ProbFM: Deep Evidential Regression for Uncertainty Decomposition in Financial Forecasting

Background

Time‑Series Foundation Models (TSFMs) have shown strong transferability and data efficiency for zero‑shot financial forecasting, but they struggle with principled uncertainty quantification. Existing approaches either impose rigid distributional assumptions, conflate epistemic and aleatoric uncertainty, or lack calibration guarantees, which limits risk‑aware decision making in finance.

Problem Definition

The paper identifies four key shortcomings of current TSFMs: (1) mixing different uncertainty sources, making it impossible to separate epistemic (reducible) from aleatoric (irreducible) uncertainty; (2) absence of a principled framework to learn optimal uncertainty representations; (3) reliance on predefined distribution forms or sampling‑based inference that is computationally expensive; (4) architectural differences that confound whether performance gains stem from the uncertainty strategy or the model architecture.

Method

ProbFM Architecture

ProbFM builds on a standard Transformer encoder for time‑series data. Input series are first partitioned into patches using a PatchTST‑style procedure, with an adaptive multi‑patch size strategy inspired by MOIRAI that selects patch lengths based on feature characteristics. Each patch receives a positional embedding that also encodes temporal information.

Patch embeddings are fed into the Transformer, which consists of multi‑head self‑attention and feed‑forward layers.

Deep Evidential Regression Head

The Transformer output is projected directly to the parameters of a Normal‑Inverse‑Gamma (NIG) distribution (μ, λ, α, β) via a linear head, enabling closed‑form predictive mean and variance.

Enhanced Loss Function

The training objective combines an evidential loss that encourages the model to allocate evidence proportional to prediction accuracy, a coverage loss that penalises prediction intervals that fail to contain the true value at a target confidence level, and a regularisation term.

Training and Optimization

ProbFM is trained in a single stage, jointly optimizing prediction accuracy and coverage loss. Optimization uses AdamW with gradient clipping, a cosine‑annealing learning‑rate schedule with warm‑up, and a fixed λ_{coverage} weight.

Inference and Uncertainty Decomposition

During inference a single forward pass yields the four NIG parameters, from which the total predictive variance is decomposed into epistemic (reducible) and aleatoric (irreducible) components. Confidence intervals are constructed from the NIG distribution.

Experiments

Dataset

The evaluation uses a cryptocurrency dataset downloaded from Stooq, containing daily log‑returns of the 11 most liquid assets from 2020‑01‑01 to 2025‑10‑03.

Baselines and Metrics

All methods share a 1‑layer LSTM with 32 hidden units as the backbone. Compared methods include MSE, Huber, Gaussian NLL, Student‑t NLL, Quantile Loss, Mixture of Distributions, Conformal Prediction, and Evidential Regression (ProbFM core). Accuracy is measured by RMSE, MAE, and Pearson correlation; probabilistic quality by CRPS, PICP, Sharpness, and Uncertainty‑Error Correlation; trading performance by risk‑adjusted returns (Sharpe, Sortino, Max‑Drawdown, Calmar, win rate).

Results

On BTC forecasting, ProbFM’s Evidential Regression achieves RMSE and MAE comparable to the baselines while slightly lower correlation, demonstrating that uncertainty decomposition does not sacrifice point‑prediction accuracy. In CRPS, Evidential Regression is competitive with Gaussian NLL, but calibration metrics (PICP, Sharpness, Unc‑Err Corr) differ across methods.

For trading, the uncertainty‑aware strategy yields an annualised Sharpe ratio of 1.33 and a Sortino ratio of 2.27, outperforming MSE and other probabilistic baselines. The model also attains the highest win rate and favourable Calmar ratio, indicating effective risk management.

Additional experiments on other cryptocurrencies (ADA, BNB, DASH, DOGE, ETH, LTC, SOL, USDC, USDT, XRP) show varying degrees of advantage for Evidential Regression, confirming its robustness across assets.

Conclusion

ProbFM demonstrates that a Transformer‑based architecture combined with deep evidential regression can provide principled, decomposed uncertainty estimates without degrading forecasting accuracy. The explicit separation of epistemic and aleatoric uncertainty enables uncertainty‑aware trading strategies that improve risk‑adjusted returns in cryptocurrency markets.