Causal Reasoning

This chapter summarizes the causal-inference literature this package builds on, and shows how probabilistic_model.probabilistic_circuit.causal and probabilistic_model.probabilistic_circuit.relational implement it. It assumes the probabilistic-circuit background from Queries and the definitions in [CVVdB20].

Interventions and the backdoor-adjustment formula

A causal query asks what happens to one variable when another is forced to a value, rather than merely observed to have it. Pearl’s do-operator [Pea09] writes this as \(P(Y \mid do(X = x))\): the distribution of effect \(Y\) once cause \(X\) is set to \(x\) by intervention, cutting \(X\) off from whatever normally determines it. This differs from conditioning, \(P(Y \mid X = x)\), whenever \(X\) and \(Y\) share a common cause: observing \(X = x\) also tells you something about that common cause, which then leaks into what you infer about \(Y\); intervening on \(X\) does not.

The backdoor criterion below is stated over a causal graph:

Definition 13 (Causal Graph)

A causal graph is a directed acyclic graph \(G = (V, E)\) whose nodes \(V\) are random variables and whose edges \(E\) represent direct causal influence: an edge \(U \to W\) means \(U\) is a direct cause of \(W\). A node \(Z\) is a descendant of \(X\) in \(G\) if \(G\) contains a directed path from \(X\) to \(Z\).

with \(X\), \(Y\), and every candidate adjustment variable among \(G\)’s nodes \(V\). This package neither constructs nor verifies \(G\): registering causal_variables, effect_variables, and an adjustment set on a CausalCircuit is the analyst’s assertion, from domain knowledge, that such a \(G\) exists and that the chosen adjustment set satisfies the backdoor criterion relative to it. What the package verifies instead is the circuit-side precondition the next section’s polytime algorithm depends on – support determinism – which is a property of the probabilistic circuit itself, not of \(G\); it is unrelated to the variable tree used later for tractability, and to the MarginalDeterminismTreeNode structure introduced below, which only groups query variables for that check and does not represent \(G\)’s edges.

Definition 14 (Backdoor Criterion)

A set of variables \(Z\) satisfies the backdoor criterion relative to an ordered pair \((X, Y)\) if:

1. no node in \(Z\) is a descendant of \(X\), and

2. \(Z\) blocks every path between \(X\) and \(Y\) that contains an arrow into \(X\).

If \(Z\) satisfies the backdoor criterion, the causal effect of \(X\) on \(Y\) is identifiable from observational data alone, and is given by

\[P(Y \mid do(X = x)) = \sum_z P(Y \mid X = x, Z = z) \, P(Z = z).\]

With an empty adjustment set (no confounding to correct for – the case for independently randomized training data), this collapses to \(P(Y \mid do(X = x)) = P(Y \mid X = x)\): the interventional and observational distributions coincide.

Tractable causal inference on probabilistic circuits

Evaluating the backdoor formula requires computing \(P(Y \mid X = x, Z = z)\) for every value \(z\) in the adjustment set’s domain, and summing. On an arbitrary joint distribution this can be as expensive as the domain of \(Z\) is large.

Definition 15 (Variable Tree)

A variable tree over a variable set \(X\) is a full, rooted binary tree whose leaves are in one-to-one correspondence with the variables in \(X\). Each internal node corresponds to the subset of \(X\) at the leaves beneath it, partitioned by that node’s two children into disjoint left and right subsets ([KVdBCD14], where it is termed a vtree).

[WK23] shows that on a structured-decomposable probabilistic circuit – one where every product unit’s children partition the scope along a shared, recursively fixed Definition 15 – backdoor adjustment is tractable in the size of the circuit whenever the circuit satisfies one additional structural property for the relevant variables: support determinism.

Definition 16 (Support Determinism)

A sum unit is support-deterministic over a variable set \(V\) if, for every pair of its children, the children’s marginal supports on \(V\) are disjoint: no assignment to \(V\) has positive probability under more than one child.

A circuit is support-deterministic over \(V\) if every sum unit that splits on any variable in \(V\) is support-deterministic over \(V\).

Support determinism over the cause and adjustment variables means each sum unit’s children already partition the world by the value those variables take – so answering “what is \(P(Y \mid X = x, Z = z)\) under this branch” never requires mixing across branches, and the backdoor sum becomes a single weighted pass over the circuit rather than one circuit evaluation per \((x, z)\) pair. [WK23] formalizes the circuit family that guarantees this – marginal-deterministic variable trees (termed md-vtrees in [WK23]), a generalization of probabilistic sentential decision diagrams [KVdBCD14] – and derives the first polytime algorithm for backdoor adjustment on such circuits. This package does not require a marginal-deterministic-variable-tree-typed circuit outright; instead it verifies the support-determinism property directly on whatever structured-decomposable circuit it is given, which is the property the polytime algorithm actually depends on.

Implementation: CausalCircuit

probabilistic_model.probabilistic_circuit.causal.causal_circuit.CausalCircuit wraps a fitted ProbabilisticCircuit together with:

• causal_variables / effect_variables – the registered \(X\) and \(Y\) of Definition 14.

• marginal_determinism_tree, a MarginalDeterminismTreeNode – a binary tree over the cause variables, built by from_causal_graph recursively bisecting the (priority-ordered) cause list. Each node records the variables in its subtree and a query_set (by default its own highest-priority variable); all_query_sets() flattens every node’s query_set in pre-order, and their union is exactly the set of cause variables that need to satisfy Definition 16. find_node_for_variable additionally lets diagnose_failure walk the tree to recover the priority a variable was registered with.

• verify_support_determinism(), which takes that union of query variables and checks Definition 16 for each of them against every sum unit in the circuit, raising SupportDeterminismVerificationResult with every violation found if any check fails. This is a validity check, not a construction step: it does not repair a circuit that fails it.

• backdoor_adjustment(cause_variable, effect_variable, adjustment_variables), which implements the formula in Definition 14 directly: with no adjustment set it truncates the circuit to each cause region and reads off the effect marginal; with one, it additionally weights each cause-region branch by \(P(Z = z)\) and sums, using the region structure verify_support_determinism already confirmed is disjoint.

Worked example: a confounded discrete circuit

Take a scene with three discrete variables: season \(\in \{\)WARM, COLD\(\}\) (a confounder), treatment \(\in \{\)HIGH, LOW\(\}\), and outcome \(\in \{\)GOOD, BAD\(\}\). treatment has no causal effect of its own; every difference in outcome is actually driven by season, which also happens to influence which treatment gets chosen – the classic Simpson’s-paradox shape. The fitted circuit is a root SumUnit over two strata:

\[\begin{split} \begin{aligned} \text{Warm stratum } (w = 0.6): \quad & P(\texttt{treatment}=\text{HIGH}) = 0.8, \quad \texttt{outcome} = \text{GOOD (deterministic)} \\ \text{Cold stratum } (w = 0.4): \quad & P(\texttt{treatment}=\text{HIGH}) = 0.3, \quad \texttt{outcome} = \text{BAD (deterministic)} \end{aligned} \end{split}\]

Structurally, each stratum is a ProductUnit of three independent factors: a point-mass leaf on season, a two-leaf SumUnit over treatment, and a point-mass leaf on outcome. The root splits on season with disjoint branches (WARM vs. COLD) – support-deterministic over season by construction. Each stratum’s inner SumUnit splits on treatment with disjoint HIGH/LOW leaves – support-deterministic over treatment too. The root itself is not a split node for treatment: both strata give treatment positive probability on both HIGH and LOW, so their marginal supports on treatment overlap rather than partition it, and _check_sum_unit_for_variable’s split-detection correctly leaves it out of the check. CausalCircuit.verify_support_determinism() passes with causal_variables = [treatment, season], effect_variables = [outcome].

Conditioning naively on treatment = HIGH recovers the spurious correlation:

\[P(\texttt{outcome}=\text{GOOD} \mid \texttt{treatment}=\text{HIGH}) = 0.8,\]

the same number backdoor_adjustment(treatment, outcome) returns with an empty adjustment set, since an empty \(Z\) cannot separate the confound from the effect. Registering season as the adjustment variable instead runs the general path, _compute_interventional_circuit_with_adjustment: it extracts season’s two disjoint regions as \(Z\)-partitions, and for each treatment region builds a ProductUnit whose effect side is a fresh SumUnit over \(P(\texttt{outcome} \mid \texttt{treatment}=x, \texttt{season}=z)\) weighted by \(P(\texttt{season}=z)\) – the backdoor sum from Definition 14 evaluated stratum by stratum:

\[ P(\texttt{outcome}=\text{GOOD} \mid do(\texttt{treatment}=\text{HIGH})) = \underbrace{0.6}_{P(\text{WARM})} \cdot \underbrace{1.0}_{P(\text{GOOD}\mid \text{HIGH}, \text{WARM})} + \underbrace{0.4}_{P(\text{COLD})} \cdot \underbrace{0.0}_{P(\text{GOOD}\mid \text{HIGH}, \text{COLD})} = 0.6, \]

and the same \(0.6\) for treatment = LOW, revealing what conditioning hid: treatment has no causal effect on outcome at all. The ProductUnit for each treatment region is itself weighted by that region’s own marginal probability – \(P(\texttt{treatment}=\text{HIGH}) = 0.6 \cdot 0.8 + 0.4 \cdot 0.3 = 0.6\) – which is what keeps the returned circuit a valid joint distribution over \((\texttt{treatment}, \texttt{outcome})\) rather than four disconnected numbers.

Relational causal reasoning

The circuits above assume a fixed variable set, decided once and for all when the circuit is fitted. That does not hold for a scene with an unknown number of objects: how many objects there are is itself a fact you might want to intervene on or condition on, but no single flat circuit has “object count” as one of its variables ahead of time – the variable set depends on which scene is being described.

RelationalProbabilisticCircuit ([ND15], “Relational Sum-Product Networks”, and [Pruser26], extended here onto circuits with the query system probabilistic_model.probabilistic_circuit.relational bridges into krrood) resolves this by grounding: given a query describing one concrete scene, it stamps out one instance of a fitted template circuit per object the query mentions, and combines the instances into a single circuit over that scene’s actual variables. Where the query leaves some aggregation statistic undetermined – e.g. it asks about “a room with some chairs” without saying how many – grounding must decide how many child instances to stamp out and how to combine them, which requires resolving the statistic to a concrete value or distribution over values (the “undetermined latent”) before grounding the exchangeable part.

Resolving that undetermined latent by marginalizing it out immediately would blend over every possible object count and discard the count itself, leaving no variable behind to register as a cause or an effect. That is fine for prediction (the blended answer is what you want, if you don’t care why), but it removes exactly the variable a causal query about it would need. GroundingMode instead gives grounding two representations that retain the latent as a variable of the grounded circuit:

• GroundingMode.SAMPLED: draw monte_carlo_sample_count values of the latent from the conditioned circuit, ground one child instance per distinct sampled value, and retain each sampled value as an additional point-valued (Dirac) leaf mounted alongside its instance.

• GroundingMode.EXACT: instead of sampling, enumerate the latent’s own exact partition as already fitted by the class circuit’s JointProbabilityTree [NPSB23] – the branches of circuit.marginal(undetermined_latents)’s root sum unit – and ground one instance per branch, retaining that branch’s own region rather than a single point.

Why retaining the latent this way preserves support determinism

This is the structural argument ExchangeablePartGrounder.attach_monte_carlo_mixture and attach_exact_partition_mixture rely on, and the reason grounding can hand a causal circuit a retained latent without breaking Definition 16:

Theorem 5 (Grounded latent retention is support-deterministic)

Let \(L\) be an undetermined latent and \(\{a_1, \dots, a_k\}\) a finite set of values (or regions) of \(L\) such that the \(a_i\) are pairwise disjoint as subsets of \(L\)’s domain. Let \(C_1, \dots, C_k\) be circuits, each grounded by conditioning on a determined statistics and one \(a_i\), and mounted so that \(C_i\) additionally carries \(L \in a_i\) as part of its scope (as a Dirac leaf at \(a_i\), for a sampled point; as the branch region itself, for an exact partition branch). Then the sum unit \(S = \sum_i w_i \cdot C_i\) is support-deterministic over \(L\).

Sketch. Support determinism over \(L\) requires every pair of \(S\)’s children to have disjoint marginal support on \(L\). \(C_i\)’s marginal support on \(L\) is exactly \(a_i\) – for the Dirac case because a point mass has no support outside its point, and for the exact-partition case because \(a_i\) is one branch of circuit.marginal(undetermined_latents)’s root sum unit. Since the \(a_i\) are pairwise disjoint by hypothesis, so are the \(C_i\)’s marginal supports on \(L\). \(\blacksquare\)

The hypothesis – that the \(a_i\) are actually pairwise disjoint – is where the two modes differ, and it is not free for the exact-partition case:

• SAMPLED deduplicates its Monte-Carlo draws before grounding (_sample_undetermined_latents), so the \(a_i\) are distinct points – trivially pairwise disjoint regardless of what the underlying distribution over \(L\) looks like. This is why SAMPLED always succeeds: disjointness of single points needs no assumption about the fitted model.

• EXACT instead needs the fitted circuit to have actually partitioned on \(L\): a sum unit is only guaranteed support-deterministic (Definition 16) over variables whose children happen to disjointly cover it, and a fitted circuit does not retain, after fitting, which variables that held for. \(L\) not being one of them can leave circuit.marginal(undetermined_latents)’s root as a single, undifferentiated branch (trivially “one region”, not a genuine partition) or, in principle, with overlapping branches – either of which would violate the theorem’s hypothesis while still superficially looking like “a sum unit over \(L\)”. _undetermined_latents_partition_disjointly checks pairwise disjointness on the marginalized circuit directly, rather than assuming it from how the circuit was fit. If the check fails, the precondition of Theorem 5 does not hold, and attach_exact_partition_mixture raises UndeterminedLatentsNotPartitionedError rather than silently grounding every branch from the same representative point (which would discard \(L\)’s correlation with the rest of the circuit while still looking, structurally, like a valid disjoint partition). Grounding catches exactly this and falls back to SAMPLED, logging a warning that the template needs refitting so that it actually splits on \(L\) for EXACT to apply.

The theorem only gives structural determinism (the property verify_support_determinism checks); it says nothing about whether the mixture weights \(w_i\) are the right ones. Getting those right is a separate, per-mounting-node computation (_node_local_latent_log_likelihoods for SAMPLED, _node_local_branch_log_probabilities for EXACT): two product nodes that mount the same exchangeable relation can correlate \(L\) differently with the rest of the scene, so each node’s weights over the same global set of branches are computed from that node’s own local marginal, not shared globally.

RelationalCausalCircuit

probabilistic_model.probabilistic_circuit.relational.causal.RelationalCausalCircuit is the factory that composes the two halves: ground() grounds a RelationalProbabilisticCircuit for a query under a chosen GroundingMode, then from_grounded_circuit() resolves the requested cause/effect/adjustment variables (by Variable or by a dotted access-path string, via resolve_variable), builds the MarginalDeterminismTreeNode, and returns a verified CausalCircuit – the same object the flat, non-relational path produces, so backdoor_adjustment and diagnose_failure work identically regardless of whether the circuit came from a fixed template or was grounded per-query.

Registering more than one relational adjustment variable at once is the one place this can get expensive: CausalCircuit’s adjustment-region extraction takes a Cartesian product across the adjustment variables’ leaf regions, and a GroundingMode.EXACT variable’s region count can grow with the training data (unlike SAMPLED, capped by monte_carlo_sample_count). RelationalCausalCircuit.adjustment_region_count_warning_threshold warns when that product is large, as a diagnostic against discovering the cost only at query time – not a guarantee, since it is read off however the adjustment variables actually got grounded.

Literature position

Work	Relational	Causal	Substrate	Tractability mechanism
[ND15] (Relational Sum-Product Networks)	Yes	No	Sum-product network	Templated grounding
[WK23] (marginal-deterministic variable trees)	No	Yes	Structured-decomposable circuit	Support determinism \(\to\) polytime backdoor adjustment
[LHB+24] (Lifted Causal Inference)	Yes	Yes	Parametric factor graphs	Domain-lifted: polynomial in domain size for a bounded-logvar fragment, non-grounding queries only
This package	Yes	Yes	Relational Sum-Product Network grounded onto a structured-decomposable circuit	Cause/effect registration: free (additive to circuit size). Adjustment registration: guarded against Cartesian-product blowup

No prior work was found combining marginal-deterministic-variable-tree-style circuit causal inference with relational grounding directly; [LHB+24] is the closest bridge, but on a different substrate (parametric factor graphs, not circuits) with its own, differently scoped tractability class (lifted over object symmetry, rather than exact per-query grounding). This package stays fully grounded per query rather than lifted: it answers exact causal queries at the relational sum-product network’s existing per-query grounding granularity, not population-scale inference over many symmetric objects at once.

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