Minimal Sufficiency: Definition & Core Concepts

• Minimal sufficiency is a principle that identifies the smallest set or representation preserving all task-relevant information while eliminating redundancies.
• It is applied in statistical inference, spatial reasoning, representation learning, and causal analysis to achieve efficiency, accuracy, and interpretability.
• Concrete implementations include minimal sufficient statistics, MSS systems in 3D reasoning, and minimal interventions in causal models.

Minimal sufficiency is a principle designating the smallest set, representation, or intervention that is just enough to achieve a specified outcome—preserving all information relevant to the task while eliminating all redundancies. The notion underpins diverse domains: in statistical inference, it grounds efficient data reduction; in spatial reasoning, it yields compact, interpretable fact sets for reasoning agents; in representation learning, it steers unsupervised models toward disentangled semantic codes; in causal analysis, it identifies the least set of manipulations that guarantee an effect. Across all contexts, minimal sufficiency involves dual requirements: sufficiency (the retained set or representation can support the correct inference or outcome) and minimality (no strict subset suffices).

1. Formal Definition and Core Properties

Across applications, minimal sufficiency is formalized by the following attributes:

• Sufficiency: The set, representation, or intervention contains all and only the task-relevant information—the designated agent, statistic, or transformation can achieve the correct outcome using precisely this material.
• Minimality: No strict subset of the set or representation is sufficient; any reduction would lose essential information and compromise correctness.

For instance, in spatial reasoning over 3D scenes, a subset $\mathcal{S}^* \subset \mathcal{W}$ is a Minimal Sufficient Set (MSS) for a question $q$ with answer $a^*$ if

$$\mathcal{R}^*\bigl(\mathcal{S}^*,\,q\bigr) = a^*, \quad \forall\,\mathcal{S}'\subsetneq\mathcal{S}^*,\,\, \mathcal{R}^*(\mathcal{S}',q)\neq a^*,$$

where $\mathcal{R}^*$ is an oracle reasoning agent and $\mathcal{W}$ is all spatial and semantic information derivable from the scene (Guo et al., 19 Oct 2025).

In minimal sufficient representation learning for unsupervised domain generalization (UDG), sufficiency is posed via information-theoretic constraints: a representation $\hat{z}_1^{suf}$ is sufficient if $I(\hat{z}_1^{suf};x_2) = I(x_1;x_2)$ (preserving all mutual information in augmentations), and minimal among all such if it minimizes $I(\hat{z}_1^{suf};x_1)$ (Pan et al., 19 Sep 2025).

In structural causal models, "weak sufficiency" is minimal: for endogenous variable $X$ and outcome $Y$, $X=x$ is weakly sufficient for $Y=y$ if intervening to $X\leftarrow x$ guarantees $Y=y$ in all contexts—without holding other variables fixed (Beckers, 2021).

Properties commonly associated with minimal sufficiency include:

• Uniqueness (up to irrelevant permutations or degeneracy) if both criteria are met.
• Guaranteed maximal data (or intervention) compression without loss of inference capability.

2. Minimal Sufficiency in Spatial Reasoning and Vision-LLMs

Minimal sufficiency in spatial reasoning is operationalized as MSS: compact, curated fact sets tailored for each query on a 3D scene. The MSSR (“Minimal Sufficient Spatial Reasoner”) system comprises a Perception Agent (PA) that programmatically queries 3D expert models and a Reasoning Agent (RA) that iteratively refines the information set:

• At each iteration, the RA prunes the current set to attain minimality, requests necessary missing information, and evaluates sufficiency.
• The PA responds only with targeted, non-redundant new facts, thus enforcing efficiency and interpretability.

The Situated Orientation Grounding (SOG) module provides robust, language-grounded directional information. Rather than continuous regression, SOG implements a multi-choice, visually grounded selection protocol that supplies just those orientation vectors needed for sufficiency and minimality.

Empirical results demonstrate a tight inverse correlation between curated set cardinality and reasoning accuracy; MSSR produces both improved accuracy and interpretable intermediate reasoning steps (Guo et al., 19 Oct 2025).

3. Information-Theoretic Minimality in Representation Learning

In unsupervised domain generalization, minimal sufficient semantic representations are defined to optimally capture semantics while filtering out nuisance variation:

• Sufficiency: Enforced via an InfoNCE loss ensuring that the learned semantic code ($s$) preserves all semantic information shared across views.
• Minimality: Promoted through a disentanglement loss that encourages all variation to be routed into a separate code ($v$), and a reconstruction loss that prevents semantic leakage into $v$.

Key results formalize this interplay using mutual information: $$\hat {z}_1^{min} = \arg\min_{\hat{z}_1^{suf}:\;I(\hat{z}_1^{suf};x_2) = I(x_1;x_2)} I(\hat{z}_1^{suf}; x_1)$$ Satisfying these objectives is shown theoretically to achieve the lowest possible upper bound on out-of-distribution error for downstream tasks whose labels depend only on shared semantics (Pan et al., 19 Sep 2025).

A summary of MS-UDG’s components is provided below:

Loss Term	Objective	Role in Minimal Sufficiency
$\mathcal{L}_{suf}$	InfoNCE (contrastive)	Satisfy semantic sufficiency
$\mathcal{L}_{min}$	Disentanglement InfoNCE	Enforce minimality (semantic/variation split)
$\mathcal{L}_{max}$	Reconstruction	Ensure all residual info routed to $v$

4. Minimal Sufficiency in Structural Causal Models

In the context of structural equation models, minimal sufficiency is instantiated as “weak sufficiency,” the minimal intervention required to guarantee a specific outcome:

• Weak sufficiency: $X=x$ is weakly sufficient for $Y=y$ if, for every exogenous context $u$, setting $X\leftarrow x$ implies $Y=y$.
• Actual weak sufficiency: The same, but required only in the actual context $u_0$.

No additional variables are held fixed, differentiating weak sufficiency from stronger “direct” or “network” sufficiency, which require interventions on larger sets. Beckers shows that weak sufficiency and its actual-context specialization are the unique minimal sufficiency clauses in his taxonomy (Beckers, 2021).

Contrasted with the Halpern–Pearl framework, which generally intervenes on both the candidate cause and additional context variables (“witnesses”), minimal sufficiency in this setting focuses solely on the candidate cause, directly embodying the NESS (“Necessary Element of a Sufficient Set”) intuition.

5. Criteria, Metrics, and Algorithmic Realizations

Minimal sufficiency is assessed and implemented using both formal criteria and algorithmic proxies:

• In spatial reasoning, sufficiency is operationalized by correctness (final answer matches ground truth using only the MSS); minimality by successful pruning (all redundant facts removed, no further reduction possible without loss of correctness) (Guo et al., 19 Oct 2025).
• In representation learning, sufficiency is enforced through mutual information maximization between representations of augmentations; minimality through mutual information minimization between residual codes and the input, and tested empirically via OOD error rates (Pan et al., 19 Sep 2025).
• In causality, minimal sufficiency is tested by checking that an intervention on the minimal set (often a singleton variable) suffices for the outcome in all or the actual context.

A generic algorithmic realization—in spatial reasoning—follows a dual-agent protocol:

Initialize S ← ∅ repeat Reasoning Agent prunes S to S_curated if S_curated sufficient: action ← "Decide" else: action ← "Request" Perception Agent queries missing info, augments S S ← S_curated until action == "Decide" return answer, S_final = S

6. Relationship to Minimal Sufficient Statistics and Theoretical Foundations

The concept of minimal sufficiency is formally analogous to minimal sufficient statistics in estimation theory, where a statistic $T(X)$ is sufficient if it retains all information in $X$ about a parameter and minimal if further coarsening loses sufficiency. Minimal sufficiency in spatial reasoning and representation learning extends this compression principle to broader tasks, focusing on preserving necessary information for question answering or semantic characterization, not just parameter estimation (Guo et al., 19 Oct 2025).

The theoretical justification for seeking minimal sufficient sets or representations includes efficiency (avoiding redundant computation or overfitting), interpretability (providing traceable reasoning steps), and, in learning, provable minimization of generalization error in transfer and OOD settings (Pan et al., 19 Sep 2025).

7. Illustrative Examples and Practical Implications

Concretely, in spatial reasoning for the question “Is the lamp to the right of the sofa?” on a 3D scene, the Perception Agent may first supply 18 facts (bounding boxes, depths), but the Reasoning Agent prunes this to just lamp coordinates, sofa coordinates, the global frame, and (after a targeted request) the relative vector. This set is both sufficient and minimal for the query (Guo et al., 19 Oct 2025).

In unsupervised domain generalization, inferring class in a new artistic style is robust only if the semantic code $s$ has been purified of style/texture cues—this is achieved only by sufficient and minimal semantic encoding as enforced in MS-UDG (Pan et al., 19 Sep 2025).

In causal analysis of the model $Y = X \wedge A$, with observed $A=1$, only $X=1$ is the unique minimal weakly sufficient set for $Y=1$ in the actual context—no further intervention or addition needed (Beckers, 2021).

A plausible implication is that minimal sufficiency, when operationalized in algorithmic or statistical agents, can simultaneously achieve efficiency, accuracy, and interpretability, provided the selection/pruning mechanism references the explicit dual criteria of sufficiency and minimality.

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Key citations:

• "Pursuing Minimal Sufficiency in Spatial Reasoning" (Guo et al., 19 Oct 2025)
• "Minimal Semantic Sufficiency Meets Unsupervised Domain Generalization" (Pan et al., 19 Sep 2025)
• "Causal Sufficiency and Actual Causation" (Beckers, 2021)