From Samples to Scenarios: A New Paradigm for Probabilistic Forecasting
Authors: Xilin Dai
Zhijian Xu
Wanxu Cai
Qiang Xu
Venue: ICLR 2026 Poster (2026)
Keywords: Probabilistic Time Series Forecasting
Probabilistic Scenarios
Time Series Analysis
Sampling-Free
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: Introduced a sampling-free paradigm, with which a linear model achieves SOTA against complex forecasting models.

Abstract: Most state-of-the-art probabilistic time series forecasting models rely on sampling to represent future uncertainty. However, this paradigm suffers from inherent limitations, such as lacking explicit probabilities, inadequate coverage, and high computational costs. In this work, we introduce **Probabilistic Scenarios**, an alternative paradigm designed to address the limitations of sampling. It operates by directly producing a finite set of {Scenario, Probability} pairs, thus avoiding Monte Carlo-like approximation. To validate this paradigm, we propose **TimePrism**, a simple model composed of only three parallel linear layers. Surprisingly, TimePrism achieves 9 out of 10 state-of-the-art results across five benchmark datasets on two metrics. The effectiveness of our paradigm comes from a fundamental reframing of the learning objective. Instead of modeling an entire continuous probability space, the model learns to represent a set of plausible scenarios and corresponding probabilities. Our work demonstrates the potential of the Probabilistic Scenarios paradigm, opening a promising research direction in forecasting beyond sampling.

Published as a conference paper at ICLR 2026
FROM SAMPLES TO SCENARIOS: A NEW PARADIGM FOR PROBABILISTIC FORECASTING
Xilin Dai Zhijian Xu Wanxu Cai
Qiang X
'ZJU-UIUC Institute, Zhejiang University
DDeparmment of Computer Science and Engineering, The Chinese University of Hong Kong "School of Software, Tsinghua University
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ABSTRACT
Most state-of-the-at aht probabilistic time series forecasting models rely on sampling represent future uncertainty. However, this paradigm suffers from inherent limi- tations, such as lacking explicit probabilities, inadequate coverage, and high com- putational costs. In this work, we introduce Probabilistic Scenarios, paradigm designed to address the limitations of sampling. It operates by directly producing finite set of (Scenario, Probability] pairs, thus avoiding Monte Carlo- like approximation. To validate this paradigm, we propose TimePrism, simple model composed of only three parallel linear layers. Surprisingly, TimePrism achieves 9 out of 10 state-of-the-eee art results across five benchmark datasets on two metrics. The effectiveness of our paradigm comes from fundamental reframing of the learning objective. Instead of modeling an entire continuous probability space, the model learns to represent a set of plausible scenarios and corresponding probabilities. Our work demonstrates the potential of the Probabilistic Scenarios paradigm, opening a promising research direction in forecasting beyond sampling.
alternative
Forecasting Beyond Sampling: A New Paradigm
Motivation
Solution
Evaluation
Limitations of Sampling
Probabilistic Scenarios
Proof-of-Concept
E.
Du.pi
To Show the Potential of the New Paradigm
TimePrism
(ii) Coverage
(iii Inference
(ii) Diverse
(iii)
v Simple Linear Model Built with Probabilistic Scenarios
Probability
(Scenario, Probabillt)
Single
Absence
Inadequacy
Cost
Scenarios
Forward Pass
Competitive against SOTAs
Figure I: Motivation, solution, and evaluation of this work. We illustrate the limitations of the prevailing sampling-based paradigm for probabilistic forecasting. In response, we introduce Probabilistic Scenarios, a new paradigm that directly produces set of (Scenario, Probability) ) pairs, and validate its potential with a simple proof-of-coneept model, TimePrism.
INTRODUCTION
Probabilistic time series forecasting is fundamental optimal decision decision-making under uncertainty, describes the likelihood of future outcomes (Gneiting & Katzfuss 20144] Hyndman & Athanasopoulos
202T). Although this problem has been studied extensively within the machine learning community, current approaches tend to rely on a predefined predictive distribution or sampling approximation (Kong et al.) [2025] Fang & Wang 2020 Lim & Zohren (02T). These strategies have led to three main categories of models: (i) Parametric Distribution Models, assumes that the predictive distribution
conforms a predefined parametric family, such as a Gaussian (Salinas et al. (020);; (ii) Generative
Models, such as diffusion-based models (Rasul al. 202I), which learn iterative process to
Equaa contribution. Corresponding author.

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generate samples from the latent distribution without explicitly defining its density function; and (iii) Structured Probabilistic Models, such as Flow-based Models (Rasul et al. |2220] Ashok et al. 2023). which learns continuous probability density field, from which trajectories are then sampled.
However, the reliance on sampling introduces challenges (Cortés et al. 2025). While the alternative of predefined distribution is not widely discussed by state-of-the-an methods for its evident inflexibility (Zhang et al.) 2024 Ashok et al. 2023), the sampling paradigm suffers from three primary limitations, as shown in Figure
(i) Probability Absence. The most significant issue is that the generated trajectories are not paired with their probability of occurrence. Although confidence intervals can be inferred from a large set of samples, this process is indirect, computationally intensive, and lacks the intuitiveness of a direct scenario ceenario-robbabllly mapping. (ii) Coverage Inadequacy. A finite set of samples may fail to represent low- now-probability, high-impact tail events. This is a critical failure for applications where preparing for rare occurrences is paramount, such as extreme weather or stock market volatility (Cortés et al.] 2025). (iii) Inference Cost. The process of generating multiple samples is often expensive, with costs scaling to the number of samples required. This expense exacerbates the two preceding issues in practical applications, limiting the reliability and utility of such forecasts (Chen& & Boccelli, 2018: Ashok et al., 2023).
To address the limitations of sampling- baaed forecasting, we introduce a new paradigm for proba- bilistic forecasting that term Probabilistic Scenarios. The objective is to produce, in a single forward pass and without reliance on sampling, a finite set of (Future Scenario, Probability) pairs that explicitly represents the predictive distribution. While some state-of-the-art deep learning models have made progress toward this goal, none have fully achieved it. Structured probabilistic models like TempFlow and TACTiSTTACTIS2 can compute probabilities, but only as a continuous density field (Rasul et al. 2020; Ashok et al. 2023; Drouin et al. 2022), not as discrete, interpretable scenarios; obtaining explicit trajectories still requires reverting to the expensive sampling. Meanwhile, al- though TimeMCL (Cortés et al. (025) produces a set of discrete scenarios, its optimization objective prioritizes scenario fidelity over probability matching.
To realize and validate the concept of Probabilistic Scenarios, we designed proof-of-conee model, TimePrism. As its name suggests, TimePrism processes the input history to generate discrete set of distinct future trajectories, which we term Scenarios, and concurrently estimates their likelihood to yield set of (Future Scenario, Probability] } pairs. Specifically, to validate the effectiveness of our paradigm, the architecture is intentionally kept simple. TimePrism is composed of only three parallel linear layers, designed end-to-en to directly produce Probabilistic Scenarios. Despite its simplicity, TimePrism achieves 9 out of 10 state-of-the-an (SOTA) results and one second-best result across five benchmark datasets on our two primary metrics.
Contributions:
We introduce a new paradigm for probabilistic time series forecasting. This paradigm addresses the limitations of sampling by reframing the learning objective from continuous probability space estimation to more structured task of learning a distribution over a set of scenarios For quantitative measurements, we establish evaluation framework with two complementary metrics and provide distinct but comparable formulations for both sampling- based models and our paradigm, serving fair standard for future research on Probabilistic Scenarios.
To show the potential of our paradigm, we introduce TimePrism, a simple linear model built within the Probabilistic Scenarios paradigm. Despite its simple structure, TimePrism still achieves competitive performance against SOTA sampling-based models, indicating a promising research direction for forecasting with Probabilistic Scenarios.
2 RELATED WORKS
Parametric Distribution Models employ neural network output the parameters of prespecified probability distribution (Wu et al.) 2020), with examples including DeepAR (Salinas et al. 2020))
GPVar (Salinas et al. 2019), and Mixture Density Networks (Li et al., 2024). The primary limitation of this approach is its reliance on a strong distributional assumption. Due to this inherent inflexibility, this approach is less commonly adopted in recent SOTA methods (Zhang al. 2024).
Generative Models represent the predictive distribution implicitly through a learned sampling process (Ho et al. 2020). For instance, Rasul al. (022) (TimeGrad) and Alcaraz & Strodthoff
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(2022) (SSSD) pioneered the of conditional diffusion models for probabilistic forecasting and imputation. Recent advancements have focused on adapting the diffusion process to the sequential
nature of time series (Gao et al. 2025 Bilos et al., )023). Another direction focuses enhancing
the conditioning mechanism (Kollovieh et al. 2023 Liu et al. 2025) and adding cross-modal visual information (Ruan et al.] 2025). Further adaptations include designing non-stationary processes (Ye)
itt al. 2025) and specializing models for tasks like imputation, such as CSDI (Tashroo et al. 1021).
Related work also employs Variational Autoencoders (VAEs), such as GP-VAE (Fortuin et 2020
While powerful, all these methods produce forecasts via an iterative sampling procedure and do not provide an explicit probability density for any given trajectory.
Structured Probabilistic Models are class of methods that learn an explicit, continuous probability density field over the forecast horizon, primarily including flow-hased models and copula-based models. Flow-based models learn a distribution transformation (Papamakarios et al., (022), with
recent applications in time series forecasting, such as TempFlow, using techniques like conditioned normalizing flows and flow matching (Rasul et al. 2020: Kollovieh et al.] (0224). Copula-based models, which have a long history in econometrics and finance (Patton| [2012] Größer & Okhrin 2022), construct joint distribution by copula (Salmon & Bouyé |008]] Bouyé et al. [2.0 Wang &
Tao 2020). With neural networks involved to model the copula (Wen & Torkkola 2019 Krupski
& Joel 2020 Mayer & Wied 2023 Toubeau et al.] 2019), recent research leads to fully neural, parmmeaaeiticc approaches like TACTiS (Drouin et al.) 2002) and its successor, TACTiS-2 (Ashok
ct al. 2023). Despite their different formulations, models in this category still rely on a sampling procedure, drawing from a continuous probability density field to obtain future trajectories.
Multiple Choice Learning (MCL) framework offers a practical path toward realizing our Probabilis- tic Scenarios paradigm (Cortés et al. 2025). MCL, introduced by Guzmán-riveraa et al. (2012). uses a Winner-Takes- (WTA) loss to train a multi-head network, where each head specializes in a different mode of the data. This approach has been successfully applied and extended in various domains, particularly computer vision and reinforcement learning (Lee et al. 2016 Rupprecht et al. 2017 7,
Tian et al. 2019 [Seo et al. [2220 Garcia et al. 2021). Recent work has further analyzed its geometric
properties and variants (Letzelter et al. 2024 2023 Perera ct al.) 2024). In probabilistic time series forecasting, TimeMCL (Cortés et al. (025) produces discrete set of scenarios. However, its score heads do not directly model a probability distribution. Consequently, to compute probabilistic metrics like Continuous Ranked Probability Score (CRPS), the original work resamples from its finite set of scenarios. The authors acknowledge their prioritization of scenario fidelity over probability matching. This design addresses Coverage Inadequacy to some extent, but fails solve Probability Absence. As reported in their work, this trade-off results in CRPS scores less competitive than SOTA models such as TACTiS-2 (Ashok et al. 2023) and TimeGrad (Rasul et 2021).
To transcend this trade-off, our Probabilistic Scenarios paradigm unifies scenario fidelity and probabil- ity matching, designed address all three limitations of sampling coherently; Probability A Absence. Coverage Inadequacy and Inference Cost.
3 THE PROBABILISTIC SCENARIOS PARADIGM
3.1 CONVENTIONAL FORECASTING PARADIGM WITH SAMPLING
We begin by formalizing the objective of probabilistic time series forecasting, Given historical context window of length L, denoted as x = (Z1,..... .I)) E RxD, where D is the number of variates, the goal is to predict the distribution of the future trajectory over a horizon T,, denoted as
y = (Vi,... )r) E R Rx The objective is to learn a model that captures the conditional probability distribution over all possible future trajectories:
P(y
(1)
Directly modeling this high-dimensional distribution is often intractable. Consequently, state-of- the-art sampling-based methods learn model, parameterized by 0, that represents this distribution, denoted as (y)))... A single forecast sample, y. is generated by sampling from this learned distribution in equation 2] The final probabilistic forecast is then represented by set of S such samples, ssumlee = {y: 1 where S is the number of samples. This set serves as an empirical Monte Carlo approximation of the true conditional distribution in equation This workflow leads to
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m
New Paradigm: Probabilistic Scenarios
m
Input
Unified Evaluation Framework
Limitations
Sampling
Metrics: Comprehensive & Fair
Inference
Trainad Madel
(i) Probability
Weighted
Weighted by Probability for Scenarios
Absence
CRPS
Equally Weighted Samples
67%
(ii) Coverage
Minimal Error Among Scenarios
33%
Distortion
Training Set (Similar History, Diverse Futures)
Ideal Model With the New Paradigm
SSeeaaio, Probability) Pains
(ii) Inference
Inadequaacy
Minimal Error Among Samples
FLOPs Inference
Cost
Figure 2: Probabilistic Scenarios Paradigm and Unified Evaluation Framework. The left panel illustrates an ideal behavior: a model trained on a dataset where similar histories lead to diverse futures should learn to output (Scenario, Probability) } pairs that reflect the empirical frequency of those futures. The right panel details our evaluation framework, which links the limitations of sampling to adapted metrics and provides distinct yet comparable formulations for both paradigms.
the mentioned limitations: the Probability A bbence for any given sample ý,, the risk of Coverage Inadequacy when the set V'amples fails to capture rare but critical events, and the high Inference Cost of generating a sufficient number of samples.
ÿ ~ Po(y)x) 3.2 NEW PARADIGM: PROBABILISTIC SCENARIOS
In light of the challenges in sampling, we explore an alternative paradigm that reframes the forecasting objective, as illustrated in Figure 2 Instead of learning a generative process to approximate continuous distribution, our paradigm aims to learn direct mapping from the historical context to a discrete, finite probability space of future scenarios. Formally, we define a model under this paradigm as function, f. that maps the history to tuple containing both the set of all future scenarios and their corresponding probabilities:
f(x) (Ypred: P)
where: Dpree b is the finite set of N predicted future scenarios, with each scenario yn E RRTDD == (p1....,Ph) is the vector of probabilities associated with the scenarios in predd- The probabilities must satisfy the axioms pm 2 0 for all n and ΣN Pr = 1.
This formulation directly yields explicit set of (Scenario, Probability] pairs in a single forward pass. It differs from the Monte Carlo approximation of equation 2 by providing a discrete probability distribution that is both interpretable and computationally efficient. In essence, this paradigm neither assumes a parametric distributional form nor requires sampling, but instead learns to end-to-an -nd generate (Scenario, Probability) pairs. This reframing of the objective simplifies the learning problem. detailed discussion and theoretical analysis are provided in the Appendix A.
3.3 UNIFIED EVALUATION FRAMEWORK
To quantitatively measure the limitations of sampling- based methods, we establish an evaluation framework by adapting two complementary metrics. We use the standard metric for overall fore- cast quality (Zhang et al. 2024, Zheng & Sun 2025), the CRPS, to assess Probability Absence. Concurrently, we use Distortion, defined as the error of the best single trajectory in set, to assess Coverage Inadequacy (Cortés et al.) 2025). For both metrics, we provide distinct but comparable formulations for the sampling-basee based and Probabilistic Scenarios paradigms.
Weighted CRPS for Probability Absence We employ the energy score formulation of CRPS, which is defined for a single ground truth observation Yg and set of forecasts as --r-yyllll - DEllyyy'lll, where y and y' are independent samples from the forecast distribution. We generalize this to our discrete, weighted scenario set. Given a set of N scenarios Ypred = {yn) J. and a corresponding probability vector p (pı,...,7) the Weighted CRPS is defined as:
CRPS() prod ya)
Σ p. P.lly Yeall ll
M
PuPsllyn - ysllı
where ||- denotes the L1 norm summed over all elements of the trajectory. We apply this formulation to both paradigms:
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Training: Lpriau Lrcon + byeob
TimePrism Structure
Firand
Trend Layer
Season Layer
Ytrand
Layer
000
Crecon
I|y ||y0r Yill
yx-11
Yea
min ned...
Mcrend
History Input
Moving Average Decompositinn
Shored = (Pealarr
baslen
N = M Future Scenarios
Prob Layer
Gooob
CrossEEtropy(tr.at)
AAAA Aowason
OOo
Y41
00O
O0O O0O Xseason
Inference:
mm P5
Probability Layer (Logits)
Season Layer
Soltmax(11
wwW
Probabilistic
mm Px
Scenarios
Figure 3: Structure of TimePrism, linear model demonstrate the potential of the Probabilistic Scenarios paradigm. The model operates in three parallel streams: after an initial decomposition, separate linear layers generate basis of M trend and K seasonal forecasts. Simultaneously, third linear layer produces the N M K logits from the undecomposed history. This architecture, built within the Probabilistic Scenarios paradigm, achieves competitive performance despite its simplicity, demonstrating the potential of the new paradigm.
For Probabilistic Scenarios, the scenarios {yn J and probabilities J are taken directly from the model's output (Dyred p).
For sampling-baser models, the evaluation is performed the generated set of S samples, Dssmples = { Each sample is assigned a uniform probability, i.e., P; = 1/5.
The CRPS directly rewards models that assign higher probabilities scenarios that are closer the ground truth, thus quantitatively measuring the impact Probability Absence.
Distortion for Coverage Inadequacy. Distortion measures the best-case performance of forecast, quantifying how well the generated set of trajectories covers the true outcome (Cortés et al. (2025). It defined the minimum error between any single trajectory in the set and the ground truth. Following the implementation in our evaluation code, we define as the minimum Root Mean Squared Error (RMSE) over the set of trajectories:
y..y Distortion()..ya) min V D Ily Yalli
where || || is the Frobenius norm. We apply this formulation follows:
For Probabilistic Scenarios, the minimization is performed over the complete set of N scenarios generated by the model, y =Yrred-
For sampling- bassl models, the minimization is performed over the set of generated samples,
y ) samples-
This metric directly assesses the diversity and reach of the generated set of futures. A lower Distortion score indicates better coverage, particularly for tail events that may be missed by a limited number of samples, thus measuring Coverage Inadequacy.
4 TIMEPRISM: A PROOF-OF-CONCEPI MODEL
4.1 DESIGN PHILOSOPHY
The primary goal of TimePrism not to introduce a new complex architecture, but to serve as a clear proof-of-concept for the Probabilistic Scenarios paradigm. We intentionally adopt minimalist design to test a core hypothesis: that the Probabilistic Scenarios paradigm can prove effective even when implemented with simple model architecture.
To this end, we construct TimePrism using only three parallel linear layers as its core learnable components, devoid of any non-linear activation functions deep, stacked layers. This deliberate
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simplicity acts controlled experiment. By stripping away architectural complexity, ensure that the model's strong performance can be directly attributed to the strengths of the paradigm itself.
4.2 TIMEPRISM ARCHITECTURE
The architecture of TimePrism is illustrated in Figure 3 It consists of three parallel streams that process the input history to generate the final set of Probabilistic Scenarios. Inspired by recent works such DLinear and FITS (Zeng al. [023) Xu al. 2023), which showed that simple architectures can effectively validate a new paradigm, we use a backbone based on decomposition and linear layers.
The model operates follows.
1) Decomposition: First, the input history x E R is separated into a trend component Xirend and a seasonal component Xscason using a moving average filter. This is a standard decomposition technique where:
X semsnn
Xxrnnd
with
Xtrend AvgPool AvgPoolPaddding(S (Padding(x))
2) Trend and Season Layers: The two decomposed components are then fed into two independent linear layers. The trend layer maps the trend component Xcrend to a set of M distinct trend forecasts, M {yrm m=11 m== Concurrently, the season layer maps the seasonal component Xyesso a set of K
distinct seasonal forecasts, K. = [ys,&Jk= The complete set of N M K future scenarios, ) pred-
is constructed by combining these two sets:
Y pred M K (yim + Vs,k Vum M, M.
3) Probability Layer: Operating in parallel to the scenario generation, a third linear layer acts as the probability module. This layer takes the original, undecomposed history input and directly produces a logits vector π E RV. Each element n, in this vector corresponds to one of the N scenarios generated via the combinatorial process in Eq. equation
4.3 TRAINING AND INFERENCE
The design of the loss function directly guided by the Probabilistic Scenarios paradigm. Specifically, the loss function, Prissm is composed of two terms, each designed to supervise one component of the target (Scenario, Probability) output. The reconstruction loss, Lreonn responsible for optimizing the fidelity of the generated Scenarios. Concurrently, the probability loss, Cprob. supervises the learning of a meaningful probability distribution over these scenarios. The coefficient of the probability term, X, is set to this work. Given the ground truth future trajectory Yg, the total loss is:
Loss Function and Training.
P Lrecun + frpub
(8)
For Scenarios: The reconstruction loss, Lreco, uses the -aaner-Ta (WTA) principle. first identifies the index n* of the scenario in pmeed that has the lowest Mean Squared Error (MSE) with the ground truth. The loss then the MSE this single "winner" scenario:
Lreson = lly yn. 112 nt = arg min
lly - ynl12
n=1...N
For Probability: The probability loss, Lproh, trains the probability layer to assign the highest probabil- ity to this winner. It is the Cross-Entropy loss between logits vector π and winner index n":
exp(TTn-)
Cprub CrossEntropyy(s n²) log
(10)
Σ== exp(IT))
In our experiments, we employ a relaxed variant of the WTA loss (Rupprecht et al. 2017) to further stabilize training. The complete formulation of this loss, including its specific implementation for the multivariate case, is provided in the Appendix C.2
Inference. During inference, the model performs a single forward pass to generate the set of N scenarios, prredd and the logits vector, æ. The logits are then converted into a valid probability vector, p. using the Softmax function as in equation II The final output of TimePrism the complete set of Probabilistic Scenarios, {(yn, p
p = Softmax(T)), where pr. = exp(ITn)
(11)
Σne1 exp(TT,)

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5 EXPERIMENTS
5.1 BASIC SETUP
Data. Following the recent benchmark for probabilistic time series forecasting, ProTS (hhang et al. 2024), we evaluate our model on five datasets: Electricity (Elec.), Exchange (Exch.), Solar(Sol.), TrafficTTraf.). and Wikipedia (Wiki.). These are benchmark datasets taken from the GluonTS library (Alexandrov et al. 2020), preprocessed exactly in prior works (Gasthaus et al. 2019) A detailed analysis of dataset properties cited from previous work provided in the Appendix C.1 Following prior work, we set the forecast horizon to 24 (hours) for the hourly datasets (Electricity, Solar, Traffic) and 30 (days) for the daily datasets (Exchange, Wikipedia) (Zhang et al., 2024). For TimePrism, the input length is set equal to the forecast horizon. Other baselines may use longer context lengths as lagged features, for which we adhere to their configurations in prior work (Cortés) al. 2025). Notably, TimePrism achieves strong performance even with less information input. For a comprehensive comparison, we also provide results in the Appendix D.2 where TimePrism uses an input length comparable to that of the baselines.
Metrics. As established in our framework, the primary metrics are Weighted CRPS and Distortion. To measure the Inference Cost, we report inference Floating Point Operations (FLOPs). While MSE and Mean Absolute Error (MAE) are not primary indicators for probabilistic forecasting, we include their definitions and normalized results the Appendix B.3 and D.I for comprehensive comparison.
Baselines. For a comprehensive comparison, we select seven models covering all three categories discussed in our related work. ETS (Hyndman al., 2008) serves as a non-neural baseline. DeepAR (Salinas et al. 2020) represents parametric distribution models. TimeGrad (Rasul et al. (021) is diffusion- based generative model. TempFlow (Rasul et al., (220)). Transformer TempHlow (Trf.Flow), and TACTiS-2 (Ashok et al., (023) are structured probabilistic models. Tempflow is implemented with Long Short-Term Memory (LSTM) backbone (Hochreiter & Schmidhuber
[197 and Trf.Flow
is implemented with Transformer backbone (Vaswani et al.] (017), as in Rasul et al. (2020). Finally,
TimeMCL (Cortés et al. (2255) represents multi-choice learning models.
Training Details. All models are trained using the Adam optimizer with initial learning rate of 10 for 200 epochs. Given its lack of hidden layers, the number of scenarios N is the primary tunable hyperparameter for TimePrism. In this section, TimePrism uses N 625 scenarios, composed of M 25 trend and K = 25 seasonal components. In practice, the number of distinct future scenarios is known priori, N be set match this number; otherwise, in our benchmark datasets, N should be set to a value large enough to allow the model to learn on its own. Further training details and analysis on N are included the Appendix C.3 To ensure fair comparison, considering that TimeMCL employs 16 hypotheses in its original implementation, we specifically include variant with N 16, denoted as TimePrism-16. This serves to validate the effectiveness of the new paradigm, demonstrating that it functions effectively with simple structure and without requiring a large number of parameters.
5.2 MAIN RESULTS
Table 1: CRPS for Probability Absence. Results on 5 benchmark datasets. We report the mean + standard deviation over 3 random seeds. The best and second results are bold and underlined.
Model
Elec.
Exch.
Sol.
Traf.
Wiki.
ETS
0.376 + 0.00
1,22 + 0.02
0.375 + 0.00
0.813 1 0.00
4.88 = 0.01
DeepAR
0.997 0.03
0.701 + 0.00
0.583 + 0.02
0.826 + 0.01
1.75 0.30
TimeGrad
0.232 0.00
0.845 0.24
0.241 + 0.00
0.162 + 0.00
0.517 0.02
TempFlow
0.316 - 0.00
0.669 0.01
0.272 0.00
0.601 0.01
1.26 0.06
Trf.Flow
0.396 + 0.08
1.17
0.280 0.02
0.607 0.01
1.71 0.12
TACTiS-2
0.299 + 0.01
0.648 + 0.03
0.236 + 0.03
0.257 0.01
0.484 + 0.00
TimeMCL
0,370 + 0.01
1.12
+22900000
0.262 0.01
0.640 + 0.03
TimePrism-16
0,414 0.12
0.611 = 0.06
0.137 0.00
0.159 + 0.02
0.654 0.01
TimePrism
0.133 + 0.02
0.468 + 0.01
0.0852 + 0.00
0.111 + 0.00
0.506 0.00
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Probability Absence and Weighted CRPS. The limitation of Probability Absence means that decision-makers cannot directly assess the likelihood of specific outcomes from sampling-bas based models. To quantitatively measure the benefit of providing explicit probabilities and evaluate the overall quality of the forecast distribution, we use Weighted CRPS. Table presents the results on five datasets, reported as the mean and standard deviation 3143), following prior work. TimePrism achieves state-of-the-art performance on four of the five datasets and secures the second-best result Wikipedia. Furthermore, detailed discussion on the applicability of TimePrism is provided in the Appendix C.1.2
three random seeds (3141, 3142,
Table 2:
Distortion for Coverage Inadequacy, Results on 5 benchmark datasets. We report the mean + standard deviation over random seeds. The best and second results are in bold and underlined.
Model
Elec.
Exch.
Sol.
Traf.
Wiki.
ETS
1.24 + 0.02
1.92 + 0.06
1.03 + 0.00
2.69 + 0.01
142 + 0.71
DeepAR
2.82 + 0.11
1.87 + 0.03
1.09 0.02
1.86 + 0.09
5.36 0.42
TimeGrad
0.731 + 0.02
1.37 0.17
0.550 :: 0.03
0.561 + 0.02
1.64
TempFlow
1.41 = 0.04
1.32
0.00
0.981 = 0.00
37.
Trf.Flow
1.70 + 0.28
1.70
0.552 0.04
1.02 0.01
63.7 8.02
TACTiS-2
0.674 0.04
0.873 0.04
0.586 0.02
0.592 0.05
1.26 0.10
TimeMCL
0.607 + 0.01
1.08 + 0.08
0.462 0.04
0.454 0.00
1.49 + 0.30
TimePrism-16 16
0.911 + 0.27
0.920 0.04
0.307 0.04
0.346 0.09
1.16 + 0.15
TimePrism
0.211 + 0.04
0.595 + 0.01
0.101 0.03
0.144 0.00
1.04 + 0.03
Coverage Inadequacy and Distortion. To assess Coverage Inadequacy, we use the Distortion metric, with results presented in Table 2 TimePrism achieves the state-of-the-art result across all five datasets, demonstrating its superior ability to generate a diverse set of scenarios that covers the ground truth. This is because our reconstruction loss, frconn allows the model not be heavily penalized for predicting a plausible but non-realized future, in datasets containing similar histories but diverse futures.
Table 3: Inference FLOPs. FLOPs required to generate S forecast samples on the Exchange dataset with batch size 1. The cost for TimeMCL and TimePrism is constant as they produce all scenarios in single forward pass.
Sampling S
DeepAR
TimeGrad TempFlow Trf.Flow
TACTiS-2 TimeMCLTimePriss
2.9 10
1.9 x 10
5.8 106
1.4 x 107
2.5 x 107
2.9 105
1.9 x 10
5.8 107
1.3 x 108
1.2 x 10
8.8 10 5.1 10
100
2.9 x 106
1.9 x 10t0
5.8 x
10 1.3 x 109 1.1 x 109
Inference Cost and FLOPs. To evaluate the Inference Cost, we compare the FLOPPs required by each model to generate a set of S samples, with results shown in Table 3 As demonstrated, the inference cost of TimePrism is constant regardless of the number of samples required, as it generates all N scenarios and their probabilities in single forward pass. In contrast, the cost for sampling- based models scales with S, forcing a direct trade-off between forecast quality and computational efficiency. TimeMCL also generates its full set of hypotheses in a single pass. However, lacking explicit probabilities, its original implementation for CRPS evaluation relies resampling from this fixed set. For a fair comparison, we also only report the single-pass FL.OPs of TimeMCI..
Overall Comparison. The CRPS and Distortion results in Tables 1 and are based on S = 100 samples for all baselines. At this sampling level, TimePrism is more efficient by one to five orders of magnitude than its most competitive counterparts (TimeGrad, TACTiS-2, and TimeMCL). The models, especially when a large
results confirm that TimePrism is more efficient than sampling- number of samples is needed, highlighting the efficiency of the Probabilistic Scenarios paradigm. This analysis details the trade-off between inference cost and forecast quality, and how our paradigm transcends it.
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Table Impact of Scenario Count (N) on Performance and Complexity. This table presents systematic ablation study across Electricity, Exchange, and Solar datasets, illustrating how model complexity and forecasting error (CRPS, Distortion) scale with the number of scenarios N.
Solar
Electricity
Exchange
N
FLOPs
CRPS
Distortion
CRPS
Distortion
CRPS
Distortion
1.0x
0.199 + 0.00 0.266 0.00 0.409 0.01 0.733 0.03 0.596 0.00 0.803 0.00
16
4.2x
0.137 + 0.00 0.307 = 0.03 0.414 0.10 0.911 0.22 0.611 0.04 0.920 0.04
256
19.9x
0.0927 + 0.000.158 + 0.01 0.162 + 0.01 0.267 0.03 0.486 + 0.00 0.666 + 0.02
625
34.8x
0.0852 + 0.000.101 = 0.03 0.133 + 0.01 0.211 0.03 0.468 0.01 0.595 = 0.01
1024
48.3x
0.0822 + 0.0m00917 +
0.010.139 0.010
0.01 0.212 0.02 0.452 + 0.00 0.583 = 0.02
5.3
IMPACT SCENARIO SET SIZE
We conduct systematic analysis investigate the trade-off between the scenario set size N and model performance. Table @ the results across three representative datasets (Electricity, Exchange, and Solar) with N ranging from to 1024.
Complexity Scaling. A key advantage of our combinatorial architecture (N M K) is its efficiency. The parameter complexity the shared basis layers (Trend and Season) scales as O(VN), while only the probability head scales linearly O(N). Consequently, the overall model complexity grows favorably between (/21/3) and ( allowing for large scenario sets.
Performance Trends. Increasing N generally leads to lower CRPS and Distortion errors, as larger discrete set approximate the continuous probability space with higher fidelity. However, observe diminishing returns: the performance gains tend to plateau around = 625. Beyond this point, the marginal benefit of adding scenarios decreases while the computational cost continues to rise. Based on this equilibrium, we adopted N 625 as the unified setting for our main experiments.
Dataset Dependence. The results also indicate that the "saturation point" varies slightly by dataset. For instance, the Solar dataset benefits more from a larger N compared to the Exchange dataset. This suggests that the optimal N is determined by the intrinsic complexity of the data, highlighting the potential for future work on adaptive mechanisms that dynamically adjust N.
5.4 VISUALIZATION AND QUALITATIVE ANALYSIS
Probabilistic Scenarios: TimePrism
200
Probability
Low
High
Common
Rare Case
Sampling Paradigm: TACTIS-2
200
100
Samples Mean
02-12 18
02 06
10
02:1100
09-25 761111
26
06
10
19:1 D8
unl1 11
10-12
on
18
86
191310 18
10-14
96
10-14
10
10:15
on
10 d6
10-15
10-15 18
10
Figure Qualitative Analysis of the New Paradigm. visual comparison between the Probabilistic Scenarios paradigm (TimePrism) and the Sampling Paradigm (TACTiS-2). The figure highlights their distinct behaviors in both common high-peak cases and a rare low-peak case, on the Solar dataset.
To visually compare the two paradigms, conduct qualitative analysis the Solar energy dataset, selecting the last variate (D 137) and identifying instances with similar histories but diverse futures. As shown in Figure 4 these instances include four Common Cases of high-peak futures

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and one Rare Case of a low-peak future. We compare TimePrism against TACTiS-2, the strongest baseline. The top panel displays the top 10 scenarios from TimePrism, with line color and thickness representing probability from low (red, thin) to high (blue, thick). TimePrism successfully captures both types of cases, assigning high probabilities to the common cases while also identifying a rare case with low probability. In contrast, the bottom panel shows that the S 100 samples from TACTiS-2 cluster around their mean (dashed line). While the envelope of TACTiS-2 samples may loosely cover both high-peak and low-peak, its forecast suffers from Probability Absence. Without explicit probabilities. the common high peak case can not be distinguished from the rare low peak case, rendering the entire set of samples uninformative for assessment.
CONCLUSIONS AND DISCUSSION
6.1 DISCUSSION
Reason of Effectiveness. The strong performance of TimePrism stems from the paradigm's reframing of the learning objective. Instead of learning to model an entire continuous probability space, the model is learning a more structured problem: a probability distribution over a discrete set scenarios. This concept parallels Vector Quantization (VQ) techniques in representation learning, most notably VQ-VAE (van den Oord et al. 2017). but applies the discretization directly to the output trajectory space rather than latent space (see Appendix |A.3 for detailed discussion). This shift reduces the required model capacity, allowing a simple linear architecture to achieve strong results.
Limitations:
Dataset Applicability. The intentionally simple structure of TimePrism, while effective for validating the paradigm, may have limitations in more complex scenarios, such as those with extremely high dimensionality, or series lacking trend or seasonal patterns.
Structural Rigidity. As a linear model, the current version of TimePrism requires fixed-length inputs and prediction horizons, limiting its flexibility in scenarios where variable-length contexts are available during inference.
Simplified Multivariate Modeling, Our current implementation utilizes a weight-sharing strategy. We believe there is significant room for improvement by incorporating more sophisticated channel- mixing mechanisms to model cross-variate relationships.
Future Works:
Models within the new Paradigm. TimePrism serves only as a proof-of-concept. The true potential of Probabilistic Scenarios lies in its application to more powerful backbones. Future work could integrate this paradigm with rate-of-the-ar architectures like Transformers, Diffusion, or Flow Matching models to unlock new levels of multivariate performance.
Refinements of the new Paradigm. The paradigm itself can be further enhanced. For instance, developing methods to adaptively determine the number of scenarios based on data complexity could improve its practical utility.
Pecision-Centriy Assessment. Metrics like CRPS and Distortion may not fully reflect the down- stream utility of probabilistic forecasts in real-world environments. In future work, decision-centric metrics can be incorporated, such tail-risk assessment and utility-based scores. Furthermore, we plan to explore the direct integration of our Probabilistic Scenarios paradigm into real-world dicision-makingg demonstrate its practical value beyond pure forecasting accuracy.
6.2 CONCLUSION
Probabilistic time series forecasting is crucial for reliable decision-making, While powerful, current SOTA methods predominantly rely on sampling, paradigm that faces limitations of Probability Absence, Coverage Inadequacy, and Inference Cost. To address these challenges, we introduced the Probabilistic Scenarios paradigm. This paradigm operates by directly producing a set of ( (Scenario, Probability] ) pairs in a single forward pass, without reliance sampling. We validated this new paradigm with TimePrism, a simple linear model. Evaluated under our unified framework, TimePrism addresses these challenges and demonstrates the potential of the new paradigm. In summary, our work provides a practical alternative to sampling and broadens the conceptual landscape of probabilistic forecasting, establishing promising foundation for future research.
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ETHIC STATEMENT
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
We comply with intellectual property agreements for all data sources. Data properly anonymized with no concerns regarding sensitive or illegal activity in our dataset.
REPRODUCIBILITY STATEMENT
The code of this work is available at:
LLM STATEMENT
We utilized a large language model to assist in polishing the grammar and phrasing of our manuscript.
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Appendix
A THEORETICAL ANALYSIS
A.1 EFFECTIVENESS OF THE PROBABILISTIC SCENARIOS PARADIGM
The empirical success of the Probabilistic Scenarios paradigm is rooted in its fundamental reframing of the learning objective. This section provides theoretical perspective why this reframing leads to more tractable and effective learning problem.
The conventional sampling- paaseddg-basedd based paradigm requires a model learn complex, bigh-dimensional conditional probability distribution, Po over the continuous space TTDD Optimizing this objective, often by maximizing the log- ligliikli log (y x), is difficult. requires the model to conrecty assign a probability density to every possible point in an infinite space, a task that demands immense model capacity.
In contrast, our Probabilistic Scenarios paradigm transforms this intractable density estimation problem into more structured, two-part learning task:
1. Scenario Representation:
The first task is to learn finite set of N discrete points, Y {yn) J. 1. that effectively represent the most meaningful regions of the true conditional distribution. This simplifies the problem from modeling the entire continuous space to finding a good discrete basis for it.
Probability Assignment: The second task is to learn a categorical distribution, p. over this
finite set of N scenarios. The objective shifts from computing density (y(gi)))) to solving large-scale classification problem: determining which of the N' representative regions the ground truth Yg is most likely fall into.
In essence, the paradigm decouples the problem of "what" can happen (the scenarios) from "how likely" it is happen (the probabilities). This structured decomposition significantly reduces the complexity of the learning problem, allowing even simple models to allocate their limited capacity efficiently and achieve strong performance.
A.2 THEORETICAL FOUNDATIONS OF TIMEPRISM
The theoretical amalysis of the Winner-Takes-AN principle in this section is inspired by the framework presented in Cortés et al. (2025) and Letzelter et al. (2024). However, we adapt and extend this analysis to our specific non-autoregressive, combinatorial architecture and our probabilistic objective, which, as we will show, provides stronger thcoretical guarantees.
A.2.1 OPTIMAL SCENARIOS VIA RECONSTRUCTION Loss
The goal of our reconstruction loss to find of scenarios that provides the best discrete approximation of the continuous space of all possible future trajectories. We formalize this in the following proposition.
Proposition 1. Assuming that the model parameters reach local minimum of the reconstruction loss, a necessary condition is that the set of N M K combined scenarios forms a Centroidal Voronoi Tessellation (CVT) of the space of fature trajectories, conditioned on the input history. Specifically, each combined scenario yiom + Ss.k converges region.
conditional mean of its corresponding Voronoi
Proof. The objective is to find the model parameters (which in turn define the scenarios) that minimize the expected reconstruction loss over the data distribution P(x, yg)). Our model is auo-eeroreersssivve so the generated scenarios (x)} are a direct function of the input history x. The expected loss is:
E [[[rocom]
Ex
min
y y. (x))))
(12)
The min operator partitions the space of future trajectories, for given x, into N Voronoi regions, o (x)) We formally define the Voronoi region for the n-th scenario as (x) = {y E R ? x
ly (n(x]] - (x))x)),,, Vj}. Each region R, (x) contains all trajectories Yg: for which the
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n-th scenario is the winner. The inner expectation can then be rewritten sum of integrals over these regions:
Byslx min
Iyy
yn (x))2
Σ
lay (xx((x)))22 x)dya
(13)
n=1 (x)
To find the optimal scenarios (x)), we take the functional derivative of the expected loss with respect to each (x) and set it to zero. The derivative only affects one term in the summation.
Following the derivation in Cortés al, (2025), the minimum is achieved when:
SR.,(x) (s (y) Elyst Yg E R (x)]
Yn (x)
(14)
(R,,(x) P(ygix)dd
This derivation holds provided that the Voronoi region has non-zero probability mass, that is, when JR. (x) p(yg(xxddy # 0. This demonstrates that for any given history x, the scenarios generated by an optimal model must be the conditional means of their respective Voronoi regions (Du et al. 1999). In geometric terms, the set of N scenarios acts as a set of centers that partition the high-dimensional space of all possible futures into N distinct regions, known as a Voronoi tessellation. Each region consists of all future trajectories that are closer to one specific scenario than to any other. Our result shows that the WTA training objective effectively drives the model to find an optimal set "cluster centers" (our scenarios) that best represent the underlying structure of the data, where "best" is defined in the sense of minimizing the expected squared error, akin to the objective in k-means clustering (Cortés et al. 2025 Arthur & Vassilvitskii 2007).
A.2.2 SCENARIO REPRESENTATION AND DISTORTION
Our reconstruction loss is designed to optimize for scenario fidelity, which directly contributes to the model's s ability to achieve low Distortion score. The core mechanism lies in how the Winner- Takes-All (WTA) objective interacts with datasets exhibiting diverse potential futures from similar histories.
Consider the gradient of the reconstruction loss, Frewm with respect to the model's parameters O. The parameters define the mapping from the input x to the entire set of scenarios )xee 0) = (x: ) The loss for a single data instance (x, Nz1) is:
Lrecon ( Iyy yn (x: 0)11
(15)
where the winner index n² is itself function of 0:
min ll yn (x: 0)1 n (0) = arg n= w=I....
(16)
Assuming the winner index n is locally constant with respect to small changes in 0, the gradient of the loss is given by the chain rule:
V Cressnn OCrcon
Tyn- (x; 0)
(17)
Oyn-
00
Crucially, for all non- non-winning scenarios where n # n," the partial derivative of the loss with respect their outputs is zero:
HLrcco Öy
Vn + n
(18)
This implies that the gradients for the parameters governing the non-winning scenarios are also zero for this specific training instance.
The direct consequence of Eq. equation 18 is that the model is not explicitly penalized for generating a plausible but non-realized scenario. In dataset containing instances of "similar histories, diverse futures," this property allows different scenarios within the set )/ to specialize in representing different potential outcomes without interfering with one another during training. For one training instance, only the parameters responsible for the winning scenario are updated to better match the ground truth. For another instance with a similar history but a different future, a different scenario may become the winner, and its corresponding parameters will be updated. This dynamic encourages the model to maintain a diverse and comprehensive set of scenarios to cover the full spectrum of possibilities observed in the training data, directly leading to lower expected Distortion.
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A.2.3 OPTIMAL PROBABILITIES VIA PROBABILITY Loss
The goal of our probability loss is to ensure that the learned probability vector p accurately reflects the true probability mass over the Voronoi regions defined by the optimal scenarios.
Proposition 2. At the global minimum of the expected probability loss, the predicted probability vector matches the true conditional probability mass function over the Voronoi regions. That is, Pn = g( gi E R,, x). where R2 is the Voronoi region of the n-th scenario.
Proof. The optimization objective for the probability loss is to minimize the expected Cross- Entropy loss. We denote the cross-entropy between two discrete distributions q and p as H(q. P) = =- -nn On log Pr.
(19) Let g(a) x) = P(y Rn, x) be the true, unknown probability that the n-th scenario is the winner for a given history x. The inner expectation corresponds to the cross-entropy between this true distribution q(x) and the model's predicted distribution p(x) = Softmax(T((x)) By the properties of cross-entropy:
B(pprob] Ex [Easiix| log Pw (x)]] Ex ({((()), p(x))
(20) Since the entropy of the true distribution H (x())) ) is a constant with respect to our model's parameters, minimizing the expected cross-entropy is equivalent minimizing the expected KL divergence. The KL divergence is non-negative and is minimized at zero if and only if p(x) = (xx) for all x. Thus, the optimal solution for our probability output is the true probability distribution over the discrete set of winner outcomes.
E (q()),, p(x))] [xx (q(x) +Fx [x(
A.2.4 PROBABILITY MATCHING AND CRPS
Our paradigm's ability achieve strong performance on the Weighted CRPS metric is rooted in its direct optimization of a true probability distribution. As established in Proposition 2, the Cross-Entropy loss drives the model's output probability vector, p Softmax(TT), to match the true conditional probability mass function over the set of optimal scenarios. The objective is to minimize the Kullback-Leibler (KL) divergence between the predicted and true discrete distributions, DKL (xL(q(x) [p(x)), where qxx) is the true distribution of winner outcomes. Since the Weighted CRPS directly incorporates the probability vector p (equation 4 a model that learns a more accurate probability distribution is expected achieve lower (better) score.
This approach provides a strong theoretical foundation for probabilistic modeling. The probability Pn for scenario Yn in our framework represents holistic assessment of the entire trajectory, conditioned on the initial history. In contrast, autoregressive multi-hypothesis models like TimeMCL (Cotéés al. 2025). where scenarios, termed hypotheses in the original work, are generated step-by- step. face a challenge in aggregating pointwise confidences into a valid trajectory-level probability. For instance, consider two scenarios over a horizon of T
Scenario A might have pointwise confidences of (0.2, 0.2), while Scenario B has 0..1. Averaging these values, as is done for evaluation in TimeMCL, would assign both scenarios an identical score 0.2. However, under the principles of conditional probability, their joint probabilities would be different (0.04 vs. 0.03), a distinction that simple averaging fails to capture. Furthermore, the set of scores produced by TimeMCL does not constitute valid probability distribution their sum is not constrained to be one.
A.3 CONNECTION TO DISCRETE REPRESENTATION LEARNING
As noted in our discussion the model's effectiveness, the Probabilistic Scenarios paradigm shares conceptual roots with discrete representation learning techniques, most notably Vector Quantized Variational AutoEncoders (VQ- VAE) (van den Oord et al. 2017). Both approaches posit that continuous spaces can be effectively approximated by a finite set of discrete vectors. However, TimePrism distinguishes itself from VQ-VAE in three fundamental aspects, tailored for the forecasting task:
VQ-VAE discretizes latent features, which serve intermediate representations. In contrast, TimePrism directly discretizes future trajectories, operating within the final output space.
Discretization Target:
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Weighted CRPS. Our implementation of the Continuous Ranked Probability Score is computed on a per-channel basis. For each variate d E (1...., ........ D). we calculate the score using the energy score formulation. Given the normalized ground truth for single channel, Yeg.a E R set of N Published as a conference paper at ICLR 2026
Nature of Codebook/Scenarros:: VQ-VAE utilizes static, global codebook shared across all imputs, where codes are fixed parameters learned from the entire dataset. Conversely, TimePrism generates dynamic set of scenarios in real-time based the imput. These scenarios function as a conditional codebook that adapts to the specific history of each time series.
Probability Modeling: VQ- VAE typically employs an implicit, two-stage approach that requires training separate prior model over discrete codes to perform sampling and probability estimation. TimePrism, however, uses an explicit, end-to-end approach featuring built-in probability head that directly outputs the probability distribution p over the generated scenarios in a single forward pass.
B METRICS
B.I IMPLEMENTATION DETAILS
This section provides detailed formulations for our primary metrics, Weighted CRPS and Distortion, clarifying how they are applied to the outputs of both the Probabilistic Scenarios and sampling- -base
paradigms.
normalized scenarios for that channel, {y",ah==1- and a corresponding probability vector for that channel, Pa = (PI,d,..., the per-channel Weighted CRPS is:
CRPS Σ Pnaalyyn, Ygaal
Σ
PnaPPjallynnd Yaalla
(21)
where || denotes the LI norm. The final reported CRPS score is the average of these per- channel scores. For sampling-based models, each of the S samples is assigned a uniform probability Pi.d 1/S for all channels.
Distortion. In contrast to CRPS, our Distortion metric is computed jointly across all dimensions to assess the quality of the entire multivariate trajectory. This aligns with its purpose of evaluating the coverage of the joint distribution. It is defined as the minimum Root Mean Squared Error (RMSF) over the set of complete multivariate scenarios:
Distortion().yg min V T Ily y.EY
Yg. |
(22)
where y' represents the set of scenarios and || || is the Frobenius norm. Note that the calculation performed normalized data as described above. For Probabilistic Scenarios, the minimization is performed over the complete set of N scenarios, y Dpeed For sampling-ber based models, is performed over the set of S generated samples, y V'samples
B.2 COMPREHENSIVENESS AND FAIRNESS
Scenarios and Probabilities. Our evaluation framework comprehensive because its two primary metrics are complementary, addressing the two core components of probabilistic scenario. The Weighted CRPS evaluates the quality of the entire predictive distribution, considering both the accuracy of the scenarios and the correctness of their assigned probabilities. Distortion, the other hand, isolates the quality of the scenario set itself by focusing solely on its best-case coverage, irrespective of probability assignments.
Per-channel and Joint Evaluation. While our per-channel CRPS formulation a standard approach (Zhang al.] (224). it is known be insensitive to errors in the correlation structure of multivariate forecast (Marcotte et al. 2023). We specifically complement this with a jointly computed Distortion metric. Because Distortion evaluates the enror the entire T D space for each scenario, is
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sensitive to the quality of the multivariate structure, thus compensating for the limitations of the per-channel CRPS.
L1 and L2 Norms. The of different norms for our two primary metrics is deliberate design choice. For Weighted CRPS, we use the LI norm, which is standard for this metric and provides robustness against outliers (Zhang et al. 2024). This is appropriate for metric assessing the overall distributional quality, where the influence of single extreme errors should be contained. For Distortion, whose sole purpose is measure the fidelity of the best available scenario, we use the L2 norm (via RMSE), aligned with related work (Cortés et al. 2023). Its higher sensitivity to large deviations is feature, more strictly penalizes a model whose best-case scenario still far from the ground truth.
Fairness. Our evaluation framework designed be fair. The Continuous Ranked Probability Score is strictly proper scoring rule, meaning it is minimized in expectation if and only if the predicted distribution coincides with the true lata-generating distribution (Zhang et al. 2024). Our Weighted CRPS, as an average of these strictly proper rules applied to the marginal distributions, inherits this property for the set of marginals. Distortion, however, not a strictly proper scoring rule as it only considers the single best scenario. For this reason, it serves as a complementary, auxiliary metric focused specifically on coverage, not as a complete measure of probabilistic quality.
B.3 COMPLEMENTARY METRICS
For more comprehensive comparison, we also report on complementary metrics: Mean Squared Error (MSE) and Mean Absolute Error (MAE). These metrics are computed on the same per-channel normalized data as our primary metries ensure a consistent evaluation scale. While these are typically used for deterministic forecasting, we include their definitions and results in the Appendix align with standard practices in recent benchmarks (Zhang et al. 2024, Cortés et al. 2025).
For our Probabilistic Scenarios paradigm, we derive single representative forecast from the set of scenarios by weighting them by their learned probabilities. For sampling- based models, this the standard mean median of the samples.
-based
Mean Squared Error (MSE). Following standard practice, the MSE is calculated based on the mean forecast, mcan-- For a set of scenarios Y pred with probabilities p. this is the expectation of the predictive distribution:
meun
Pnyn
(23)
#=1
The MSE score is then the average of the per-channel Mean Squared Errors:
t Ily D T
1
MSE
G Fmman,ddl2
(24)
Mean Absolute Error (MAE). The MAE calculated based on the median forecast, Vmedian- which is the 0.5-quantile of the predictive distribution. For set of scenarios Ppred with probabilities p. the weighted median is computed for each point in the trajectory. The MAE score is then the average of the per-channel Mean Absolute Errors, where || |/1 denotes the LI norm:
D
1 T
MAE Σ
Ilyer, Smmciianalll)
(25)
D
C DATA AND EXPERIMENT DETAILS
C.I DATA ANALYSIS
C.1.1 DATASET PROPERTIES
We evaluate our approach on five widely-used benchmark datasets sourced from the GluonTS library (Alexandrov et al. 2020), with preprocessing consistent with recent work (Coréés al. 2025). As
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Table Dataset characteristics and properties.
Dataset
Dim. D
Domain X'
Freq.
Time Steps
T
Trend
Seasonality Non-Gaussianity
Sol.
137
R+
Hour
7,009
24
0.1688
0.8592
0.5004
Elec.
370
R+
Hour
5,833
24
0.6443
0.8323
0.3579
Exch.
8
R+
Day
6,071
30
0.9982
0.1256
0.2967
Traf. 963
10,
Hour
4,001
24
0.2880
0.6656
0.2991
Wiki.
2,000
N
Day
792
0.5253
0.2234
0.2751
summarized Table 5 these datasets span multiple domains and exhibit diverse characteristics in terms of dimensionality (Dim. D), data domain (A'), frequency, and length. To further characterize the data within the forecast horizon (T'), we include three quantitative indicators from a recent benchmark, ProbTS (Zhang et al. 2024):: trend strength (Fr), seasonality strength (Fs), and Non-Gaussianity. This selection allows for a comprehensive evaluation across a spectrum of time series properties, from low to high dimensionality and from strong periodicity to wend-dominated behavior.
Electricity (Elec.) contains the hourly power consumption of 370 clients. It exhibits strong seasonality (Fs 0.83) due to daily and weekly human activity patterns, along with noticeable trend (Fr = 0.64).
Exchange (Exch.) records the daily exchange rates of cight currencies. As is common with financial data, it is heavily dominated by trend (Fr 0.99) and shows very weak scasonality
(Fs = 0.13).
Solar (Sol.) consists of the hourly solar power output from 137 locations. It has the strongest seasonality (Fs = 0.86) our benchmark due to the clear day-night cycle, but a very weak underlying trend (Fr 0.17). also displays the highest non-Gaussianity.
Traffic (Traf.) measures the hourly occupancy rates of 963 road sensors. It shows moderate patterns, coupled with relatively weak trend
seasonality (Fs 0.67) driven by daily
(Fy = 0.29).
Wikipedia (Wiki.) contains the daily page views for 2000 Wikipedia articles. As the most high-dimensional dataset, its series are characterized by moderate trend (Fr = 0.53) but weak seasonality (I's 0.22).
C.1.2 APPLICABILITY OF THE PROPOSED TIMEPRISM
Our proof-of-cooeoo ofoo--of-concept model, TimePrism, is built upon backbone that decomposes the time series into trend and seasonal components. As shown in Table all five benchmark datasets exhibit significant presence of either trend or seasonality, providing a solid foundation for this decomposition-bun architecture perform well.
However, it is crucial to distinguish the contributions of the paradigm from those of the specific backbone. The remarkable performance of TimePrism, achieving out of 10 state-of-the-art results, primarily attributable to the fundamental shift in the learning objective introduced by the Probabilistic Scenarios paradigm. By transforming the complex task of continuous density estimation into more structured problem of learning a discrete distribution over a combinatorial scenario space, the paradigm itself simplifies the learning challenge. The decomposition backbone merely provides simple yet effective way to generate the initial candidate scenarios for this paradigm.
Consequently, while the current implementation of TimePrism might be less suitable for datasets where both trend and seasonality are weak, this does not diminish the validity of the underlying paradigm. The Probabilistic Scenarios framework itself makes no assumptions about the data's 's characteristies and be integrated with more advanced backbones better suited for different data characteristics in future work.
C.2 IMPLEMENTATION DETAILS OF PROPOSED TIMEPRISM
This section provides the exact formulations for the loss functions used to train TimePrism in the multivariate setting. The total loss, Lerims the sum of reconstruction loss and probability loss.
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For the multivariate case, the loss is computed per-channel basis and then averaged across all D channels.
For each channel d {L,.... D}, we first identify the channel-specifie winner index, nz:
n2 arg min lyyg.. Vn.all2
(26)
w=1...N
The total reconstruction loss, incorporating the Relaxed- WTA mechanism, is the average of the per-channel relaxed losses:
D (1-c)
Croon = D d=1
Lrgid
Σ Ln.d
(27)
N
where Lnnd 11y Vn,al12 is the MSE for the n-th scenario the d-th channel.
Similarly, the total probability loss is the average of the per-channel Pross-Entropy losses, where each channel's probability distribution is optimized against its own winner:
Cprob
CrossEntropyy Ta, ni)
(28)
D
d=1
The following subsections provide detailed motivation for the two key components of these loss functions: the Relaxed-
Miixeed-WAAA WTA mechanism and the per-channel, weight-sharing design.
C.2.1 RELAXED WINNER- TEKER-TTKKS-AAA ALL Loss
The motivation for the relaxed variant in Eq. equation 27 addresses potential issue the standard WTA objective. In the standard formulation (c 0), non-winning scenarios receive zero gradient for a given training instance. This can lead to parameter stagnation if certain scenarios are consistently not selected winners across the dataset. By providing small, non-zero gradient to all non-winning scenarios (controlled by the hyperparameter e 0.01 in our work), the relaxed loss ensures that all parameters in the scenario- gentratingg layers receive continuous updates, promoting more robust and stable optimization (Rupprecht et al. 2017
C.2.2 WEIGHT SHARING
To maintain the structural simplicity and lightweight nature TimePrism, we adopt a weight-sharing strategy for handling multivariate time series. Instead of learning a separate set of parameters for each of the 1) variates, the three linear layers in our model (Trend, Season, and Probability layers) share their weights across all variates. This design significantly reduces the total parameter count (Zeng et al. 2023).
As detailed in Eq, equation 28 TimePrism learns separate probability distribution (parameterized by πT) over the shared set of scenarios for each channel, rather than explicitly modeling the joint probability distribution. However, the use of weight sharing allows the model to implicitly learn cross-chts chassechannel relationships during training. Because the weights the linear layers are shared, the gradient used to update them is aggregation of the gradients from all D channels. This forces the model to learn a basis of trend and seasonal components, along with their probabilistic mappings, that is collectively useful for the entire multivariate system. Thus, while the model fully decoupled across channels during inference, the training process is coupled, enabling the simple architecture to capture implicit cross-channel structures. This design choice directly explains the model's 's performance on the nigh-dimensional (2000 variates) Wikipedia dataset. The weight- searing assumption is less likely to hold datasets with high channel heterogeneity, where each series may follow distinct pattern. The observed lower performance on this specific dataset is therefore an expected consequence of our intentionally simple, weight-sharing design, rather than a flaw in the Probabilistic Scenarios paradigm itself.
C.3 TRAINING PROCEDURE
Baseline Configurations. The configurations for all baseline models, including DeepAR, TimeGrad, TempFlow, Transformer TempFlow, TACTiS-2, and TimeMCI, adhere the experimental setups
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Table MAE. Results on five benchmark datasets, reported as the mean + standard deviation over three random seeds. Lower is better. The best result is in bold, and the second best is underlined.
Model
Elec.
Exch.
Sol.
Traf.
Wiki.
ETS
0.577 + 0.00
1.90 0.05
0.558 + 0.00
1,21 + 0.00
4.81 0.13
DeepAR
1.12 0.02
1.09 0.01
0.921 = 0.04
1.36 0.05
3.83 0.96
TimeGrad
0.369 + 0.00
1.33 = 0.35
0.383 = 0.00
0.278 0.00
1.03 = 0.03
TempFlow
0.633 + 0.12
1,73 = 0.27
0,451 + 0.03
1.02 + 0.00
2.74 + 0.19
Trf.Flow
0,633 + 0.12
1.73 = 0.27
0,451 + 0.03
1.02 + 0.00
2.74 + 0.19
Tactis2
0,467 + 0.03
1.02 + 0.03
0.388 + 0.03
0.420 = 0.02
0.944 0.01
TimeMCL
0.519 + 0.00
1.38 0.21
0.431 + 0.04
0.438 0.03
1.23 + 0.09
TimePrism 0.171 + 0.03
0.666 0.01
0.0832 + 0.01
0.144 0.00
0.995 + 0.01
established in prior work (Cortés et al., 2025 and other training details. In this work, TimeMCL is configured with N 16 scenarios, consistent with its original implementation (Cortés et al., 2025). We deem this a fair comparison because TimeMCL's autoregressive structure computationally intensive. Even with only 16 scenarios, its inference FLOPs (8.8 x 10%) are an order of magnitude higher than TimePrism's 's with 625 scenarios (5.1 10 The original work presents two variants, relaxed-WTA (r-WTA) and Manealed-WTI, (a- WTA). Based on their reported results in Table I of their work, the r-WTA variant achieved stronger performance (3 place and second-place results versus second-place results for WTA) Therefore, we use the more competitive r-W r-WAA variant as our baseline. All other configurations for TimeMCL are kept identical to the original work.
encompassing model architecture, hyperparameters,
Batch Size and Scaler. Following the setup in Cortés et al. 2025). all baselines are trained with batch size 200, with the exception of TimeGrad, which uses batch size of 100 on the Wikipedia dataset due to memory constraints. For TimePrism, we use a batch size of 100 for all datasets except Wikipedia, for which batch size of 50 is used. While TimePrism has very low inference FLOPs, our intentionally simple implementation is not optimized for memory efficiency, necessitating slightly smaller batch size on high-dimensional datasets. The data scaler configurations for all baseline models are identical to those used in Cortés et al. (2025). For TimePrism, we use the 'mean std' scaler for the Exchange dataset and the "mean' scaler for all other datasets.
Proposed TimePrism Configuration. The number of scenarios N in TimePrism is automatically factorized into the two closest integers for the number of trend (M) and seasonal (K) components. In our main experiments, N is set to 625, corresponding Given the hourly (24) and daily (30) frequencies of our datasets, we set the decomposition kernel size 7. An analysis of the effect of different values of N on performance is provided in subsequent appendix.
configuration of M 25 and K 25.
Historical Context Length. Nominally, for datasets sourced from GluonTS, the input look back length is often set equal to the prediction horizon T (Zhang et al. 2024 Cortés et al.] [025 Alexandrov al. 2020). However, in practice, some models, like TimeMCI. are designed use longer history by incorporating lagged features. Modifying these structural designs to only use an input of length would be complex and potentially unfair. We therefore adhere to their established configurations. In contrast, our implementation of TimePrism requires only a look-back window of length T'. It is noteworthy that TimePrism achieves strong results even with less historical information, highlighting the potential of the new paradigm. For comprehensive comparison, also provide results in subsequent appendix where TimePrism uses the full available history as input, which we term "Full History".
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Table MSE. Results on five benchmark datasets, reported as the mean + standard deviation over three random seeds. Lower is better. The best result is in bold, and the second best is underlined.
Model
Elec.
Exch.
Sol.
Traf.
Wiki.
ETS
0.519 + 0.01
3.96 + 0.30
0.455 0.00
2.09 + 0.01
550 + 47.70
DeepAR
1,42 0.07
1.47 = 0.04
1.19 0.07
1.87 = 0.06
11.5 4.00
TimeGrad
0.278 = 0.01
2.43 = 1.28
0.361 + 0.00
0.190 = 0.00
1.84 = 0.10
TempFlow
3.84 + 3.21
4.73 + 2.14
0.463 + 0.07
1.04 = 0.01
676 + 302.55
Trf.Flow
3.84 + 3.21
4.73 + 2.14
0.463 + 0.07
1.04 0.01
676 + 302.55
Tactis2
0.366 + 0.03
1.23 + 0.07
0.365 0.06
0.368 + 0.02
1.34 0.19
TiMeMCL
0.393 0.01
2.46 + 1.11
0.542 0.13
0.319 0.02
13.7 18.85
TimePrism 0.104 0.02
0.712 0.09
0.0769 0.01
0.0983 0.01
1.28 0.02
D ADDITIONAL EXPERIMENTS
D.1 RESULTS OF COMPLEMENTARY METRICS
MAE. Table 6 presents the results for the Mean Absolute Error, reported as the mean + standard deviation over three random seeds (3141, 3142, 3143). As both MAE and our primary metric, CRPS, based the L1 norm, the overall ranking of the models shows similar pattern. TimePrism achieves the best performance on four out of five datasets and the second best on Wikipedia, reinforc- ing the conclusions from our main results and demonstrating its strong performance in terms of the median forecast.
MSE. The Mean Squared Error results presented in Table reported the d: standard deviation over three random seeds (3141, 3142, 3143). As a metric based on the L2 norm, the MSE more sensitive to large errors or outliers. The results show a consistent pattern where TimePrism outperforms all baselines across all five datasets, demonstrating the robustness of its mean forecast even under a stricter, squared-error evaluation.
D.2 EXPERIMENTS ON HISTORY LENGTH CONFIGURATION
As discussed in the main text, some baseline models, such TimeMCL (Cortés et al.) 2025).
structurally designed to utilize a historical context longer than the nominal forecast horizon T' by incorporating lagged features. Modifying these established architectures only T' would be complex and potentially unfair. It is noteworthy that the main results for TimePrism are achieved using only this nominal input length T,, demonstrating the potential of the new paradigm even with less information.
input of length
For more direct comparison, present an additional experiment in Table where TimePrism uses the full available history, a variant we term "Full History" (Full His.). The length of this history set be comparable to the total context available the baselines' data processing modules as in Cortés et al. (025). The results show that using longer history does not consistently improve TimePrism's performance;
some cases, the scores are similar slightly worse, though still highly competitive. This is not perfectly fair comparison, engincering capabilities to extract value from long lagged inputs, while our simple linear model the full history directly. For such simple architecture, much longer input sequence introduce noise without a sophisticated mechanism to filter it, which explains why more data does not necessarily lead to better performance.
other models designed with feature
This highlights a potential direction for future work, where more advanced feature engineering or model structures could be integrated within our paradigm to better leverage longer historical contexts.
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Table 8: Main Results on Primary Metries with Full History. Comparison of Weighted CRPS and Distortion on five benchmark datasets. Lower is better. The best result is in bold, and the second best underlined. TimePrism (Full His.) refers to our model using the full historical context for more direct comparison with baselines.
Elec.
Exch.
Sol.
Traf.
Wiki.
Model
CRPS
Dis.
CRPS
Dis.
CRPS
Dis.
CRPS
Dis.
CRPS
Dis.
ETS
0.376
1.23
1,23
1.98
0.374
1.03
0.815
2.69
4.88
142
DeepAR
0.993
2.79
0.698
1.89
0.607
1.11
0.829
1.82
1.41
4.88
TimeGrad
0.230
0.720
0.739
1.33
0.237
0.587
0.163
0.540
0.516
1.62
TempFlow
0,449
1.73
0.988
1.55
0.278
0.555
0.613
1.01
1.81
71.6
Trf.Flow 0,449
1.73
0.988
1.55
0.278
0.555
0.613
1.01
1,81
71.6
Tactis2 0,285
0.637
0.641
0.919
0.222
0.567
0.243
0.55
0.481
1.37
TimeMCL
0.375
0.603
1,30
1.10
0.301
0.485
0.251
0.455
0.624
1.32
TimePrism
0.148
0.237
0.456
0.588
0.0835
0.140
0.109
0.140
0.508
1.01
TimePrism 0.210
0.475
0,461
0.594
0.224
0.295
0.184
0.346
0.505
1.025
(Full His.)
Table 9: Generalizability Analysis with Transformer Backhone. Comparison of Primary Metrics (CRPS and Distortion) across five datasets. TimePrism-iT represents the iTransformer (Liu et al.
2023)
architecture adapted our Probabilistic Scenarios paradigm. All experiments use Seed 3141.
Elec.
Exch.
Sol.
Traf.
Wiki.
Model
CRPS
Dis.
CRPS
Dis.
CRPS
Dis.
CRPS
Dis.
CRPS
Dis.
DeepAR
0.993
2.79
0.698
1.89
0.607
1.11
0.829
1.82
1.41
4.88
TimeGrad
0.230
0.720
0.739
1.33
0.237
0.587
0.163
0.540
0.516
1.62
TempFlow
0.449
1.73
0.988
1.55
0.278
0.555
0.613
1.01
1.81
71.6
Trf.Flow
0.449
1.73
0.988
1.55
0.278
0.555
0.613
1.01
1.81
71.6
Tactis2 0.285
0.637
0.641
0.919
0.222
0.567
0.243
0.55
0.481
1.37
TimeMCL
0.375
0.603
1.30
1.10
0.301
0.485
0.251
0.455
0.624
1.32
TimePrism 0.148
0.237
0.456
0.588
0.0835
0.140
0.109
0.140
0.508
1.01
TimePrism-iT
0.330
0.600
0.454
0.681
0.164
0.245
0.201
0.371
0.756
1.425
D.3 PARADIGM GENERALIZABILITY. ADAPTING TO TRANSFORMER ARCHITECTURES
To more rigorously validate that the superior performance of our method stems from the proposed Probabilistic Scenarios paradigm rather than solely the specific linear architecture of TimePrism, we conducted a controlled study adapting a distinct, complex architecture to our framework. We selected one of the state-of-the-art Transformer-based time series models, iTransformer (Liu et al. (023). as the backbone.
Experimental Setup. We developed a variant named TimePrism-iT,, where the linear encoder of TimePrism is replaced by the inverted Transformer structure from Liu et al. (2023). Crucially, to demonstrate the "out-of-the-box" applicability and robustness our paradigm, we did not perform extensive hyperparameter tuning for TimePrism-iT. Instead, we applied generally consistent config- uration across all datasets. This setup serves as a rigorous stress test to verify if the paradigm can yield performance gains without relying
cechitecture-speciii optimization.
Results Analysis. The comparative results are presented in Table Despite being an unoptimized implementation, TimePrism-iT demonstrates remarkable performance. It outperforms standard
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Table 10:
The results of models in datasets from GIFT-Eval (Aksu et al. 2024 and fev-bench Shchur et al. 2025). The best result in each column is in bold.
Dataset
UCI
Hosp.
Hier.
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Distortion
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Distortion
ETS 0.450
0.811
0.585
1.30
0.990
4.03
0.782
2.23
Tactis2
0.605
0.787
0.583
1.20
0.623
1.58
0.614
1.13
TimeMCL
0.359
0.449
0.722
1.13
1.08
1.33
0.771
0.597
TimePrism 0.261
0.394
0.565
1.06
0.602
1.03
0.907
1.11
baselines on 6 out of 10 metrics across the five datasets. Notably, on the Exchange dataset, TimePrism- iT achieves CRPS of 0.454, slightly surpassing even the original linear TimePrism (0.456).
D.4 EXTENDED EVALUATION ON ADDITIONAL BENCHMARKS
To provide a more comprehensive evaluation of our proposed paradigm, we extended our experiments to include four datasets selected from two latest benchmarks: Gift-Eval (Aksu et al. 2024) and fey-bench (Shchur et al. 2025). These datasets were chosen to cover diverse domains: Hierarchical Sales (Retail, abbr. Hier.), M-DENSE (Mobility, abbr. M-Den.), Hospital Admissions (Healtheare, abbr. Hosp.), and UCI Air Quality (Nature, abbr. UCI).
Experimental Setup. For these experiments, we selected the numerical baseline ETS and the two most competitive neural models from Table | and Table 2 namely TimeMCL and Tactis2, for comparison. All models were evaluated using a random seed of 3141 ensure reproducibility.
Results Analysis. As shown in the additional results in Table 10 TimePrism maintains its strong per- formance across these new domains. Regarding the M-DENSE dataset, we observed that TimePrism exhibits relatively higher distortion. We hypothesize that the nature of this dataset may be more suitable for RNN backbones, as both Tactis-2 and TimePrism perform suboptimally on this dataset, while MCM remains competitive. This is not limitation of our new paradigm, but rather consequence of TimePrism's simple structure. Nevertheless, achieving SOTA results out of 18 metrics across 9 datasets still demonstrates the effectiveness of the TimePrism model and highlights the potential of the new paradigm.
D.5 PROBABILITY CALIBRATION DIAGNOSTICS
To rigorously assess the reliability of the probabilities assigned by TimePrism, we employ two standard diagnostic tools: the Reliability Diagram (Coverage vs. Nominal Confidence) and the Probability Integral Transform (PIT) Histogram.
Methodology. Since TimePrism outputs tuple (Dpred. p) consisting of finite set of scenarios Ppred= = y. j and their associated probabilities p = (PI,..., ,PN), we compute these metries as follows:
PIT Histogram: For a ground truth observation Ygt. the PIT value is the cumulative probability of scenarios that are less than or equal to the observation: PIT = E Pn. I[y,, S Yg). For perfectly calibrated model, the distribution of PIT values over the test set should approach a Uniform distribution U(0, 1), resulting in a flat histogram.
Reliability Diagram: We calculate the empirical coverage for varying nominal confidence levels E [0, 1]. The prediction interval for level is constructed by aggregating the scenarios yn, with the highest probabilities until their cumulative sum reaches a. If the model is well-calibrated, the curve should align with the diagonal y x.. Curves above the diagonal indicate under-confidence (conservative), while curves below indicate over-confidence.
Analysis. We performed these diagnostics on two representative datasets: Exchange and Solar. The results visualized in Figure
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Figure 5: Calibration Diagnostics. The diagnostics show different behaviors across datasets: (a)()) Exchange dataset demonstrates near-perfect calibration; (c)d) Solar dataset exhibits slightly conservative profile to ensure robust tail coverage.
Exchange Dataset (Fig. 5a & 5b): The diagnostics indicate near-perfect calibration. The PIT histogram is remarkably flat, and the Reliability Diagram closely follows the ideal diagonal line. This suggests that for stable financial data, TimePrism accurately estimates the true uncertainty distribution.
Solar Dataset (Fig, 5c & 5d): The diagnostics exhibit a slightly conservative profile. The PIT histogram shows mild hump shape, and the Reliability curve lies slightly above the diagonal. This behavior is expected and often desirable for highly stochastic, multimodal data like Solar energy. It indicates that TimePrism tends to widen its predicted scenario distribution to safely encompass multimodality and potential outliers. This "conserva- tive" strategy ensures robust coverage of low-probability, high-impact tail events without becoming over-contident, aligning with our design goal of prioritizing coverage adequacy.
D.6 ADDITIONAL VISUALIZATION AND QUALITATIVE ANALYSIS
Window Selection Rule. To provide fair and insightful qualitative comparison, we developed systematic rule for selecting the windows to be visualized. For given dataset and variate, we first select query window from a recent part the historical data. We then search through the entire history to find the five past windows that are most similar to this query window, based on Euclidean distance. To ensure that the selected windows represent distinct, non-overlapping events, we enforce minimum temporal separation between them. This greedy, iterative process allows to identify set of instances where the model repeatedly faced with a similar historical context, providing a controlled setting to analyze its predictive behavior.
D.6.1 VISUALIZATIONS ON OTHER DATASETS
To further demonstrate the applicability of our paradigm, we provide additional qualitative results for TimePrism on the Electricity and Traffic datasets in F The top panel showcases forecasts for the Electricity dataset. Across the selected windows, the model successfully generates diverse set of scenarios that cover the volatile and complex patterns of power consumption, assigning higher probabilities (thicker, blue lines) to the most plausible outcomes. The bottom panel of Figure displays the results for the Traffic dataset. Here, the model also produces a sharp and well-calibrated set of scenarios that effectively captures the distinct peaks and troughs characteristic of traffic flow
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Figure 6: Qualitative Analysis on Electricity and Traffic Datasets. Visualization of TimePrism's s probabilistic scenarios on two additional benchmark datasets. The top row shows forecasts for the Electricity dataset, and the bottom row shows forecasts for the Traffic dataset.
data. These visualizations further confirm that the Probabilistic Scenarios paradigm can generate meaningful forecasts across different domains and data characteristics.
D.6.2 FULL COMPARISON ON SOLAR
We now present a full visual comparison of all neural network-based baselines against TimePrism on the Solar dataset. We select two representative variates for this analysis: the first (D = 1) and the last (D 137). The figures display the top 10 scenarios from TimePrism, with line color and thickness representing probability from low (red, thin) to high (blue, thick), and 100 samples from each baseline model. The historical context is shown with gray background, while the future prediction horizon has a light green background.
Figure 7 shows the results for the first variate of the Solar dataset. Across the five selected windows, we observe several Common Cases of high-peak solar generation, along with two Rare Cases (third and fourth from the left) that exhibit more volatile or lower-peak behavior. For the common cases, TimePrism correctly assigns high probabilities (thicker, blue lines) to scenarios that accurately match the ground truth. Crucially, for the rare cases, it successfully identifies and covers these less frequent patterns while correctly assigning them lower probabilities (thinner, redder lines). In contrast, the sampling-basee based models, including the strong baseline TACTiS-2, tend produce cloud of samples centered around an average forecast. This often results in mean forecast that matches neither the common nor the rare cases well, and the sample envelope may fail to adequately cover the true outcome in the rare cases, demonstrating the limitations of Probability Absence and Coverage Inadequacy.
Figure 8 presents the analysis for the last variate of the dataset. This example provides clear distinction between four Common Cases and one Rare Case (far right). TimePrism again demonstrates the strength of the Probabilistic Scenarios paradigm: it allocates the majority of its probability mass to accurately predict the common high-peak cases, while still generating a low- low-probability scenario that correctly captures the rare low-peak future. The sampling-basee models, however, struggle with this scenario. Their samples tend to cluster around a mean that represents an uninformative compromise between the high and low peaks. This visually exemplifies how forecast lacking explicit probabilities can fail to provide actionable insights for decision making, especially when preparing for rare but critical outcomes.
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Figure 7: Qualitative Analysis on Solar (D=1). A visual comparison of forecasts from all neural network- based models the first variate of the Solar dataset. The figure highlights performance both common high-peak cases and two rare cases.
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Figure 8: Qualitative Analysis on Solar (D=137), A visual comparison on the last variate of the
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Reviews:
[{"review_id": "8NC420kqVT", "reviewer": "Reviewer_zH2j", "summary": "This paper tackles the problem of probabilistic time series forecasting, challenging approaches based on sampling. Indeed, the authors argue that sampling-based methods (parametric models, generative models, flow-based models) suffer from three issues: absence of probabilities associated with sampled trajectories, failure to cover edge cases, and inference cost. As a remedy, the authors propose TimePrism, an algorithm based on the winner-takes-all training paradigm that consists in three linear layers modelling respectively, scenarios of trend, seasonality, and the associated probabilities. Empirically, the results show that the proposed algorithm performs better than the considered baselines in 5 datasets.", "questions": "- See Weaknesses.\n- I have a question that might be more related to the winner-takes-all paradigm than to your specific method. How do you make sure that the selected best scenario is different so that different weights from the last layer can get a learning signal? Also in datasets where we only observe one realization of the target (and not very similar inputs either), what makes the different scenarios diverse and prevent them from collapsing into the observed sample?\n- Can you please elaborate more on the differences between your approach and that of TimeMCL? Skimming through the TimeMCL paper, I can see that the loss formulation has a very similar construction, based on a WTA term and a cross entropy term maximizing the probability of the selected scenario, which to me seems similar if not identic to what's proposed in your paper.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "- The paper is clearly written and well motivated.\n- The suggested training paradigm is innovative in the field of probabilistic time series forecasting. And given that it consists of only linear layers, it opens up a new horizon of lightweight methods that can challenge the recent rise of Foundation models that are efficient in the zero shot mode.", "weaknesses": "- My main concern with the paper is the evaluation scope which is limited to 5 datasets, some of which have already been criticized in the literature. For instance, [1] shows how many methods become equivalent when looking at a relative scale of the MSE metric for datasets such as Electricity and Exchange. Furthermore, [2] showed that even a constant no-change baseline yields a very strong baseline in the Exchange dataset. Given the simplicity of the proposed algorithm, I believe that the paper will highly benefit from a more comprehensive evaluation in other benchmarks such as Gift-eval [3] and fev-bench [4]. \n- Another concern I have is about the Coverage Inadequacy aspect, which is evaluated by the authors using the error of the best scenario (or best sample for the baselines). I agree that this metric is relevant to evaluate coverage, however, I don't think this means that your method is able to \"represent low-probability, high-impact tail event\" as stated in the introduction. This would have been true if the authors could show what is the percentage of the selected best scenarios that actually are the least probable. Maybe a histogram of the best scenario ranks by probabilities can help further elaborating this aspect.\n- Finally, for the probability absence evaluation metric, the authors assign uniform probability to all samples coming from the sampling-based models. I think that this maneuver is making these models at a disadvantaged position compared to the proposed method. For instance, for parametric models, even without access to true probabilities (as these are 0 for single data points of continuous random variables), one can use the value of the density function on samples as a proxy to their probability, then normalize so that everything sums up to 1. This is just a suggestion and depending on how samples are generated for each baseline there might be smarter ways to make this comparison a bit more fair.\n\n[1] Brigato, Lorenzo, et al. \"Position: There are no champions in long-term time series forecasting.\" _arXiv preprint arXiv:2502.14045_ (2025).\n\n[2] Beck, N., Dovern, J. & Vogl, S. Mind the naive forecast! a rigorous evaluation of forecasting models for time series with low predictability. _Appl Intell_ **55**, 395 (2025)\n\n[3] Aksu, Taha, et al. \"Gift-eval: A benchmark for general time series forecasting model evaluation.\" _arXiv preprint arXiv:2410.10393_ (2024).\n\n[4] Shchur, Oleksandr, et al. \"fev-bench: A Realistic Benchmark for Time Series Forecasting.\" _arXiv preprint arXiv:2509.26468_ (2025).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ki0cbelDTE", "reviewer": "Reviewer_Ca7z", "summary": "The paper argues that sampling-based probabilistic forecasting has three shortcomings—no explicit probabilities per sample, coverage gaps for tails, and high inference cost—and proposes a new paradigm that predicts a finite set of {Scenario, Probability} pairs directly in one pass. As a proof-of-concept, TimePrism (three parallel linear layers for trend/seasonality and a probability head) achieves 9/10 SOTA across five benchmarks using CRPS (probability-weighted) and Distortion (best-case coverage) as a unified evaluation.", "questions": "- Is it possible to provide calibration diagnostics (e.g., PIT, coverage vs. nominal) for the predicted scenario probabilities?\n\n- Can authors explore soft assignment or entropy-regularized variants to reduce mode collapse and improve probability learning?\n\n- Is it possible to add a discussion on how to grow (N) adaptively at inference based on data complexity, and its computational trade-offs?\n\n- Are you planning to examine multivariate extensions (shared/global scenario bases vs. per-dimension) and correlation structure?\n\n- Is it possible to include decision-centric metrics (e.g., tail-risk costs) to emphasize the benefit of explicit probabilities?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Framing probabilistic forecasting as explicit discrete scenario prediction with calibrated probabilities is a compelling conceptual shift. While related threads exist (MCL/“winner-takes-all” heads; quantile/mixture models), combining explicit scenario probabilities with a simple linear architecture that performs competitively is surprising and potentially impactful for decision-making contexts where scenario weights matter. The evaluation mapping that compares sampling models (uniform weights) and scenario models (predicted weights) under CRPS and Distortion offers a fairer perspective on probability absence and coverage.\n\n- The model is intentionally simple: trend and seasonal linear layers produce a cartesian set of (N=M!\\times!K) scenarios, and a probability layer outputs logits (\\pi), trained with argmin-assignment + cross-entropy (a form of hard assignment akin to WTA), with reconstruction loss for the closest scenario. The training objective is clearly specified.\n\n- The critique of sampling (probabilities/coverage/cost) is accurate, and the metrics (CRPS weighted by (p_n); Distortion as min-RMSE over the set) operationalize those points. Conceptual reframing with practical benefits (explicit probabilities, one-pass inference) with a competitive accuracy with extreme simplicity, brings strong efficiency advantages.\n\n- Claims of broad efficiency gains appear well supported: TimePrism is orders faster than diffusion/flow/copula models evaluated with (S=100) samples while staying competitive on accuracy.", "weaknesses": "I see two technical caveats: (i) probability calibration is central but the paper does not show strong calibration diagnostics (e.g., reliability diagrams of event probabilities across horizons); (ii) the discrete scenario set size (N) trades computation vs. fidelity—there is some ablation showing performance improves with (N), but more systematic experiments would help to understand it better.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ymyPGi3dl7", "reviewer": "Reviewer_VcU5", "summary": "This paper introduces a novel paradigm for modeling the probabilistic future evolution of time series by outputting a discrete set of scenario-probability pairs. To demonstrate this paradigm's efficacy, the authors propose a simple model, TimePrism. The work is positioned as a proof-of-concept that the proposed discretization approach is a viable and effective direction for probabilistic forecasting.", "questions": "The authors should directly address the specific concerns raised in the \"Weaknesses\" section above. Further, there are additional questions that raise my concerns, but I would not directly count them as \"weakness\", which I put in this section:\n \n1. Dead Code Issue: The discrete scenario (\"code\") representation may suffer from a \"dead code\" problem, analogous to the issue in VQ-VAE. Since Figure 3 indicates that only the selected scenario receives gradients, a scenario may never be updated if it is never the best match for any forecast. In particular, a scenario gets update ONLY if a randomly intialized forecast exceed an already-updated scenarios. The authors should investigate and report the code usage rate across datasets. An analysis or experiment confirming that this is not a prevalent issue, or a modification to mitigate it, would strengthen the paper significantly.\n\n2. Interpretation of MSE Results: In the appendix, the MSE scores for short-term horizons (e.g., 24, 30) appear relatively low. Could the authors clarify why this is the case? Providing a baseline from a well-established forecasting model (e.g., DLinear, Koopa) for these horizons as a point of reference would help contextualize these results and furthur clarification of any potential differences in evaluation metrics is helpful.\n\n3. Connections to Discretization Literature: The core idea of discretization has clear parallels in other fields, most notably in VQ-VAE. The paper would benefit from a more thorough discussion of these connections, explicitly outlining the similarities and, more importantly, the distinctions between this work and prior art in discretized representations. This would better position the novelty of the contribution within the broader machine learning landscape.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Simple yet Effective Paradigm: The core modeling paradigm is elegantly simple and, as demonstrated, highly effective. While discretization is a classical machine learning technique, its application to this specific problem is well-motivated. The empirical results strongly suggest this is a worthwhile research direction.\n\n2. Strong Empirical Performance: The proposed TimePrism model, serving as a proof-of-concept, achieves state-of-the-art performance on key metrics (CRPS, Distraction). This makes it a compelling new method that is effectively plug-and-play for probabilistic forecasting tasks.", "weaknesses": "The Proof-of-Concept Gap: The primary weakness is the indirect link between the success of the TimePrism model and the proposed paradigm itself. As noted in recent literature [1, 2], simpler models can often outperform complex ones in time series forecasting. Therefore, TimePrism's strong performance could be attributed to its specific architecture rather than the paradigm shift.\nTo more directly validate the paradigm's contribution, the authors should provide a more controlled ablation. This could take one of two forms (preferably the second):\n\n1. A study showing that the TimePrism architecture, when used under a previous paradigm (e.g., predicting a continuous distribution), performs worse.\n\n2. A study adapting a selected existing method to the new discrete paradigm, demonstrating a performance lift attributable to the paradigm itself.\n\nWithout such a study, the claim of the paradigm's superiority remains suggestive rather than conclusive.\n\n[1] Zeng, Ailing, et al. ‘Are Transformers Effective for Time Series Forecasting?’ arXiv [Cs.AI], 2022, arxiv.org/abs/2205.13504. arXiv.\n[2] Lin, Shengsheng, et al. ‘SparseTSF: Modeling Long-Term Time Series Forecasting with 1k Parameters’. arXiv [Cs.LG], 2024, arxiv.org/abs/2405.00946. arXiv.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jkIK1IadLA", "reviewer": "Reviewer_NW9F", "summary": "The paper introduces a new paradigm for time-series forecasting based on the philosophy that predicting directly interpretable scenarios (along with their probabilities) is more effective and relevant than sampling-based methods. This is because (i) sampling-based methods don't directly provide probabilities for each scenario (ii) they are computationally costly at inference, especially when they require many samples to have a good coverage of the target distribution (iii) and they may fail to capture rare (yet critical) events. The proposed approach, TimePrism, is a follow-up of TimeMCL from Cortés et al. (2025). \n\nThe model is non-autoregressive, uses only three linear layers in their architecture (using shared weights across each dimension, and applied in parallel), making it extremely light, and showing strong CRPS and Distortion performance. Theoretical foundations of TimePrism are demonstrated, showing in particular that its non-autoregressive structure makes it easier to establish the conditional quantizer property (Proposition 1) and to capture scenario probabilities (Proposition 2) compared to TimeMCL.", "questions": "See the questions in the weaknesses above.\nWhat are the main limitations of the method? Limitations such as the fact that you can’t handle inputs/prediction of variable length during inference, and that weight sharing may fail to model cross-channel relationships during inference, should be acknowledged in the main paper in the next revision.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The approach is well motivated for practical applications and conceptually simple. It shows a good trade-off between quality and inference computational cost, and the foundations of the approach based on Multiple Choice Learning are well discussed.", "weaknesses": "To make sure I understood well, Tables 1 and 2 use $S = 100$ samples for each model, and $N = 625$ predictions (used for categorical sampling) for TimePrism ? It is very important to add a row in the table where TimePrism uses $N = 16$ (e.g., with $M = 4$ and $K = 4$), because otherwise it is not fair with respect to TimeMCL, which uses only $16$ scenarios (especially for Distortion comparison). I agree that the computational cost of TimePrism is smaller, but the Distortion comparison needs to include a comparison with the same number of hypotheses for completeness.\n\nAlso, in the whole paper, TempFlow and Transformers TempFlow gives the same numbers, which is odd. You need to double-check your code pipeline to verify/correct this potential bug.\n\n*Typos/issues:*\n\nL755: $R_{n}(x)$ should be formally defined.\n\nEq. (13): the left-hand side expectation should be written completely.\n\nEq. (19) / (20): You used two different notations for the cross-entropy. \n\nMore rigour is required in the introduction of the theoretical tools. In particular, the assumptions in Propositions 1 and 2 should be stated explicitly: you did not prove the convergence, but *necessary* conditions of convergence (i.e., if the model reaches a local optima, see e.g., Asm A.3 in Cortés et al., 2025, or Proposition 3.1 in Du et al., 1999).\n\nI am optimistic about the quality of the paper. I think the paper deserves acceptance, provided that the authors can show that these weaknesses and issues will be properly solved in the next revision.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)