transfer task across cell types.  
- Author responses were thorough, with additional experiments (e.g., use of alternative embeddings, calibration with ECE, statistical significance tests) that strengthened the empirical case.  
- Reviewers generally acknowledged the technical soundness and practical significance of the contribution.

**(c) Weaknesses**  
- The paper remains technically dense and will require improvements in exposition for accessibility to both ML and biology audiences.  
- Some methodological choices, such as the Gaussian assumption and encoder design, could have been further motivated relative to alternatives like Poisson-based models or set transformers.  
- Calibration improvements, while demonstrated, are incremental in some metrics compared to strong baselines (e.g., GEARS with modified uncertainty scores).  
- Generalization beyond the three main datasets is not fully explored, leaving open questions about broader applicability.

**(d) Key reasons for decision**  
Most reviewers were ultimately supportive, with three (YtU5, HczP, tFuR) recommending acceptance and one (UDmR) moving from borderline reject to a borderline accept after the rebuttal. Reviewer Z96M maintained reservations, but the author rebuttal provided additional ablations, specifications, and statistical tests that addressed many concerns. The consistent trajectory of the reviews and the quality of the rebuttal indicate that the work represents a meaningful contribution despite clarity issues. The core methodological advance, uncertainty-aware prediction of perturbation responses, is judged to be significant enough to merit acceptance.

**(e) Discussion and changes**  
The author–reviewer discussion was constructive and led to concrete improvements. Reviewers’ concerns about clarity, scope of ablations, and calibration metrics were met with new experiments (e.g., encoder ablations, cross-cell-type transfer, ECE metrics, statistical tests). While Reviewer Z96M remained less convinced, others expressed satisfaction with the responses and confirmed that their concerns were resolved. The rebuttal strengthened the empirical case and clarified limitations, particularly regarding distributional assumptions and generalization. Overall, the balance of the discussion shifted positively, supporting an acceptance decision.

**Decision: Accept (Poster).**  
Overall justification: The paper presents a well-motivated and empirically supported contribution addressing uncertainty quantification in single-cell perturbation prediction, with sufficient strengths to outweigh remaining concerns about clarity and scope.
Decision: Accept (poster)