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Exposure Bias as Epistemic Underidentification in Recursive Forecasting

This paper reframes exposure bias in recursive forecasting as an epistemic underidentification problem caused by partial observability, demonstrating that models fail to generalize to self-generated states and proposing that incorporating provenance information into induced-state corrections can mitigate these errors.

Original authors: Riku Green, Zahraa S. Abdallah, Telmo M Silva Filho

Published 2026-06-12

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Riku Green, Zahraa S. Abdallah, Telmo M Silva Filho

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are teaching a robot to predict the weather.

The Old Way (The "Teacher" Problem)
Usually, we train the robot by showing it real history: "Yesterday it was sunny, today it rained." We force the robot to look at the real past to guess the future. This is called "Teacher Forcing." The robot learns perfectly when the teacher is present.

But in the real world, the robot has to predict the future on its own. It can't look at the real past anymore; it has to look at its own previous guesses. If the robot guesses "It will rain tomorrow," and then uses that guess to predict "It will be wet the day after," a small mistake on day one can snowball into a huge disaster by day ten. This is what experts call Exposure Bias.

The Paper's New Idea (The "Identity Crisis")
Most people think this happens just because the robot is looking at "fake" data instead of "real" data (a distribution shift).

This paper argues there is a deeper, more confusing problem: Epistemic Underidentification.

Here is the analogy:
Imagine you are a detective trying to solve a crime. You have a photo of a suspect (the "state").

Scenario A: You see the photo in a police file (Real Data). In this context, the photo belongs to a harmless baker.
Scenario B: The robot generates a fake photo that looks exactly the same (Induced State). But because it was generated by the robot's own mistakes, in this specific context, that same photo actually belongs to a criminal.

The paper proves that if you only train the robot to recognize the photo in the police file (Scenario A), it will fail when it sees the exact same photo in the robot's own generated world (Scenario B). The robot doesn't know how the photo was made. It only sees the picture, not the story behind it.

Because the robot doesn't know the "origin story" of the data it is looking at, it can't know the right answer. It's an identity crisis: the same input leads to two different correct answers depending on where it came from.

The Solution: Adding a "Provenance" Tag
The authors suggest giving the robot a little "name tag" or "provenance" label.

If the data came from the real world, the tag says "Real."
If the data came from the robot's own prediction, the tag says "Fake."

Even if the picture looks identical, the tag tells the robot: "Hey, this is a fake picture, so the answer is different than if it were real."

What They Found
The researchers tested this on three different datasets (like weather patterns and energy usage) and found:

The Robot Drifts Away: As the robot predicts further into the future, the data it looks at becomes increasingly different from the real data it was trained on. It enters a "fake world" where the rules are slightly different.
It's a Different Job: Trying to fix the robot's mistakes on this "fake world" data is a completely different task than fixing it on real data. Sometimes, retraining the robot on its own fake data helps; sometimes, it makes things worse. It depends entirely on the specific dataset.
Changing the Path: When they taught the robot to use its own "Fake" tag (Provenance), it didn't just get better at guessing the next step. It also started making different guesses that led to a "better" fake world for the next step. It's like a driver who, by knowing they are driving a practice car, learns to take a safer route that avoids potholes they would have hit otherwise.

The Bottom Line
The paper says we shouldn't just think of this problem as "the robot is looking at the wrong data." We should think of it as "the robot is looking at data it doesn't fully understand because it's missing the backstory."

By adding a simple tag that tells the robot where the data came from, we can help it solve the puzzle. However, the paper warns that this tag isn't a magic wand; it only works if the tag actually contains useful information for that specific problem. It turns the problem from a simple "mismatch" into a lesson on reasoning under uncertainty.

Technical Summary: Exposure Bias as Epistemic Underidentification in Recursive Forecasting

Problem Statement
Recursive multi-step forecasting is traditionally framed as a distribution shift problem (covariate shift), where models trained on observed histories (via teacher forcing) are deployed on self-generated predictions, leading to compounding errors known as exposure bias. This paper argues that this framing is incomplete. The authors posit that under conditions of partial observability, noise, or state truncation, recursive forecasting is not merely a mismatch between training and test distributions, but fundamentally an epistemic underidentification problem.

Specifically, the paper contends that a one-step Bayes-optimal predictor, trained on observed contexts, may fail to uniquely identify the correct behavior for the recursive rollout process. When a model generates its own future states (induced states), the numeric value of these states alone may be insufficient to determine the correct local target. The "correct" next step may depend on the provenance of the state (e.g., whether it was observed or self-generated), information that is lost when the state is truncated to a numeric representation. Consequently, the deployed recursive predictor operates in a regime where the local target is underidentified by the state representation alone.

Methodology and Theoretical Framework
The authors formalize this problem using a framework of induced states ( $Z_h$ ) and provenance variables ( $P_h$ ).

Induced States ( $Z_h$ ): The numeric state at a specific rollout depth $h$ , derived from the predictor's own outputs.
Provenance ( $P_h$ ): Information describing how the state was formed (e.g., observed vs. model-generated).

The paper establishes three theoretical components:

Underidentification Theorem: The authors prove that even if two recursive predictors are identical on the observed support (achieving the same one-step Bayes risk), they can induce different multi-step forecasts once they query self-generated states outside that support. The one-step objective does not constrain behavior on these induced states.
Error Decomposition: The error in recursive forecasting is decomposed into three non-negative terms:
- Teacher-forcing/Rollout Mismatch: The inability of a predictor trained on observed data to generalize to the induced-state task.
- Representation–Class Gap: The approximation error of the chosen function class on the induced representation.
- Provenance Gap: The recoverable information loss incurred by omitting provenance data from the state input.
Provenance Utility: Theoretically, conditioning on provenance can strictly reduce Bayes risk on induced states if the expected target differs based on how the state was generated (i.e., $E[Y|Z, P] \neq E[Y|Z]$ ).

Empirical Evaluation
The authors test these theoretical claims on three numerical forecasting datasets (ETTh1, MG, Weather) using Multi-Layer Perceptrons (MLPs) and GRUs. The experiments are designed to validate four specific predictions:

Distinct Regime: Does rollout enter a state regime distinct from observed training data?
- Result: Linear probes trained to distinguish observed contexts from teacher-forced induced states show increasing accuracy with rollout depth, confirming that recursive deployment moves the model into a distinct state regime.
Distinct Local Task: Is the task of predicting on fixed induced states different from the original one-step task?
- Result: When induced states are frozen and retrained, performance varies significantly across datasets. On some datasets (MG, Weather), relearning on fixed induced states improves performance; on others (ETTh1), it degrades it. This suggests the induced-state task is not merely a shifted version of the original task but a structurally different problem.
Provenance Utility: Does adding provenance information help?
- Result: A binary provenance tag (observed vs. generated) provides conditional gains. In the "frozen-state" setting, gains are limited, suggesting the simple binary encoding captures only a fraction of the theoretical potential. However, in closed-loop settings, provenance-aware methods show promise on specific datasets.
Closed-Loop Dynamics: Does correction help by changing the states visited, not just by improving local prediction?
- Result: Closed-loop correction (using Scheduled Sampling with Provenance) improves overall rollout performance. Crucially, the authors show that models trained with correction generate induced states that are "easier" for other predictors to handle, indicating that correction improves the induced-state regime itself, not just the local predictor.

Key Contributions

Reframing Exposure Bias: The paper recasts exposure bias from a simple train-test distribution mismatch to a problem of epistemic underidentification arising from insufficient state representation in partially observed systems.
Formal Decomposition: It provides a rigorous decomposition of recursive forecasting error into mismatch, approximation, and provenance information gaps.
Empirical Evidence of Distinct Regimes: It demonstrates that recursive rollout enters a distinct induced-state regime and that the local corrective task on these states is heterogeneous and dataset-dependent.
Mechanism of Correction: It shows that the benefits of recursive correction arise not only from better local fitting but from altering the trajectory of induced states visited during deployment.

Significance and Claims
The paper claims that understanding exposure bias requires acknowledging that recursive deployment creates a prediction problem where the local target is not determined by the numeric state alone. The significance lies in shifting the narrative from viewing exposure bias solely as a covariate shift to treating it as reasoning under self-induced epistemic uncertainty.

The authors modestly note that their specific binary provenance encoding is a minimal signal and that gains are conditional rather than uniform. They do not propose a universal fix but argue that future method design should focus on exploiting information about how states are formed (e.g., rollout depth, uncertainty estimates, or structured traces) to resolve the aliasing between observed and induced states. The work serves as a testbed for epistemic uncertainty in sequential learning, suggesting that improving recursive forecasting requires addressing the underidentification of the deployed recursive map.

Technical Summary: Exposure Bias as Epistemic Underidentification in Recursive Forecasting

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