Multiscale Causal Framework

• Multiscale Causal Framework is a method to extract macro-level causes by grouping micro-level observations into equivalence classes based on intervention outcomes.
• It employs algorithms that estimate conditional distributions and cluster similar response patterns to design cost-effective experimental interventions.
• The approach applies to fields like neuroscience and climate science, offering a hierarchical, interpretable analysis of high-dimensional data.

A multiscale causal framework formalizes how causal relationships can be discovered, represented, and analyzed across multiple hierarchical levels of description, especially in systems where only micro-level measurements are available but macro-level causal relations are scientifically relevant. This approach enables the automatic construction of macro-variables and their causal connections from high-dimensional micro-variable datasets, leveraging intervention theory, information theory, and modern clustering and classification algorithms. In its canonical form, such a framework not only identifies the fundamental "units" of causation (macro-variable causes and effects) but also systematically relates micro-level manipulations to macro-level causal structure, providing algorithms for variable discovery and experimental design strategies that minimize intervention cost. Applications span domains from neuroscience (e.g., neural population responses) to climate science, and the approach generalizes classical causal modeling by decoupling the definition of macro-variables from a priori experimental knowledge.

1. Theoretical Foundation and General Principles

The central paradigm is the automatic discovery of macro-level causal variables from micro-level observations by positing that causal relations between macroscopic features are supervenient on equivalence classes of microstates. Two microstates $i_1, i_2$ in the input space $I$ are considered causally equivalent if they yield identical distributions over output states $J$ under intervention:

$$i_1 \sim_{(I)} i_2 \iff \forall j \in J,\, P(j\mid \operatorname{man}\{i_1\}) = P(j\mid \operatorname{man}\{i_2\})$$

and similarly for output states:

$$j_1 \sim_{(J)} j_2 \iff \forall i \in I,\, P(j_1 \mid \operatorname{man}\{i\}) = P(j_2 \mid \operatorname{man}\{i\})$$

where $\operatorname{man}\{i\}$ denotes an atomic manipulation setting micro-variable $I$ to $i$.

Macro-level variables are then defined as the equivalence classes (fundamental causal partitions) determined by these relations. These define the fundamental cause $C$ and effect $E$ at the finest resolution at which causal description remains lossless. Coarser macro-variables (subsidiary causes/effects) can be constructed by merging clusters under stricter coarsening conditions, encapsulating independent mechanisms.

The generative model for such systems, or "ml-system", incorporates possible confounding:

$$P(J, I) = \sum_{h} P(J \mid I, H=h)\,P(I \mid H=h)\,P(H=h)$$

but the only structurally relevant features for macro-level causality are those invariant under intervention.

2. Micro-to-Macro Causal Abstraction

The transition from micro-level measurements to macro-level causes and effects is driven by an operationally grounded equivalence: two micro-inputs belong to the same causal macro-class if intervening on either produces indistinguishable effect distributions. In mathematical terms, macro-variables arise as minimal sufficient statistics of the micro-variables with respect to the downstream causal effect.

This process is distinct from classical dimension reduction or clustering; the equivalence classes must preserve response distributions under intervention, not just observational similarity. The algebraic relationship ensures the causal model at the macro-level is both minimal (irreducible) and lossless for intervention-based queries, differing from mere empirical aggregation or regression.

3. Algorithms for Macro-Variable Discovery

The framework provides explicit algorithmic processes for extracting fundamental causes/effects:

Algorithm 1: Learning Fundamental Causal Variables

• Input: Interventional dataset $$\mathcal{D}_{csl} = \{ (i_1, j_1),..., (i_n, j_n) \}$$ with $j_k \sim P(J \mid \operatorname{man}\{i_k\})$.
• Density Estimation: Estimate $P(J \mid \operatorname{man}\{i\})$ for each microstate $i$ (dimension reduction and kernel density estimation may be applied).
• Vector Construction: For each $i \in I$, compute the effect vector $$\operatorname{Eft}_{mic}(i) = [P(j_1 | \operatorname{man}\{i\}),...,P(j_n | \operatorname{man}\{i\})]$$.
• Clustering: Cluster the vectors via a flexible approach (e.g., Dirichlet Process Gaussian Mixture Model).
• Merging: Merge clusters with indistinguishable macro-level (partitioned) causal behavior.
• Classification: Train classifiers on the macro-variable labels to generalize partitions outside the observed intervention set.

Additional algorithms extend to extraction of subsidiary variables—coarser macro-variables—by further merging under independence constraints, thus uncovering decomposable causal mechanisms.

4. Efficient Experimental Design

Experimental design leverages the "Fundamental Causal Coarsening Theorem" (fCCT), which states that, except for a set of measure zero, the observational partition (obtained from $P(J \mid I)$ without interventions) refines the fundamental causal partition. The practical implication is that:

1. Observational data can be used to form preliminary equivalence classes (observational clusters).
2. Interventions are then required on only a single representative per observational cluster, drastically reducing the necessary number of manipulations to correctly deduce the fundamental causal partition.

This design enables causal discovery in high-dimensional domains with intervention cost proportional not to $|I|$ but to the number of observational classes, which is typically much smaller.

5. Multiscale Causal Structure and Subsidiary Variables

The multiscale aspect arises from recognition that causal descriptions may be organized hierarchically: the fundamental partitions admit coarsenings corresponding to subsidiary causes/effects when these groupings respect independent mechanisms. For example, in cases where horizontal and vertical bars in an image elicit independent neural population responses, partitioning along each yields subsidiary variables explaining independent effects.

Algorithmically, subsidiary variables are discovered by merging causal classes only when this does not introduce ambiguity in effect, ensuring manipulation of a subsidiary cause is interpretable and non-interacting. This systematic discovery of a multilevel hierarchy of causal variables supports the analysis of systems with nested or modular mechanisms.

6. Illustrative Neuroscience Example

A simulated neuroscience experiment solidifies the framework's principles:

• Inputs: Images with possible presence of horizontal bar (h-bar), vertical bar (v-bar), both, or neither, plus pixel-level noise.
• Outputs: Population neural activity with the top half encoding pulse (probability 0.8 with h-bar), or 30 Hz rhythm (probability 0.8 with v-bar); the bottom half manifests a distractor rhythm, independent of the stimulus.
• Ground truth macro-variables: The fundamental cause $C$ comprises four classes—only h-bar, only v-bar, both, neither. The fundamental effect $E$ mirrors these with neural response patterns—pulse only, rhythm only, both, neither.
• Results: The framework correctly recovers $C$ and $E$ by clustering the estimated conditional distributions, discarding spurious distractor patterns due to their invariance under intervention. Subsidiary variables elucidate more focused queries (e.g., isolating the causal path specific to neural pulses from the h-bar).

This example demonstrates the method's ability to extract intuitive macro-level descriptions from noisy, high-dimensional data.

7. Implications and Applications

The multiscale causal framework formalized in this context enables:

• Objective construction of macro-level causal variables tailored to high-dimensional measurement settings.
• Theoretical guarantees on minimal experimental cost for causal variable extraction.
• Hierarchical analysis of causal structure via the discovery of subsidiary causal variables, revealing decomposable or modular causal structure.

Potential applications span neuroscience (neural coding and brain imaging), medicine (biomarker discovery), climate science (spatiotemporal modeling), and other fields requiring interpretable mechanisms from micro-measurements.

The framework's main innovation is the formal connection—via equivalence on intervention-induced effect distributions—between microstates and macro causal variables, supported by robust, algorithmic, and experimentally efficient variable discovery and multiscale causal analysis.