Predictive Sufficiency in Forecasting Models

• The principle of predictive sufficiency is a design paradigm that retains only the future-relevant information while discarding irrelevant details in model representations.
• It underpins methods in sequence modeling, world models, and fairness by enforcing conditional independence and minimality to ensure precise forecasting.
• Practical applications show improved forecasting performance and robustness, although additional constraints are needed when extending to control or intervention tasks.

The Principle of Predictive Sufficiency denotes a family of closely related ideas rather than a single canonical doctrine. Across recent work in sequence modeling, world models, probabilistic forecasting, causal representation learning, fairness, and neural-network theory, the shared claim is that a representation should preserve exactly the information needed for a specified predictive task while discarding information that is irrelevant to that task. What counts as “sufficient,” however, depends on the target: future observations, future latent representations, calibrated probabilities, outcome-relevant causal features, group-conditional label consistency, or a complete predictive state for control or planning (Wang et al., 5 Aug 2025, Kim, 3 May 2026, 0806.0813, Zhao et al., 2024, Shen et al., 14 Jul 2025).

1. Foundational definitions and conceptual scope

A common modern formulation comes from sequential prediction. In "Rethinking Selectivity in State Space Models: A Minimal Predictive Sufficiency Approach" (Wang et al., 5 Aug 2025), the principle is stated as follows: the ideal hidden state $h_k$ should be the Minimal Predictive Sufficient Statistic of the observed history $U_{1:k}$ for the future $Y_{k:\tau}$. The sufficiency condition is

$Y_{k:\tau} \perp U_{1:k} \mid h_k,$

equivalently

$I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$

and the minimality condition requires $h_k$ to be the most concise among sufficient statistics (Wang et al., 5 Aug 2025).

In latent-state world modeling, the same idea is expressed in POMDP language. With history

$h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$

a latent state is predictively sufficient for a chosen future target if

$p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$

or, more generally, if the same equality holds for future embeddings rather than pixels (Kim, 3 May 2026).

In temporal probabilistic models, sufficiency is defined differently but compatibly. Given a conditional distribution $P(Z\mid X,Y)$, the marginals over $X$ and $U_{1:k}$0 are sufficient for predicting $U_{1:k}$1 if any two joint distributions on $U_{1:k}$2 with the same marginals induce the same marginal distribution on $U_{1:k}$3. This holds if and only if the conditional distribution is additively separable,

$U_{1:k}$4

with conditional separability extending the result to overlapping parent sets and hierarchical decompositions (Pfeffer, 2013).

In probabilistic forecasting, sufficiency is a relation between forecasting schemes. If $U_{1:k}$5 and $U_{1:k}$6 are forecasting schemes with $U_{1:k}$7, then $U_{1:k}$8 is sufficient for $U_{1:k}$9 when

$Y_{k:\tau}$0

This means $Y_{k:\tau}$1 contains at least as much predictive information about $Y_{k:\tau}$2 as $Y_{k:\tau}$3 (0806.0813).

In fairness, the term is used in an explicitly conditional-probability sense. The sufficiency rule is

$Y_{k:\tau}$4

or equivalently

$Y_{k:\tau}$5

so that the true outcome is conditionally independent of the protected attribute once the prediction is known (Zhao et al., 2024).

In neural-network representation theory, a transformation $Y_{k:\tau}$6 is sufficient for $Y_{k:\tau}$7 with respect to $Y_{k:\tau}$8 if

$Y_{k:\tau}$9

almost surely, equivalently $Y_{k:\tau} \perp U_{1:k} \mid h_k,$0. This formulation makes predictive sufficiency identical to preservation of the conditional target distribution (Shen et al., 14 Jul 2025).

Domain	Sufficient object	Formal criterion
Sequence models	Hidden state $Y_{k:\tau} \perp U_{1:k} \mid h_k,$1	$Y_{k:\tau} \perp U_{1:k} \mid h_k,$2
World models	Latent state $Y_{k:\tau} \perp U_{1:k} \mid h_k,$3	$Y_{k:\tau} \perp U_{1:k} \mid h_k,$4
Temporal probabilistic models	Parent marginals	$Y_{k:\tau} \perp U_{1:k} \mid h_k,$5 depends only on marginals
Forecast evaluation	Forecasting scheme $Y_{k:\tau} \perp U_{1:k} \mid h_k,$6 for $Y_{k:\tau} \perp U_{1:k} \mid h_k,$7	$Y_{k:\tau} \perp U_{1:k} \mid h_k,$8
Fairness	Predictor $Y_{k:\tau} \perp U_{1:k} \mid h_k,$9	$I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$0
Neural representations	Layer output $I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$1	$I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$2

Older theoretical lineages already anticipated this predictive reading. In algorithmic prediction, the minimal description $I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$3 of observed data $I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$4 is treated as a predictively sufficient representation for the universal mixture, since

$I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$5

and the best computable model is the shortest extra description beyond the data needed to construct $I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$6 (Stiffelman, 2014). In inductive logic and Papangelou processes, predictive sufficiency appears as a count-based postulate: prediction at location $I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$7 depends only on the local count $I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$8, and strong versions characterize Poisson and Pólya point processes (Rafler et al., 2013).

2. Information-theoretic and optimization formulations

The most explicit information-theoretic operationalization appears in Minimal Predictive Sufficiency State Space Models. Because exact constraints such as

$I(U_{1:k};Y_{k:\tau}) = I(h_k;Y_{k:\tau}),$9

are intractable, the principle is approximated by combining a prediction loss with a mutual-information-based minimality regularizer: $h_k$0 Here $h_k$1 is a standard forecasting loss and $h_k$2 penalizes information retained from the past that does not improve prediction (Wang et al., 5 Aug 2025).

This construction is deliberately distinguished from the classical Information Bottleneck. The paper states that predictive sufficiency seeks maximal compression under the hard constraint of no predictive loss, rather than an explicit trade-off between compression and predictive information. Under sufficient expressivity and a tight variational bound on mutual information, the global minimizer is shown to satisfy approximate sufficiency and minimality, and to become approximately invariant to non-causal perturbations (Wang et al., 5 Aug 2025).

The same paper operationalizes the principle in state-space models by training a selective SSM backbone, a prediction head, and an auxiliary decoder used to estimate an upper bound on $h_k$3. This produces MPS-SSM and related regularized models such as MPS-Mamba, MPS-DLinear, and MPS-PatchTST. On ETT, Weather, Electricity, Traffic, and Exchange benchmarks, the method is reported as either best or second-best in MSE and MAE for most horizons, with especially pronounced gains on long-horizon and noisy tasks; robustness under injected impulse noise improves monotonically as the regularization strength $h_k$4 increases, though the paper also reports a clean-accuracy versus robustness trade-off (Wang et al., 5 Aug 2025).

A separate but convergent optimization tradition starts from regret and proper scoring rules. In "Divergence and Sufficiency for Convex Optimization" (Harremoës, 2017) and "Proper Scoring and Sufficiency" (Harremoës, 2015), many prediction and decision problems are cast as maximizing a convex value function $h_k$5 over a state space, which induces a regret

$h_k$6

that is, a Bregman divergence when $h_k$7 is differentiable. The decisive claim is that if such regret satisfies a sufficiency condition—equivalently locality or monotonicity in the finite-dimensional $h_k$8-algebra setting—then it must be proportional to information divergence, and the associated local strictly proper scoring rule must be logarithmic (Harremoës, 2017, Harremoës, 2015).

Forecast evaluation gives another information-theoretic decomposition. For a strictly proper score and a forecasting scheme $h_k$9 with $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$0 and climatology $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$1,

$h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$2

The second term is resolution and the third is reliability. If $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$3 is sufficient for $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$4, then the resolution of $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$5 cannot exceed that of $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$6, providing a formal link between predictive sufficiency and forecast refinement (0806.0813).

These strands support a common interpretation: predictive sufficiency is not merely retention of predictive information, but retention under a compression or decision criterion that rules out irrelevant structure. This suggests why recent work repeatedly couples sufficiency claims to minimality, invariance, locality, or regret.

3. Sequential prediction, latent states, and world models

In world-model research, predictive sufficiency is explicitly treated as one sufficiency constraint among several. "Latent State Design for World Models under Sufficiency Constraints" (Kim, 3 May 2026) defines latent-state design as choosing a compressed statistic $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$7 of history $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$8 that preserves what a downstream function needs. For prediction, the relevant requirement is future-target sufficiency: $h_t=(o_1,a_1,\ldots,o_t), \qquad z_t=\phi(h_t),$9 For predictive embeddings such as JEPA-style models, the design goal is to retain only the aspects of the world needed to predict future structure, suppress nuisance variation, and avoid reconstructing every sensory detail (Kim, 3 May 2026).

The paper’s functional taxonomy distinguishes six latent-state roles: predictive embedding, recurrent belief state, object/causal structure, latent action interface, grounded planning interface, and memory substrate. Predictive sufficiency is central only to the first role. The paper’s summary matrix describes the prediction role as preserving future-relevant structure, discarding pixel nuisance, and enabling stable forecasts (Kim, 3 May 2026).

A major contribution of this literature is the insistence that predictive sufficiency is not a universal adequacy criterion for world models. Proposition 2 states that predictive sufficiency does not imply control sufficiency: $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$0 do not follow merely from accurate prediction of future observations. Two histories can induce the same predictive distribution over observations while differing in reward or reachability consequences (Kim, 3 May 2026). Proposition 3 adds that passive predictive sufficiency under the behavior policy does not identify counterfactual dynamics: $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$1 under logged actions does not guarantee correct representation of

$p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$2

This distinction is used to explain why passive video prediction can fail at intervention queries and planning (Kim, 3 May 2026).

The paper therefore draws a strong task-relative conclusion: an actionable world model is the one whose state construction matches the task, not the one that preserves the most information. This is an explicit rejection of the idea that “more information” is automatically better. For prediction-oriented roles, predictive sufficiency is the correct target; for control, planning, memory, or causal reasoning, other sufficiency notions are required (Kim, 3 May 2026).

This task-relativization also clarifies recent sequence-modeling practice. JEPA-style predictive embeddings, V-JEPA, and related latent predictors can be strong at future representation prediction and semantic transfer, yet may require additional action-conditioned dynamics, value-equivalent objectives, bisimulation losses, or planner-aware geometry before they become useful planning states. The paper identifies the evaluation mistake as “treating representation probes or video prediction as planning evidence” (Kim, 3 May 2026).

4. Neural representations and temporally decomposed systems

A distinct line of work asks whether standard neural-network layers are themselves sufficient statistics. "A Graph Sufficiency Perspective for Neural Networks" (Shen et al., 14 Jul 2025) models a layer as a graph variable

$p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$3

where neurons compute pairwise functions between the input $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$4 and learned anchor points $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$5. Under compact support, point separation, continuity in the anchor, and dense anchors, the map $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$6 becomes injective as $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$7, and

$p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$8

The paper states that asymptotic sufficiency therefore holds in the infinite-width limit and remains preserved throughout training under a homeomorphism-style condition on gradient updates (Shen et al., 14 Jul 2025).

The same framework covers inner products, Euclidean and squared distances, universal kernels, cosine similarity under an extra condition, fully connected layers, general pairwise functions, ReLU and sigmoid activations, and convolutional neural networks. For finite-width networks, the paper replaces injectivity by region separation: if the support of $p(o_{t+1:T}\mid h_t)=p(o_{t+1:T}\mid z_t),$9 can be partitioned into regions on which $P(Z\mid X,Y)$0 is constant, then any representation that separates those regions is sufficient. For $P(Z\mid X,Y)$1-region-separated distributions, the resulting sufficiency is approximate (Shen et al., 14 Jul 2025).

This neural sufficiency theorem has a close analogue in older temporal probabilistic models. "Sufficiency, Separability and Temporal Probabilistic Models" (Pfeffer, 2013) proves that, for a child variable $P(Z\mid X,Y)$2 with parents $P(Z\mid X,Y)$3, the marginal distributions over $P(Z\mid X,Y)$4 and $P(Z\mid X,Y)$5 suffice to determine the marginal distribution of $P(Z\mid X,Y)$6 if and only if

$P(Z\mid X,Y)$7

The paper generalizes this to conditional separability and to self-sufficient families of subsystems in dynamic Bayesian networks. In that case, exact prediction of future subsystem marginals can be carried out by propagating subsystem marginals rather than the full joint state (Pfeffer, 2013).

These results give two structural routes to predictive sufficiency. One route is injective or asymptotically injective representation: preserve all predictive information by preserving input identity. The other is selective quotienting: collapse inputs only within regions that already share the same $P(Z\mid X,Y)$8. The former is associated with dense high-dimensional embeddings; the latter with minimal predictive partitions. The temporal-model results also show a limit: observations can break separability, so exact monitoring generally remains hard even when exact prediction is tractable (Pfeffer, 2013).

5. Causal, fairness, and application-specific reinterpretations

Some papers use the phrase in a more causal than statistical sense. In multimodal representation learning, "Seeking the Sufficiency and Necessity Causal Features in Multimodal Representation Learning" (Chen et al., 2024) defines a high-quality predictive feature as one with high Probability of Necessity and Sufficiency (PNS). Under exogeneity and monotonicity,

$P(Z\mid X,Y)$9

This formulation treats a representation as predictively sufficient and necessary when interventions on it would reliably turn the outcome on and off. The paper operationalizes this with modality-invariant and modality-specific latent variables, complement extractors, and losses that increase $X$0, decrease $X$1, and impose monotonicity-like constraints (Chen et al., 2024).

A closely related medical-imaging application appears in "Medical Image Quality Assessment based on Probability of Necessity and Sufficiency" (Chen et al., 2024). There, hidden features with high PNS are taken to be more causally linked to image quality and therefore more robust under OOD shift. The framework uses a standard backbone $X$2, a predictor $X$3, a complement feature extractor $X$4, and a loss

$X$5

The complement loss pushes the complement representation away from the target label, and the monotonicity term aligns training with Pearl-style monotonicity assumptions. On AS-OCT quality assessment, the paper reports improved in-distribution F1 and improved OOD accuracy in the more difficult Limited-only setting (Chen et al., 2024).

In algorithmic fairness, the principle is explicitly conditional on predictions rather than latent features. "Enhancing Fairness through Reweighting: A Path to Attain the Sufficiency Rule" (Zhao et al., 2024) defines the sufficiency rule as

$X$6

and measures deviations from it by the sufficiency gap

$X$7

The paper uses a bilevel reweighting objective whose outer loss is IRM-based, arguing that invariance of $X$8 across groups implies sufficiency of $X$9 across groups. Empirically, the method lowers $U_{1:k}$00 on Toxic Comments from $U_{1:k}$01 to $U_{1:k}$02 and on CelebA from $U_{1:k}$03 to $U_{1:k}$04, while largely retaining predictive accuracy (Zhao et al., 2024).

These causal and fairness usages broaden the principle substantially. Instead of asking only whether a representation preserves future information, they ask whether it preserves causally essential information, or whether predictions have the same label meaning across groups. This suggests that “predictive sufficiency” is increasingly serving as an umbrella notion for several target-conditioned invariances, provided the target is clearly specified.

6. Limits, misconceptions, and current debates

A recurrent misconception is that predictive sufficiency is a universal standard of model quality. The world-model literature rejects this explicitly. Predictive sufficiency is necessary for forecasting, but it does not imply control sufficiency, planning sufficiency, causal sufficiency, or memory sufficiency (Kim, 3 May 2026). A latent state can support accurate future-observation forecasts while discarding distinctions that matter for reward, action choice, or counterfactual intervention.

A related misconception is that structural relevance or causal-parent selection automatically yields a complete predictive state. "Structural Causal Discovery and Predictive Sufficiency in High-Dimensional Dynamical Systems" (AlMomani et al., 30 May 2026) studies this directly in next-hour precipitation prediction. Projection-based entropic regression identifies a stable six-variable informational backbone and yields strong short-horizon discrimination, with AUC approximately $U_{1:k}$05 at $U_{1:k}$06, but the resulting variable set still shows limited calibration, limited intensity prediction, and limited fixed-threshold event-detection skill. The paper concludes that structural relevance does not imply predictive closure: the selected variables are a stable informational backbone, but not a complete predictive state (AlMomani et al., 30 May 2026).

A third misconception is that preserving more information is always beneficial. The latent-state taxonomy argues the opposite: each sufficiency constraint specifies what the latent must retain and, by implication, what it should discard. For prediction, the right compression target is future-relevant structure; retaining pixel nuisance or reward-irrelevant detail is not automatically helpful (Kim, 3 May 2026).

There is also a statistical-inferential debate over what counts as predictively sufficient evidence. In the critique of the Strong Likelihood Principle, sampling-based frequentist inference is said to depend essentially on the relevant sampling distribution, so likelihood proportionality alone is not sufficient for p-values, significance levels, or confidence procedures. This position distinguishes predictive or error-based adequacy from mere likelihood equivalence (Mayo, 2013). A plausible implication is that “predictive sufficiency” in inferential statistics cannot be reduced to any single likelihood-based criterion without specifying the sampling design.

Across these literatures, the most stable conclusion is task-relativity. Predictive sufficiency is best understood as a target-indexed design principle: preserve exactly the information needed for the predictive task at hand, and no more than necessary. When the target is future forecasting, this leads to minimal predictive sufficient statistics, stable probabilistic forecasts, and representation compression. When the target is control, intervention, fairness, or calibrated scientific inference, additional constraints become indispensable (Wang et al., 5 Aug 2025, Kim, 3 May 2026, AlMomani et al., 30 May 2026).