arxiv: 2605.06355 · v1 · submitted 2026-05-07 · 💻 cs.LG · stat.ML

Recognition: unknown

Order-Agnostic Autoregressive Modelling with Missing Data

Ignacio Peis, Jes Frellsen, Pablo M. Olmos

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:54 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords order-agnostic autoregressive modelsmissing dataimputationmissingness mechanismsactive information acquisitiondeep generative modelsconditional density estimation

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The pith

Order-agnostic autoregressive models can be trained directly on incomplete data to impute values and choose which ones to query next.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reinterprets order-agnostic autoregressive models as tools for missing data problems. It shows that their usual training on complete records already performs a form of imputation when entries are missing completely at random. The authors then supply a training procedure that works on partially observed records under arbitrary missingness mechanisms. The resulting model supplies both imputations and a way to pick the most informative missing entries to observe for downstream tasks. On real-world datasets the approach beats standard imputation methods used before modeling.

Core claim

Order-agnostic autoregressive models trained on fully observed data implicitly perform imputation under a missing-completely-at-random mechanism, and a new missingness-aware training objective extends the same models to general missingness mechanisms, yielding both improved imputation and amortized selection of informative variables to observe.

What carries the argument

The MO-ARM training objective, which computes the conditional likelihood only over observed variables while respecting the known missingness pattern during both training and amortized conditional sampling.

If this is right

Imputation remains accurate even when most entries are missing.
The same trained model can sequentially select which missing variables to observe next for prediction or inference.
No separate imputation stage is required before downstream use of the generative model.
Performance gains appear consistently across real-world datasets with varying missingness patterns.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same missingness-aware likelihood trick could be applied to other autoregressive or flow-based generative models.
End-to-end training pipelines become feasible in domains where missingness arises by design rather than accident.
Active querying could be combined with budgeted data collection in settings such as sensor networks or medical records.

Load-bearing premise

That the missingness mechanism is known or can be modeled so the likelihood can be restricted to the observed entries without introducing bias.

What would settle it

On a benchmark with non-MCAR missingness, a two-stage pipeline of baseline imputation followed by a standard order-agnostic model matches or exceeds MO-ARM imputation accuracy and active-acquisition performance.

Figures

Figures reproduced from arXiv: 2605.06355 by Ignacio Peis, Jes Frellsen, Pablo M. Olmos.

**Figure 1.** Figure 1: Graphical models used throughout the paper. Boxes denote collections of variables. The view at source ↗

**Figure 2.** Figure 2: Image inpainting comparison. The leftmost panel shows incomplete images at different rates. view at source ↗

**Figure 3.** Figure 3: Average test imputation performance at a 50% missing rate across the nine UCI tabular view at source ↗

**Figure 4.** Figure 4: Sequential active information acquisition with MO-ARM and baselines on UCI datasets. view at source ↗

**Figure 5.** Figure 5: Per-dataset average test imputation RMSE of MCAR missing data at a 50% missing rate view at source ↗

**Figure 6.** Figure 6: Per-dataset average test imputation MAE of MCAR missing data at a 50% missing rate view at source ↗

**Figure 7.** Figure 7: Per-dataset average test imputation accuracy of MCAR missing data at a 50% missing rate view at source ↗

**Figure 8.** Figure 8: Per-dataset average test imputation RMSE of MNAR missing data at a 50% missing rate view at source ↗

**Figure 9.** Figure 9: Per-dataset average test imputation MAE of MNAR missing data at a 50% missing rate view at source ↗

**Figure 10.** Figure 10: Per-dataset average test imputation accuracy of MNAR missing data at a 50% missing view at source ↗

read the original abstract

Order-Agnostic autoregressive models have demonstrated strong performance in deep generative modeling, yet their use in settings with incomplete data remains largely unexplored. In this work, we reinterpret them through the lens of missing data. First, we show that their standard training procedure on fully observed data implicitly performs imputation under a missing completely at random mechanism, resulting in robust out-of-sample imputation performance in settings with high missingness. Second, we introduce the first principled framework for training them directly on incomplete datasets under general missingness mechanisms. Third, we leverage their amortized conditional density estimation to perform active information acquisition, i.e., sequentially selecting the most informative missing variables for downstream prediction or inference. Across a suite of real-world benchmarks, our Missingness-Aware Order-Agnostic Autoregressive Model (MO-ARM) consistently outperforms established imputation baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper offers a framework for order-agnostic AR models on incomplete data plus active acquisition, but the MCAR reinterpretation and bias-free extension to general missingness look under-supported without explicit mechanism handling.

read the letter

The main point is that this work reinterprets standard order-agnostic autoregressive training as implicit MCAR imputation on complete data, then claims a new direct training method for arbitrary missingness plus an active acquisition procedure that picks informative variables sequentially. It reports better imputation results than baselines on real benchmarks via the MO-ARM model. That combination of ideas is the actual novelty here, and it targets a practical gap where separate imputation steps are usually bolted on after the fact. The amortized conditionals do lend themselves naturally to acquisition, which is a reasonable use of the model structure. The abstract positions the reinterpretation as a clean starting point that already gives robust out-of-sample imputation even under high missingness. That part reads as a useful observation if it holds. The soft spot is the extension claim. Standard order-agnostic objectives work on fully observed tuples with random permutations or averaged conditionals; they do not automatically marginalize over arbitrary missingness patterns. For MNAR cases where missingness depends on the unobserved values themselves, simply reinterpreting the objective does not guarantee the learned conditionals stay unbiased unless the missingness process is modeled or the likelihood is adjusted accordingly. No derivation or explicit likelihood expression appears in the abstract to confirm consistency outside MCAR, and the outperformance numbers lack error bars or ablation details that would show whether the gains come from the framework or from other factors. The citation pattern and lack of circularity look fine on the surface. This is for people working on deep generative models or applied pipelines that must handle incomplete tabular or sequential data. A reader already using autoregressive density estimators would see the most direct value in the acquisition angle and the integrated training. It is coherent enough on its own terms to warrant referee time, even if the theoretical grounding needs tightening and the experiments need more scrutiny. I would send it to review rather than desk reject.

Referee Report

3 major / 2 minor

Summary. The paper reinterprets order-agnostic autoregressive models (ARMs) for missing data settings. It claims that standard training on fully observed data implicitly performs imputation under an MCAR mechanism, introduces the Missingness-Aware Order-Agnostic Autoregressive Model (MO-ARM) as a principled framework for direct training on incomplete data under general missingness mechanisms, and applies the amortized conditional densities for active information acquisition. Empirical results across real-world benchmarks show consistent outperformance over established imputation baselines.

Significance. If the reinterpretation and training framework hold without introducing bias under non-MCAR mechanisms, the work would provide a novel extension of order-agnostic ARMs to incomplete data, enabling better imputation and sequential variable selection in generative modeling. The amortized inference for active acquisition is a potentially useful contribution if the empirical gains are robust.

major comments (3)

[Abstract / §3] The central reinterpretation that standard order-agnostic ARM training on complete data implicitly imputes under MCAR (abstract and likely §3) requires an explicit derivation of the training objective or likelihood that demonstrates this equivalence. Order-agnostic objectives typically average over permutations of fully observed conditionals without masking; without the precise marginalization or expectation shown, it is unclear whether this holds mechanistically or reduces to a post-hoc view.
[§4] The extension of the framework to general missingness mechanisms (including MNAR) without bias is load-bearing for the MO-ARM claim. The training objective must correctly condition only on observed variables while respecting the missingness process; if the paper's likelihood or loss (likely in §4) does not explicitly model or marginalize the missingness indicator, the learned conditionals p(x_i | x_<i, observed) risk bias under MNAR dependence. A theorem or counterexample simulation under controlled MNAR would be needed to confirm consistency.
[Experiments] The outperformance claim on real-world benchmarks is central but needs stronger statistical grounding. The experiments section should report per-dataset error bars, significance tests against baselines, and ablations isolating the effect of the missingness-aware objective versus standard ARM training or simple masking.

minor comments (2)

[§2 / Notation] Notation for observed vs. missing variables and the missingness mechanism should be introduced early and used consistently to avoid ambiguity in the methods.
[Related Work] The related work section could more explicitly contrast MO-ARM with other generative imputation approaches (e.g., VAEs or diffusion models with missing data handling) to clarify novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment point by point below, agreeing where revisions are warranted to strengthen the claims and providing plans for the revised manuscript.

read point-by-point responses

Referee: [Abstract / §3] The central reinterpretation that standard order-agnostic ARM training on complete data implicitly imputes under MCAR (abstract and likely §3) requires an explicit derivation of the training objective or likelihood that demonstrates this equivalence. Order-agnostic objectives typically average over permutations of fully observed conditionals without masking; without the precise marginalization or expectation shown, it is unclear whether this holds mechanistically or reduces to a post-hoc view.

Authors: We agree that an explicit derivation is needed to make the reinterpretation rigorous rather than interpretive. The manuscript presents the claim based on the fact that, under MCAR, the marginal likelihood over observed variables matches the complete-data order-agnostic objective. In the revision we will add a formal derivation in §3 that explicitly shows the equivalence via expectation over the missingness mask, demonstrating the marginalization step that connects the permutation-averaged conditionals to the MCAR imputation objective. revision: yes
Referee: [§4] The extension of the framework to general missingness mechanisms (including MNAR) without bias is load-bearing for the MO-ARM claim. The training objective must correctly condition only on observed variables while respecting the missingness process; if the paper's likelihood or loss (likely in §4) does not explicitly model or marginalize the missingness indicator, the learned conditionals p(x_i | x_<i, observed) risk bias under MNAR dependence. A theorem or counterexample simulation under controlled MNAR would be needed to confirm consistency.

Authors: The MO-ARM objective in §4 maximizes the conditional likelihood only over observed entries given the mask, which is the standard approach for ignorable missingness and does not require explicit modeling of the missingness indicator. For MNAR we acknowledge potential bias if the missingness depends on unobserved values. To address the concern we will add a clarifying discussion of assumptions in the revised §4 and include a controlled MNAR simulation experiment (with known missingness functions) to empirically evaluate consistency, reporting any observed bias. revision: yes
Referee: [Experiments] The outperformance claim on real-world benchmarks is central but needs stronger statistical grounding. The experiments section should report per-dataset error bars, significance tests against baselines, and ablations isolating the effect of the missingness-aware objective versus standard ARM training or simple masking.

Authors: We agree that stronger statistical support is required. In the revised experiments section we will report per-dataset standard deviations (error bars) across multiple random seeds, include paired statistical significance tests (t-tests or Wilcoxon signed-rank) against all baselines, and add ablation experiments that compare MO-ARM directly to (i) standard ARM training with naive masking and (ii) other controlled variants to isolate the contribution of the missingness-aware objective. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper reinterprets standard order-agnostic autoregressive training as implicit MCAR imputation, introduces a new MO-ARM framework for general missingness, and reports empirical benchmark outperformance. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims rest on explicit modeling extensions and external benchmarks rather than renaming or tautological equivalence. The reinterpretation is presented as a derived insight, not an input assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract. The central claims rest on the reinterpretation of existing training procedures and the introduction of a new framework whose internal assumptions are not detailed here.

pith-pipeline@v0.9.0 · 5440 in / 1175 out tokens · 49212 ms · 2026-05-08T12:54:36.450253+00:00 · methodology

discussion (0)

Reference graph

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Guidelines: 26 • The answer [N/A] means that the paper does not involve crowdsourcing nor research with human subjects

Institutional review board (IRB) approvals or equivalent for research with human subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or ...