Parallel world graph

As explained in Causal AI

A parallel world graph is a graph representation that places variables from multiple counterfactual worlds together without necessarily permitting the same separation relations as a counterfactual graph.

Suppose an online-game player had low engagement and spent less than $50. What would that player have spent if engagement had been high? The observed purchase and the hypothetical purchase belong to different possible worlds. A parallel world graph places both worlds in one model so the question can be expressed and evaluated. section 8.4.2

Start with a causal DAG and duplicate its endogenous variables for each world. Keep the same causal structure and connect the copies through shared exogenous variables in a structural causal model (SCM). Exogenous variables represent causes left outside the explicit model; functional assignments determine endogenous values from their parents. With two worlds, the result is also called a twin-world graph. Potential outcomes are values of the same variable across those worlds. section 8.4.2 section 8.4.1

Figure 8.9 In the parallel world graph, we use the exogenous variables in an SCM to unite the duplicate causal DAGs across worlds. The result is a single SCM with duplicate endogenous variables.

The duplicated DAGs share exogenous nodes, making the cross-world connection visible. section 8.4.2

For a counterfactual purchase query, use the graph in three steps:

• Abduction: use observed engagement and purchases to infer a distribution over exogenous variables. section 9.5
• Action: intervene on engagement in the hypothetical world, replacing its usual causal assignment. section 9.5 section 9.1.3
• Prediction: propagate the inferred exogenous distribution through the hypothetical world to obtain a purchase distribution. section 9.5

Without an intervention, corresponding endogenous values remain equal across worlds under consistency. An intervention can instead change downstream hypothetical outcomes. This exact equality creates a caution: ordinary d-separation or d-connection on a parallel world graph does not, in general, establish dependence or independence across worlds. A counterfactual graph is a derived graph on which those separation relationships hold. section 8.4.2 section 9.1.3

Adding a third world

A streaming-service executive can condition on actual deal outcomes and attrition, posit that a successful new-franchise deal would have raised engagement in a second world, then ask about attrition if a competing deal had failed in a third. Each added world makes its hypothetical conditions explicit; the answer still depends on the stated assumptions. section 8.4.4

Parallel-world models can be constructed by hand with pgmpy or Pyro. For a DAG-based transformation, y0 produces parallel world graphs; ChiRho provides a TwinWorldCounterfactual handler. section 11.6.1

How Causal AI covers it

How deep it goes

Works through SCM-based graph construction and inference steps, while deferring valid cross-world separation reasoning to counterfactual graphs.

Examples

Uses an online-game purchase question and a streaming-service deal scenario spanning three possible worlds.

What it assumes

Assumes a causal DAG and counterfactual question, while defining the SCM components needed to connect worlds.

What you can do after

Apply it via pgmpy, Pyro, y0, or ChiRho to represent multi-world counterfactual models.

How it connects

Links causal DAGs, structural causal models, potential outcomes, ideal interventions, counterfactual inference, and counterfactual graphs.

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Related concepts

• d-separationcontrasts with · D-separation is a graphical criterion for determining whether a conditioning set blocks all relevant paths between variables in a graph, implying conditional independence under the graph’s assumptions.
• counterfactual reasoninguses · Counterfactual reasoning examines hypothetical outcomes under conditions, actions, or policies different from those that actually occurred.
• causal graphical modelkind of · A causal graphical model is a probabilistic or structural model whose graph represents causal relationships and whose mechanisms support reasoning about interventions.
• possible worlduses · A possible world is a hypothetical state of affairs representing how a causal system might have unfolded under different conditions.
• structural causal modeluses · A structural causal model represents causal relationships among variables through equations that determine each variable from other variables and exogenous factors.

This explainer was generated by Manning's knowledge graph from the text of Causal AI. The section links point to the parts of the book it draws on.