Minimal Predictive Sufficiency Principle

• Minimal Predictive Sufficiency is an information-theoretic framework that retains only the necessary predictive information from data while discarding irrelevant details.
• It unifies concepts from minimal sufficient statistics, causal state reconstruction, and Bayesian model selection to enable lossless dimensionality reduction in predictive tasks.
• MPS enhances applications in machine learning and decision-making by optimizing model construction, data collection, and reducing computational complexity.

Minimal Predictive Sufficiency (MPS) is an information-theoretic principle that prescribes, for a given predictive task, retaining only that information from observed data which is truly needed to predict future or target variables, discarding all “independent fat”—variables, features, or statistical structure unrelated to the relevant prediction. The principle encompasses and unifies ideas from minimal sufficient statistics, complexity theory, Bayesian model selection, causal state reconstruction, data informativeness in optimization, and selectivity in sequential and machine learning models. MPS both formalizes lossless dimensionality reduction in probabilistic systems and elevates “task-aware minimality” to a unifying criterion for model construction, data collection, and explanation.

1. Information-Theoretic Definition and Core Formalism

Minimal Predictive Sufficiency arises from the classical notion of sufficient statistics, generalized to prediction and information preservation. Given random variables $X$ (data) and $Y$ (target), a statistic $T(X)$ is sufficient for $X$ about $Y$ if $X$ and $Y$ are conditionally independent given $T(X)$:

$I(X:Y\,|\,T(X)) = 0,$

i.e., $T(X)$ captures all information in $Y$0 relevant to predicting $Y$1. The minimal sufficient statistic $Y$2 is then the sufficiency-preserving statistic with minimal entropy:

$Y$3

For both variables, the central result is that simultaneously replacing $Y$4 and $Y$5 by their respective minimal sufficient statistics $Y$6, $Y$7 preserves all their mutual information:

$Y$8

and this is strictly optimal—no less complex representations can retain the dependency without information loss. This “trims the independent fat”: all idiosyncrasy in either $Y$9 or $T(X)$0 that does not affect their joint distribution is discarded (James et al., 2017).

2. Exemplars: Stochastic Processes, Causal States, and Mixed-State Geometry

Applied to stationary stochastic processes, MPS specializes to the computational mechanics framework. For process $T(X)$1, with past $T(X)$2 and future $T(X)$3, define

• Forward causal states: $T(X)$4—the minimal sufficient statistic of the past about the future, grouping all pasts yielding the same future predictive distribution.
• Reverse causal states: $T(X)$5.

The process’s excess entropy $T(X)$6 is exactly

$T(X)$7

manifesting MPS as an equality between predictive information and the mutual information between (generally low-dimensional) effective states (James et al., 2017). When the set of causal states is uncountably infinite, as in infinitary processes, “nearly maximally predictive” features are constructed via $T(X)$8-coarse graining of the mixed-state simplex. The coding cost and number of near-minimal feature classes then scale with the box-counting and information dimensions of the mixed-state support, giving sharp tradeoffs for predictive representation complexity (Marzen et al., 2017).

3. MPS in Model Selection and Bayesian Inference

In finite-data Bayesian modeling, MPS operationalizes the idea of selecting the prior and model complexity that maximize the information potentially learnable from the data—before data is seen. Given a parametric family $T(X)$9 and candidate prior $X$0, the MPS prior maximizes the expected information gain (mutual information) between parameters and data:

$X$1

At finite data, the optimal prior concentrates on discrete boundary submanifolds in parameter space, yielding lower-dimensional effective models—the model class is “compressed” along irrelevant/sloppy parameter directions. Unless infinite data is available (where Jeffreys prior emerges), MPS automatically applies Occam’s razor, discarding undetectable complexity and focusing only on those components of the model space resolvable by the data budget (Mattingly et al., 2017). This discrete atomic structure contrasts sharply with the pathology of Jeffreys prior on “hyper-ribbon” parameter manifolds of scientific models—where Jeffreys is uninformative or even misleading.

4. MPS in Complexity and Algorithmic Prediction

The principle extends to the algorithmic and computational framework via Kolmogorov complexity. Given a data string $X$2 of length $X$3, define its minimal program $X$4 of length $X$5. $X$6 can be decomposed as $X$7, where $X$8 is the minimal extra description (the “model part”), and $X$9 is recoverable from $Y$0 and $Y$1. The ideal (but noncomputable) predictive model is

$Y$2

where $Y$3 is the minimal description of $Y$4 relative to $Y$5. This formalizes MPS as minimal additional description: only information not already found in the observed data is retained in the model, with practical computable methods bounded in error by the two-part code length (Stiffelman, 2014).

5. MPS in Decision-Focused Data Informativeness

In decision-making and optimization, particularly linear programming under uncertainty, MPS dictates that only those “directions” in cost space (or data) that determine the optimal decision need to be known. Let $Y$6 be the feasible polyhedron, $Y$7 the (uncertain) cost, and $Y$8 the uncertainty set. A data set $Y$9 (i.e., linear queries or measurements) is MPS if, together with prior knowledge $X$0, it suffices to always recover the optimal decision for every $X$1:

$X$2

This leads to a geometric construction: the minimal sufficient $X$3 is a basis for the space of reachable solutions or the relevant “face-changing” directions in the LP. Algorithms exist to find minimal $X$4 under query constraints, ensuring no redundant data is collected (Bennouna et al., 17 Feb 2026).

6. Applications: Machine Learning, Explanations, and Selectivity

In modern sequence modeling, the MPS principle supplies a first-principles regularizer for state-space models and other architectures. For a latent state $X$5, the objective is

$X$6

where $X$7 enforces predictive sufficiency and $X$8 penalizes retention of extraneous past information. This has been operationalized in MPS-SSM, achieving superior predictive power and robustness, and can regularize any architecture with an internal representation. Empirical results on time series and noisy environments confirm the principled tradeoff between compression and prediction fidelity (Wang et al., 5 Aug 2025).

MPS also underpins task-specific explanation methods. In explainable AI, it underlies feature attribution approaches such as the Path-Sufficient Explanations Method (PSEM), which seeks sequences of feature sets that are minimal yet sufficient for preserving the model’s behavior on a given input, with additional stability and fidelity guarantees (Luss et al., 2021).

7. Schematic and Interpretation

Conceptually, MPS can be visualized as binning or coarse-graining of high-dimensional objects to identify equivalence classes sharing the same predictive or decision-relevant properties. Only distinctions altering the target (future, optimal decision, classification, etc.) are preserved; all irrelevant information—statistically, structurally, or algorithmically independent—is trimmed away. This yields a lossless (or nearly lossless, in the presence of infinite or uncountable support) reduction to the essential “spine” of the data-task relationship, maximizing efficiency in storage, inference, and scientific understanding (James et al., 2017, Marzen et al., 2017).

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In summary, the Minimal Predictive Sufficiency principle is a rigorous, unifying framework for identifying the irreducible core of information needed for prediction or decision, and for guiding model selection, data acquisition, representation learning, and explanation across statistical, computational, and algorithmic domains.