Surjective Independence of Causal Influences

• SICI is a Bayesian network model where a surjective mapping groups multiple parent nodes into mechanism blocks, allowing within-block interactions and independence across blocks.
• It reduces the exponential growth of conditional probability tables by lowering the elicitation burden and minimizing parameter complexity in complex systems.
• Variants like LS-SICI, DS-SICI, and US-SICI demonstrate different placements of stochasticity, offering flexible trade-offs between deterministic and probabilistic modeling.

Surjective Independence of Causal Influences (SICI) is a local Bayesian-network structure model in which a child node $Y$ is influenced not by a bijective family of one-parent–one-mechanism pathways, as in standard independence of causal influences (ICI), but by a surjective mapping from parents to a smaller set of mechanism nodes $\mathbf{M}=\{M_1,\dots,M_m\}$, $m\le n$. In this formulation, several parents may share a mechanism, so dependencies are permitted within parent blocks while independence assumptions are retained across mechanisms. The model was introduced to reduce conditional-probability-table burden and expert-elicitation burden without imposing the stronger assumption that every parent acts through its own unique independent mechanism (Drury et al., 29 Sep 2025).

1. Formal definition and local Bayesian-network structure

SICI is defined for a child node $Y$ with parent set

$\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$

In a fully general discrete Bayesian network, parameterising the local conditional distribution $p(y\mid \mathbf{x})$ requires

$N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$

where $s_i$ is the number of states of parent $X_i$ and $s_c$ is the number of states of $\mathbf{M}=\{M_1,\dots,M_m\}$0. SICI addresses this exponential growth by introducing mechanism nodes

$\mathbf{M}=\{M_1,\dots,M_m\}$1

as an augmented local structure between $\mathbf{M}=\{M_1,\dots,M_m\}$2 and $\mathbf{M}=\{M_1,\dots,M_m\}$3 (Drury et al., 29 Sep 2025).

The formal innovation is the surjection

$\mathbf{M}=\{M_1,\dots,M_m\}$4

Because $\mathbf{M}=\{M_1,\dots,M_m\}$5 is surjective, every mechanism receives at least one parent, but a mechanism may receive more than one parent. For each mechanism $\mathbf{M}=\{M_1,\dots,M_m\}$6, the parent block assigned to it is

$\mathbf{M}=\{M_1,\dots,M_m\}$7

These blocks form a partition of $\mathbf{M}=\{M_1,\dots,M_m\}$8. The interpretation is that parents in the same block are allowed to interact before affecting the child, whereas the resulting mechanisms are arranged so that an ICI-type assumption is more plausible across $\mathbf{M}=\{M_1,\dots,M_m\}$9 (Drury et al., 29 Sep 2025).

The local structure is defined by four conditional independences: $m\le n$0

$m\le n$1

$m\le n$2

$m\le n$3

These conditions localize the role of mechanism nodes. Once $m\le n$4 is known, parents carry no additional information about $m\le n$5; different mechanisms are conditionally independent given the full parent configuration; the added mechanism nodes are local modeling devices; and each mechanism depends only on its assigned parent block (Drury et al., 29 Sep 2025).

The paper also notes a planarity or ordering condition under which the SICI graph can be drawn planar by ordering parents and mechanisms so that

$m\le n$6

and whenever $m\le n$7, all parents between them map to the same mechanism. This is presented as a drawing or representation convenience rather than a substantive axiom (Drury et al., 29 Sep 2025).

2. Relation to ICI and earlier causal-independence formalisms

SICI is explicitly a relaxation of standard ICI. In standard ICI, each parent $m\le n$8 influences $m\le n$9 through its own mechanism $Y$0, so the parent–mechanism relation is bijective. In SICI, the bijection is replaced by the surjection $Y$1, so independence is assumed across grouped mechanisms rather than across singleton parents. If every block has size one, then $Y$2 and $Y$3 is bijective; in that case SICI reduces to the usual ICI/PICI setup (Drury et al., 29 Sep 2025).

This places SICI within a longer line of causal-independence research. Heckerman’s temporal formulation defined causal independence through conditional independences of the form

$Y$4

capturing the principle that when exactly one cause changes and the others are held fixed, the effect update depends only on the previous effect and that cause’s transition (Heckerman, 2013). Heckerman and Breese later gave an equivalent atemporal characterization in terms of nested functional decomposition, with cause-specific latent contributions

$Y$5

combined by

$Y$6

and, in the equal-function case, the common combiner $Y$7 must be associative and commutative (Heckerman et al., 2013).

Zhang and Yan formulated ICI generatively through parent-specific contribution variables $Y$8 and a commutative, associative binary operator $Y$9 such that

$\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$0

with the induced factorization

$\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$1

where $\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$2 (Zhang et al., 2013). Meek and Heckerman generalized the one-cause-per-mechanism restriction through causal interaction models, in which a mechanism may have several causes while mechanism families remain conditionally independent given the causes; SICI is closely aligned with that multi-cause-mechanism perspective, though the newer SICI formalism is explicitly organized around the surjection $\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$3 and blockwise local Bayesian-network parameterisation (Meek et al., 2013).

A central distinction is that SICI explicitly introduces surjectivity at the parent-to-mechanism level. Earlier ICI work does not formulate a condition named “surjective independence,” and several precursor papers state that they do not discuss surjectivity explicitly (Heckerman, 2013). This suggests that SICI’s novelty is not the general idea of independent causal contributions, but the replacement of parentwise independence by blockwise independence through a many-to-one mapping (Drury et al., 29 Sep 2025).

3. SICI variants and factorisations

The SICI paper defines three variants according to where stochasticity is placed in the augmented local structure (Drury et al., 29 Sep 2025).

Lower-Stochastic SICI (LS-SICI) uses deterministic upper relations

$\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$4

where $\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$5 may be OR, AND, XOR, MAX, or a composition of such operators, together with a stochastic lower model $\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$6. The induced local approximation is

$\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$7

Here multiple parent configurations may collapse to the same mechanism configuration, so several rows of $\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$8 share the same child distribution. The paper emphasizes that this shifts burden from quantitative elicitation to qualitative judgments about parent groupings and deterministic activation conditions (Drury et al., 29 Sep 2025).

Double-Stochastic SICI (DS-SICI) makes both levels stochastic. Each mechanism has a conditional distribution

$\mathrm{Pa}(Y)=\mathbf{X}=\{X_1,\dots,X_n\}.$9

and the child has

$p(y\mid \mathbf{x})$0

Using the law of total probability and the SICI conditional independences, the local distribution becomes

$p(y\mid \mathbf{x})$1

This is the most general SICI variant. Dependence within blocks is represented by the block-level mechanism CPTs, and approximate independence across blocks is represented by the product over mechanisms (Drury et al., 29 Sep 2025).

Upper-Stochastic SICI (US-SICI) uses stochastic upper relations

$p(y\mid \mathbf{x})$2

but a deterministic lower relation

$p(y\mid \mathbf{x})$3

Then

$p(y\mid \mathbf{x})$4

This mirrors the standard ICI formula, but with block-level mechanisms in place of singleton-parent mechanisms (Drury et al., 29 Sep 2025).

For comparison, the paper recalls the standard deterministic-lower ICI expression

$p(y\mid \mathbf{x})$5

and the PICI form

$p(y\mid \mathbf{x})$6

SICI generalizes these by replacing singleton conditioning $p(y\mid \mathbf{x})$7 with block conditioning $p(y\mid \mathbf{x})$8, and by allowing $p(y\mid \mathbf{x})$9 (Drury et al., 29 Sep 2025).

4. Parameterisation, elicitation, and illustrative constructions

SICI is presented as a middle ground between an unrestricted CPT and standard ICI. A full CPT is exponential in the number of parents; ICI is typically linear in $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$0 for many canonical models; SICI lies between them. If $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$1, the lower model $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$2 has a reduced parent set, and the upper mechanism CPTs are smaller local objects than the original full $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$3 CPT. In LS-SICI, the burden can be especially low because the upper relations are deterministic, leaving only $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$4 to quantify (Drury et al., 29 Sep 2025).

The paper proposes a practical modeling sequence. One starts from the child $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$5 and its known parent set $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$6, chooses a partition

$N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$7

introduces one mechanism node $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$8 per block, selects LS-SICI, US-SICI, or DS-SICI, parameterises upper relations by deterministic functions or conditional distributions, parameterises the lower relation directly or via an ICI-based approximation on $N_Y=\left(\prod_{i=1}^n s_i\right)(s_c-1),$9, and then recovers $s_i$0 using the appropriate SICI formula (Drury et al., 29 Sep 2025). The guiding principle for the partition is to place parents in the same block when ICI is believed to fail between them, while seeking a partition for which ICI is more plausible across mechanisms.

A canonical example is the surjective noisy-OR. In standard noisy-OR with binary variables, each parent has its own mechanism and its own inhibitor. In the SICI version, parents are partitioned into blocks, each block shares one mechanism $s_i$1, each mechanism has one inhibitor $s_i$2, the block-to-mechanism rule is $s_i$3, and the child remains

$s_i$4

In the paper’s six-parent example, standard noisy-OR requires 6 quantitative inhibitor parameters, whereas the surjective noisy-OR uses 3 because there are $s_i$5 blocks; in general the saving is $s_i$6 such parameters (Drury et al., 29 Sep 2025).

A second example embeds Hassall’s binary interpolation rule inside SICI. For binary parents and child,

$s_i$7

The paper shows that this can be represented as a PICI model with binary mechanisms

$s_i$8

and

$s_i$9

yielding

$X_i$0

The SICI role is structural: once parents have been partitioned and mechanisms introduced, such ICI-based approximation methods can be applied to the reduced lower model $X_i$1 (Drury et al., 29 Sep 2025).

This reuse of ICI machinery has antecedents. Zhang and Yan showed how ICI factorizations can be exploited algorithmically in clique tree propagation through deputation and $X_i$2-homogeneous factorizations (Zhang et al., 2013). A plausible implication is that SICI inherits not only representational flexibility but also a pathway for reusing established local-structure and inference approximations on the lower mechanism layer, provided the grouped-mechanism abstraction is substantively defensible.

5. Theoretical scope, neighboring concepts, and limitations

SICI is methodological rather than theorem-heavy, but several formal claims are explicit. First, ICI is a special case of SICI: if the best surjection is actually a bijection, SICI reduces to ICI or PICI depending on where stochasticity is placed. Second, SICI enlarges representational power relative to ICI because multiple parents may feed a common mechanism. Third, the LS-SICI, DS-SICI, and US-SICI factorisations follow from the law of total probability together with $X_i$3 and the product decomposition of $X_i$4 across mechanisms. Fourth, although the paper does not give a universal closed-form parameter-count theorem, it repeatedly states the comparative point that full CPTs are exponential in $X_i$5, whereas SICI can significantly reduce the parameter space by moving the lower model from $X_i$6 parents to $X_i$7 mechanisms and breaking the upper level into smaller local objects (Drury et al., 29 Sep 2025).

The model is also situated relative to broader causal-independence literature. Meek and Heckerman’s causal interaction models already allowed a mechanism to have several causes while retaining independence across mechanisms; this is conceptually very close to SICI’s grouped-mechanism interpretation (Meek et al., 2013). Heckerman’s temporal representation and Heckerman–Breese’s functional decomposition remain relevant because they formalize order-insensitive modular combination of cause-specific contributions, but they do not introduce a surjective parent-to-mechanism mapping (Heckerman, 2013, Heckerman et al., 2013). The 2025 SICI formulation can therefore be read as a principled local-structure reformulation rather than as a replacement for the older theory of causal independence.

Several neighboring notions should be distinguished from SICI. The paper on the intersection property of conditional independence studies when statements such as

$X_i$8

can be combined into

$X_i$9

subject to support-connectivity conditions (Peters, 2014). That result concerns the combination of conditional independence statements and causal identifiability in additive noise models, not local Bayesian-network parameterisation by grouped mechanisms. Likewise, the generalized symmetric tequila problem studies settings in which independent causes, conditional independence of effects given causes, and pairwise effect independence coexist in a symmetric binary model, but this is a classification of influence–independence coexistence rather than a local-structure model with a surjection $s_c$0 (Steel et al., 2013). Agosta’s CICI analysis concerns inter-causal independence between parents after conditioning on a child state in collider structures such as noisy-or, not SICI’s parent-to-mechanism partitioning (Agosta, 2013).

The limitations stated for SICI are practical and conceptual. Its success depends critically on selecting a good surjection $s_c$1, yet the paper does not provide a full automated or formally optimal method for doing so. Mechanism nodes often do not represent explicit real-world variables, which can make direct elicitation of $s_c$2 awkward, especially in DS-SICI and US-SICI. DS-SICI may remain elicitation-heavy, and LS-SICI may be too coarse because deterministic upper mappings force multiple CPT rows to share the same child distribution. The paper also does not provide a comprehensive theory of identifiability, parameter learning algorithms specialized to SICI, inference complexity reductions, structure-search procedures for partitions, or asymptotic guarantees (Drury et al., 29 Sep 2025).

6. Significance and prospective research directions

The practical significance of SICI lies in its reformulation of elicitation. Rather than assessing a full $s_c$3, experts can make two kinds of judgments: qualitative judgments about which parents belong in the same mechanism block and what deterministic or semantic relation governs that block, and smaller quantitative judgments about $s_c$4 or about stochastic mechanism nodes. The paper argues that this better matches expert thinking in domains with sparse data, many interacting factors, and substantial reliance on elicited knowledge (Drury et al., 29 Sep 2025).

From a historical perspective, this suggests a shift from parentwise modularity to pathway-level modularity. The classic ICI literature focused on independent contributions $s_c$5 or $s_c$6 combined by a deterministic operator such as OR or addition (Heckerman, 2013, Zhang et al., 2013). SICI retains the mechanism-centered view but treats the parent-to-mechanism correspondence as many-to-one rather than one-to-one. A plausible implication is that SICI should be understood not as a rejection of independence-of-causal-influences models, but as a blockwise relaxation designed for cases in which experts regard some parent interactions as intrinsic to a shared pathway.

Open directions are explicitly indicated. One is systematic learning or elicitation of the surjection $s_c$7 from data and/or experts. Another is further investigation of SICI analogues of earlier ICI subclasses such as amechanistic, decomposable, multiply decomposable, and temporal ICI. A third is the development of specialized inference and learning theory. The current paper presents SICI as a structural wrapper that broadens the domain in which ICI-style approximations can be defended, rather than as a complete inferential framework (Drury et al., 29 Sep 2025).

In summary, SICI designates a local Bayesian-network structure in which a surjective parent-to-mechanism map partitions the parent set into interacting blocks, independence is asserted across the resulting mechanisms, and the child depends on parents only through those mechanisms. Its principal contribution is to preserve much of the tractability and elicitation efficiency associated with ICI while admitting controlled within-block interactions that standard ICI excludes (Drury et al., 29 Sep 2025).