# Expectations vs. Realities: The Cost of MSE-Optimal Forecasting Under Conditional Uncertainty

arXiv: https://arxiv.org/abs/2606.04342 · HTML: https://mlsanity.com/paper/2606.04342/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 31 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Deep Time Series Models: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2407.13278/) (2024)
- [Learning the Pareto Front with Hypernetworks](https://mlsanity.com/paper/2010.04104/) (2020)
- [Exploring Progress in Multivariate Time Series Forecasting: Comprehensive Benchmarking and Heterogeneity Analysis](https://mlsanity.com/paper/2310.06119/) (2023)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [TSFM-Bench: A Comprehensive and Unified Benchmark of Foundation Models for Time Series Forecasting](https://mlsanity.com/paper/2410.11802/) (2024)
- [ProbTS: Benchmarking Point and Distributional Forecasting across Diverse Prediction Horizons](https://mlsanity.com/paper/2310.07446/) (2023)
- [Deep Learning for Time Series Forecasting: A Survey](https://mlsanity.com/paper/2503.10198/) (2025)
- [AutoXPCR: Automated Multi-Objective Model Selection for Time Series Forecasting](https://mlsanity.com/paper/2312.13038/) (2023)
- [Time-Series Classification for Dynamic Strategies in Multi-Step Forecasting](https://mlsanity.com/paper/2402.08373/) (2024)
- [Epistemic Error Decomposition for Multi-step Time Series Forecasting: Rethinking Bias-Variance in Recursive and Direct Strategies](https://mlsanity.com/paper/2511.11461/) (2025)

## Built on by (most cited first, 3 of 3)

- [Exposure Bias as Epistemic Underidentification in Recursive Forecasting](https://mlsanity.com/paper/2606.12990/) (2026)
- [Forecast Collapse in Time-Series Foundation Models](https://mlsanity.com/paper/2608.14106/) (2026)
- [Deployment-Side Adaptiveness in Multi-Horizon Volatility Forecasting](https://mlsanity.com/paper/2606.27688/) (2026)

## Abstract (arXiv)

Multi-step time series forecasting (MSF) is commonly evaluated using point-wise error metrics such as mean squared error (MSE), implicitly treating the conditional mean as a sufficient target. We show that this can be misleading under conditional uncertainty, where the conditional expectation becomes unrepresentative of typical realized values at longer horizons. We formalize this effect through a conditional uncertainty gap and prove that whenever this gap is nonzero, no deterministic predictor can simultaneously minimize MSE and match the marginal distribution of realized futures. This establishes a fundamental, model-agnostic trade-off between point accuracy and marginal realism in MSF evaluation. Using controlled stochastic dynamical systems and nine real-world forecasting benchmarks, we empirically characterize the resulting accuracy--realism frontier and quantify the practical cost of MSE-only model selection. As conditional uncertainty increases with forecast horizon, the attainable set expands into a pronounced Pareto front, separating MSE-optimal but under-dispersed predictors from methods that trade accuracy for realistic marginal variability. textbfAcross benchmarks, we find that small relaxations in MSE (boldsymbol≤ 5%) frequently unlock disproportionate gains in marginal realism, with median improvements of 17.3% and gains exceeding 30% in some datasets. We further show that common forecasting strategies systematically occupy different regions of this frontier: direct multi-output predictors concentrate near the accuracy-optimal extreme, while recursive strategies and sample-based inference favors marginal realism. Together, these results expose a structural failure mode of MSE-based evaluation in long-horizon forecasting and recast strategy and inference selection as navigation of an unavoidable accuracy--realism trade-off.