The resource theory of causal influence and knowledge of causal influence

Abstract: Understanding and quantifying causal relationships between variables is essential for reasoning about the physical world. In this work, we develop a resource-theoretic framework to do so. Here, we focus on the simplest nontrivial setting -- two variables that are causally ordered, meaning that the first has the potential to influence the second, without hidden confounding. First, we introduce the resource theory that directly quantifies causal influence of a functional dependence in this setting and show that the problem of deciding convertibility of resources and identifying a complete set of monotones has a relatively straightforward solution. Following this, we introduce the resource theory that arises naturally when one has uncertainty about the functional dependence. We describe a linear program for deciding the question of whether one resource (i.e., state of knowledge about the functional dependence) can be converted to another. Then, we focus on the case where the variables are binary. In this case, we identify a triple of monotones that are complete in the sense that they capture the partial order over the set of all resources, and we provide an interpretation of each.

• The paper establishes a resource theory for deterministic causal influence (RTCaus) by proving that functions can be ordered via the monotone M(f)=log2|Im(f)|, quantifying influence in bits.
• It extends the framework to uncertain mechanisms (RTKnowCaus) by modeling epistemic uncertainty over functions with probability distributions and characterizing resources with parameters (|α|, β, |γ|).
• Operational examples, geometric visualization, and comparisons with Shannon theory illustrate how free operations dictate resource convertibility and guide causal inference applications.

Resource-Theoretic Approaches to Causal Influence and Knowledge of Causal Influence

Introduction

The paper "The resource theory of causal influence and knowledge of causal influence" (2512.11209) develops two resource-theoretic frameworks for analyzing causal relationships between classical variables, focusing on scenarios where either the functional dependence between two variables is fully known or only partially characterized by a probability distribution over such functions. The authors employ and extend the formalism of resource theories, which provides a powerful and compositional means of quantifying, manipulating, and comparing causal influence. This analysis is situated within the broader context of causal inference, inference about mechanisms, and foundations of information theory.

The Resource Theory of Causal Influence (RTCaus)

Definitions and Structure

In the scenario where the functional dependence from $X$ to $Y$ is deterministic and fully known, the resource objects are single functions $f: X \to Y$. Free resources and free operations are defined so as not to introduce any new causal influence: free functions correspond to outputs $Y$ that are independent of $X$ (reset maps), and free operations are local pre- and post-processing (sequential composition) of $f$.

The transformation preorder is induced by the existence of a free operation converting one function into another. The main result is the complete characterization of this preorder:

For arbitrary finite $X$ and $Y$, a function $f$ can be converted to $g$ under free operations if and only if $|Im(f)|\geq |Im(g)|$ (Proposition). This yields a total ordering on the equivalence classes of functions, fully characterized by the monotone $M(f)=\log_2|Im(f)|$—the number of bits of causal influence.

Binary Case

For the binary case ($|X|=|Y|=2$), there are four functions: identity ($I$), flip ($F$), and resets ($R_0$, $R_1$). The causally disconnected functions $R_0$ and $R_1$ are the only free resources, while $I$ and $F$ form the unique nonfree equivalence class.

The Resource Theory of Knowledge of Causal Influence (RTKnowCaus)

Formalism and Motivation

RTKnowCaus models situations where the precise functional relation between $X$ and $Y$ is unknown, and only a probability distribution $P_F$ over $F(X\to Y)$ is available. This arises naturally in the presence of unobserved variables ("flag variables") influencing which function is applied, generalizing the operational scenario to epistemic uncertainty about causal mechanisms. The resource objects are thus probability distributions over functions; the free objects are mixtures with support only on causally disconnected functions.

Free Operations and Resource Convertibility

Free operations are local stochastic pre- and post-processings on the function, possibly correlated via shared randomness (a common cause). Importantly, resource convertibility is assessed by a linear program: the downward closure (set of freely reachable resources) from a given $P_F$ is a convex polytope whose vertices correspond to extremal free operations.

Figure 2: The downward closure polytope $\mathcal{P}$ for a given resource, capturing the set of resources accessible by free operations.

Complete Characterization in the Binary Case

For the case of binary variables and functions, the set of resources forms the probability simplex over the four functions ($I$, $F$, $R_0$, $R_1$). The resource—the probability distribution $P_B$—can be parametrized by three parameters: $\beta$ (total weight on causally connected functions), $\alpha$ ("polarization" between $I$ and $F$), and $\gamma$ ("polarization" between $R_0$ and $R_1$). The authors rigorously demonstrate:

The triple $(|\alpha|, \beta, |\gamma|)$ constitutes a canonical form and a complete set of monotones classifying the pre-order of resources in RTKnowCaus (Lemma, Proposition).

Level sets of these monotones have direct geometric and operational interpretations. Notably, $\beta$ quantifies the probability of causal connection (the total resourceful weight), $|\alpha|$ represents the maximal bias distinguishing between distinct causal mechanisms ($I$ vs $F$), and $|\gamma|$ quantifies the polar bias between the two causally disconnected mechanisms. An additional composed monotone, $M_{|\gamma|, \beta}$, captures the trade-off between increasing certain polarizations and sacrificing resourceful weight.

Figure 4: Geometric illustration of level sets for the monotone $M_{\beta}$ within the tetrahedron of bit-to-bit function distributions.

Non-Totality of the Pre-order

Unlike RTCaus, RTKnowCaus exhibits a non-total pre-order; there exist pairs of resources that are not related by free conversion in either direction. This intrinsic non-totality arises from the interplay between the monotones: not all monotone improvements can be simultaneously realized via free operations.

Geometric Visualization

The parameterization enables a geometric visualization: the simplex of distributions over four functions (a tetrahedron) is cut by planes of constant monotone value, and downward closure polytopes are explicit subsets (Figure 1).

Figure 3: Visualization of level sets for the $|\alpha|$ monotone within the probability simplex.

Figure 5: Visualization of level sets for the composed monotone $M_{|\gamma|,\beta}$, showing how downward closure is shaped by operational trade-offs.

Operational Examples and Interpretations

The theoretical structure is motivated and elucidated via operational scenarios, particularly communication channels with channel state uncertainty. For instance, two distinct distributions $P_B$ giving rise to the same conditional distribution $P_{Y|X}$ can have fundamentally different capacities for causal inference if the flag variable becomes accessible in post-selection or side-channel information is available. The authors provide detailed analysis of post-selection tasks and their quantification by the derived monotones.

Extensions, Connections, and Categorical Structure

Generalization to Higher Cardinality

The authors generalize monotone construction to arbitrary finite cardinalities of $X$ and $Y$, introducing weight functions $\beta_k$ for image size $k$ and showing that their cumulative sums yield resource monotones, albeit not a complete set.

Relationship to Other Theories

The resource-theoretic approach is contrasted with classical Shannon theory, which analyzes channels as stochastic maps $P_{Y|X}$ and ignores epistemic knowledge of the mechanism; the authors demonstrate that distinct resources in RTKnowCaus can collapse to the same (often free) resource in the Shannon setting.

The relation to the classical average causal effect (ACE) is also analyzed. The ACE, as a function only of the stochastic map, provides only a lower bound on the true causal influence that can be realized in RTKnowCaus for a given $P_{Y|X}$.

Categorical Perspective

Both resource theories admit a natural formulation in the language of symmetric monoidal categories and combs, facilitating further generalization to quantum or generalized probabilistic frameworks.

Implications and Outlook

This work provides a systematic, rigorous, and operationally relevant foundation for understanding causal influence and knowledge thereof within the resource-theoretic paradigm. The explicit identification of complete monotone families enables fine-grained ordering and classification of causal knowledge resources, allowing, for instance, for the discrimination of "resourcefulness" even among mechanisms (or ignorance thereof) that are observationally indistinguishable.

This framework is expected to serve as a foundation for:

• Extensions to more elaborate causal graphs (arbitrary DAGs, confounded systems).
• Compositional and categorical generalizations encompassing quantum causal models, where the counterpart of a probabilistic distribution over functions remains an open problem.
• Formal connections to information-theoretic tasks, such as secure communication, channel identification, and causal discovery algorithms.
• Refinements of resource theory tasks in (classical and quantum) device-independent causal inference and foundations.

Conclusion

The paper establishes two resource-theoretic structures for classical causal inference: RTCaus for deterministic mechanisms and RTKnowCaus for uncertainty over causal maps. Both are fully characterized (for binary cases) by operational monotones and provide a unifying algebraic and geometric perspective. This formalism exposes the subtlety missed in coarser information-theoretic frameworks and operationally grounds the quantification and manipulation of causal knowledge. These results provide a principled basis for future advances in causal reasoning, generalized resource theories, and their extensions to quantum and AI applications.