ProbFM: Probabilistic Time Series Foundation Model with Uncertainty Decomposition

AI summary

Overview

Research area: Time series forecasting foundation models, probabilistic uncertainty quantification, and quantitative finance (cryptocurrency return forecasting).

Technical level: Advanced. The paper assumes familiarity with transformer architectures, Bayesian inference over distribution parameters, and Deep Evidential Regression.

Scope: The paper introduces ProbFM, a transformer-based probabilistic framework that applies Deep Evidential Regression with Normal-Inverse-Gamma priors to single-variate, single-step time series forecasting, and validates it against alternative uncertainty quantification approaches on cryptocurrency returns.

What This Paper Is About

Existing Time Series Foundation Models can forecast reasonably well, but they handle uncertainty poorly: they either fix a distributional form in advance, mix different sources of uncertainty together, or apply calibration after the fact rather than during learning. This matters most in finance, where a decision-maker needs to know whether a prediction is uncertain because the model has not seen enough data or because the market is inherently noisy. The paper's goal is to build a model that separates those two uncertainty types in a single forward pass, and to test that mechanism fairly by holding the architecture constant.

Key Contributions

1. First application of DER to TSFMs. The authors apply Deep Evidential Regression with Normal-Inverse-Gamma priors to time series foundation model architectures, adapting the approach of Amini et al. (2020) to sequential data with explicit epistemic-aleatoric decomposition.
2. Integrated coverage optimization. A coverage loss is combined with the standard evidential training objective so that prediction interval reliability is optimized directly, without post-hoc calibration or arbitrary binning.
3. Evidence annealing schedule. Built on the regularization annealing approach of Sensoy et al. (2018), an evidence annealing schedule directly controls evidence accumulation during training to prevent overconfidence in early phases (called out as a guard against evidence collapse).
4. Controlled empirical evaluation. Five probabilistic methods (DER, Gaussian NLL, Student's-t NLL, Quantile Loss, and Conformal Prediction) are compared on a consistent LSTM backbone, isolating the uncertainty quantification strategy from architectural differences.
5. Financial decision-making validation. Uncertainty-aware trading is used to show how epistemic-aleatoric decomposition supports risk management by filtering high-uncertainty predictions.

Main Findings

• Competitive point accuracy. On BTC, Evidential Regression (ProbFM core) reached RMSE 0.045, MAE 0.03, and correlation -0.536. The MSE baseline recorded RMSE 0.044, MAE 0.03, correlation -0.309; Huber 0.044 / 0.029 / -0.394; Gaussian NLL 0.044 / 0.029 / -0.192; Student-t NLL 0.045 / 0.03 / -0.529; Quantile Loss 0.044 / 0.029 / -0.395; Adaptive Conformal (MSE) 0.044 / 0.03 / -0.314. The paper concludes that adding uncertainty quantification does not sacrifice predictive accuracy.
• Competitive but not best CRPS. Evidential Regression recorded CRPS 2.65 on BTC, compared with 2.21 for Gaussian NLL, 2.26 for Adaptive Conformal (MSE), and 3.17 for Student-t NLL. CRPS is reported as NA for MSE and Huber.
• Distinct calibration behavior, with under-coverage at 95%. Evidential Regression achieved PICP (95%) of 0.46, versus 0.95 for Gaussian NLL, 0.95 for Student-t NLL, 0.81 for Adaptive Conformal (MSE), and 0.75 for Quantile Loss. The paper frames this as "distinctly different calibration characteristics" rather than as a calibration success.
• Much sharper intervals. Sharpness (95%) was 3.9 for Evidential Regression, against 15.22 (Gaussian NLL), 15.48 (Student-t NLL), 9.67 (Adaptive Conformal (MSE)), and 7.92 (Quantile Loss).
• Uncertainty-error correlation of zero. The uncertainty-error correlation was 0 for Evidential Regression, versus 0.12 (Gaussian NLL), 0.11 (Student-t NLL), 0.17 (Quantile Loss), and 0.09 (Adaptive Conformal (MSE)).
• Best risk-adjusted trading metrics on BTC. Evidential Regression recorded Annual Sharpe 1.33, Annual Sortino 2.27, Max Drawdown -15.14 bps, Calmar 3.04, and Win Rate 0.52. Comparators: MSE baseline 0.9 / 1.52 / -15.89 / 1.98 / 0.51; Huber 0.59 / 0.93 / -15.14 / 1.35 / 0.51; Gaussian NLL 0.49 / 0.77 / -15.14 / 1.12 / 0.5; Student-t NLL 0.7 / 1.08 / -15.89 / 1.41 / 0.51; Quantile Loss 1.13 / 1.9 / -15.14 / 2.6 / 0.52; Adaptive Conformal (MSE) 0.98 / 1.64 / -15.14 / 2.25 / 0.51.
• Drawdown is essentially flat across methods. Maximum drawdown was -15.14 bps for most methods (Evidential Regression, Huber, Gaussian NLL, Quantile Loss, Adaptive Conformal) and -15.89 bps for MSE and Student-t NLL. The paper argues the performance differences therefore come from prediction quality and uncertainty-aware position sizing, not from avoiding extreme losses.
• Predicted returns are more dispersed than actual returns. Figure 2 shows that, across cryptocurrency assets using Evidential Regression, predicted returns exhibit significantly wider distributions than actual returns for most volatile cryptocurrencies.

Methodology in Plain English

The model forecasts one step ahead for a single univariate series. Raw input is split into patches, with the patch size chosen automatically from the data frequency using the mapping proposed by Woo et al. (2024); patches are linearly projected and given sinusoidal positional encoding. A standard transformer (multi-head attention plus a SwiGLU feed-forward network) produces representations, and a dedicated head maps those representations into the four parameters of a Normal-Inverse-Gamma distribution: the mean, a precision term, and two parameters governing the inverse-gamma part. Softplus activations plus additive constants keep each parameter in its valid domain.

The key idea is that rather than predicting the parameters of one fixed distribution, the model predicts a distribution over those parameters. That extra layer of uncertainty is what allows the total predictive variance to be split into an epistemic part (reducible with more data) and an aleatoric part (irreducible noise). Both are closed-form expressions of the four NIG parameters, so everything is obtained in one forward pass with no sampling and no calibration set.

Training combines three terms: an evidential negative log-likelihood, a regularization term that penalizes high confidence when predictions are wrong, and a coverage term that penalizes the gap between the target prediction-interval coverage probability and the achieved one. The whole objective is optimized with AdamW, gradient clipping (max norm between 1.0 and 5.0), and cosine annealing with warmup over 5-15% of training steps. An evidence annealing factor starting at 0 and rising linearly to 1.0 over 10-20% of training steps prevents the model from becoming overconfident before it has learned meaningful patterns.

For the controlled comparison, the authors deliberately swapped out the transformer for a 1-layer LSTM with 32 hidden dimensions and 0.1 dropout, so that only the loss function and output head differed across methods. Each method's head was sized to its needs: one output for MSE, one for Huber, two for Gaussian NLL, three for Student-t NLL, multiple quantiles for Quantile Loss, mixture weights/means/variances for mixture models, an MSE base plus a calibration set for conformal prediction, and four NIG outputs for evidential regression. Location parameters were bounded with a 3.0 × tanh transform, keeping predictions within [-3, 3] in standardized units (about ±3 standard deviations).

Data came from 11 of the most liquid cryptocurrencies on Stooq — ADA, BNB, BTC, DASH, DOGE, ETH, LTC, SOL, USDC, USDT, and XRP — spanning January 1, 2020 to October 3, 2025 (subject to each asset's availability). Targets were 1-day log returns, r_t = log(P_t / P_{t-1}), with a 50-step lookback window, and targets scaled to percentage (multiplied by 100) for training. Sequences were built per symbol and then combined while preserving symbol identity. For trading evaluation, the strategy goes long when the predicted return is above 0 and short when it is at or below 0, with PnL = signal × actual_return × 100 in basis points. A separate uncertainty-aware variant executes trades only when predicted total uncertainty falls below the 75th percentile threshold. Metrics were computed on the test set using the best checkpoint selected by validation loss, with predictions denormalized before trading metrics were calculated.

Why This Matters

Impact on research. The paper's main methodological argument is about experimental design: because recent foundation models differ in both architecture and uncertainty treatment, it is hard to tell which innovation drives which gain. Holding the backbone constant while varying only the loss and head is a clean way to attribute results. The paper also claims a first — evidential regression with NIG priors inside a TSFM — which opens a route to uncertainty quantification that needs neither distributional pre-specification nor sampling at inference.

Real-world applications:

• Risk-aware position sizing. Epistemic and aleatoric estimates can scale exposure rather than just producing a point forecast.
• Trade filtering. Skipping trades in high-uncertainty regimes, as tested with the 75th-percentile uncertainty threshold.
• Portfolio allocation. The paper describes uncertainty-based position sizing outperforming equal-weighted baselines by adjusting capital allocation to prediction confidence.
• Interval reporting for downstream consumers. Closed-form prediction intervals from the Student-t predictive distribution can be surfaced directly, without a separate calibration pass.

Industry relevance. All three authors are affiliated with JPMorganChase (Palo Alto and London), and the framing throughout is quantitative finance — where knowing whether uncertainty reflects insufficient data or inherent market randomness changes how a position should be sized. The paper carries an explicit JPMorganChase disclaimer stating that it is not investment research or advice.

Future Directions

• Multi-horizon and multivariate extension. The authors state that the framework generalizes by replacing the NIG prior with a Normal-Inverse-Wishart prior, where the mean becomes a vector in R^H for H-step-ahead forecasting and the covariance becomes an H × H matrix capturing dependencies across time steps and variables. They deliberately restricted this paper to the single-variate, single-step case as a foundation.
• Scaling the evidential approach to full foundation model architectures. The controlled study used a 1-layer LSTM backbone; the paper presents the transformer architecture as the intended framework, and the natural next question is how DER behaves at foundation-model scale and pre-training.
• Reconciling the calibration results. Evidential Regression reached PICP (95%) of 0.46 despite the integrated coverage loss, alongside the sharpest intervals (3.9) and zero uncertainty-error correlation. Whether this reflects a genuine trade-off between sharpness and coverage, or a tuning issue with the coverage weight, is left open.
• Quantifying the value of decomposition specifically. The paper argues epistemic-aleatoric decomposition drives trading gains, but the reported trading table does not isolate decomposition from total-uncertainty filtering. Separating those effects would sharpen the central claim.

Target Audience

Quantitative researchers and machine learning engineers working on probabilistic forecasting, especially those building or evaluating time series foundation models. It is also relevant to quantitative finance practitioners interested in uncertainty-aware execution, position sizing, and portfolio allocation, and to methodologists studying evidential learning, calibration, and the design of controlled comparisons across uncertainty quantification strategies. Readers without a background in Bayesian uncertainty decomposition or transformer architectures will find the methodology sections demanding.

Note on completeness: The main text reports detailed results for BTC and states that forecasting accuracy and trading results for the other ten cryptocurrencies (ADA, BNB, DASH, DOGE, ETH, LTC, SOL, USDC, USDT, XRP) appear in the Appendix; those appendix result tables are not included in the provided content. Related work and an explicit limitations section are also referenced as being in the Appendix but are not present in the provided text. Table 1 lists Mixture of Distributions as an evaluated approach, but it does not appear in the BTC results tables. Maximum drawdown is reported in bps; other monetary or capital-scale figures are not reported.

Authors’ abstract

Time Series Foundation Models (TSFMs) have emerged as a promising approach for zero-shot financial forecasting, demonstrating strong transferability and data efficiency gains. However, their adoption in financial applications is hindered by fundamental limitations in uncertainty quantification: current approaches either rely on restrictive distributional assumptions, conflate different sources of uncertainty, or lack principled calibration mechanisms. While recent TSFMs employ sophisticated techniques such as mixture models, Student's t-distributions, or conformal prediction, they fail to address the core challenge of providing theoretically-grounded uncertainty decomposition. For the very first time, we present a novel transformer-based probabilistic framework, ProbFM (probabilistic foundation model), that leverages Deep Evidential Regression (DER) to provide principled uncertainty quantification with explicit epistemic-aleatoric decomposition. Unlike existing approaches that pre-specify distributional forms or require sampling-based inference, ProbFM learns optimal uncertainty representations through higher-order evidence learning while maintaining single-pass computational efficiency. To rigorously evaluate the core DER uncertainty quantification approach independent of architectural complexity, we conduct an extensive controlled comparison study using a consistent LSTM architecture across five probabilistic methods: DER, Gaussian NLL, Student's-t NLL, Quantile Loss, and Conformal Prediction. Evaluation on cryptocurrency return forecasting demonstrates that DER maintains competitive forecasting accuracy while providing explicit epistemic-aleatoric uncertainty decomposition. This work establishes both an extensible framework for principled uncertainty quantification in foundation models and empirical evidence for DER's effectiveness in financial applications.