Bayesian Structural Causal Inference

Modelling the Fish and the Tank

Based on CausalPy Documentation

The Metaphor

The Pet Shop Problem

Imagine a causal analyst as a pet-shop owner introducing a new fish to one of their many aquariums.

• We ask: “Does this new fish survive?”
• We usually focus on the fish:
  • Is it healthy?
  • Is it a hardy species?
  • What are its intrinsic properties?

The Systemic View

But the fish’s survival depends less on its intrinsic properties than on the tank.

• The pH balance.
• The predators.
• The temperature.
• The hidden currents.

The Insight: When we ask “What is the causal effect?”, we are not asking about an isolated variable. We are asking: “In which world are we operating?”

Methodological Philosophies

Two Schools of Thought

Minimalism

(Design-Based)

• Goal: Isolate the effect by minimizing assumptions.
• Tools: RCTs, Instrumental Variables, Diff-in-Diff.
• The Vibe: “Don’t model the tank; just find a clean experiment within it.”

Maximalism

(Structural)

• Goal: Explicitly parameterize the system.
• Tools: Bayesian Structural Models (SCMs).
• The Vibe: “Model the physics of the tank to understand how the fish survives.”

The Great Trade-Off

Why choose Structural Maximalism?

The Cost: Risk of Misspecification. If you get the physics of the tank wrong, your answers will be wrong.

The Reward: Transparency. Every assumption becomes an explicit, testable component rather than an implicit background condition. We trade robustness for a map of the mechanisms.

The Bayesian Workflow

Step 1: Inference (Backwards)

First, we must understand the “physics” of the aquarium.

We infer the most plausible state of the world (\(w\)) conditioned on the observable data (\(X, T, O\)).

\[P(w | X, T, O)\]

We are asking: Given the data we see, what must the causal graph look like?

Step 2: Prediction (Forwards)

Once the “world” is defined by our posterior distribution, we move forwards.

We simulate Counterfactual Worlds.

\[P(Y^* | w, do(T))\]

We are asking: In a world defined by these physics, what happens if we intervene?

Identification & Structure

The Problem of Endogeneity

In the real world, the propensity to take a treatment is often predicted by the same factors that determine the outcome.

• The Bias: The model attributes outcome variation from unobserved factors (\(U\)) to the treatment (\(T\)).
• The Result: We confuse the “currents” of the water with the “movement” of the fish.

Priors as Structure

How do we solve this without a Randomized Controlled Trial?

We use Information to constrain the Structure.

By placing tight priors on the correlation parameters (e.g., \(\rho\)), we “regularize” the latent correlation. We effectively limit how much endogeneity is allowed to distort the inference.

We use our prior knowledge of the tank to constrain our estimates of the fish.

Variable Selection in Joint Models

Beyond Prediction

In standard regression, variable selection (Lasso, Ridge) is a Predictive tool. * Goal: Prune the noise to prevent overfitting.

In Joint Structural Models, variable selection becomes a Structural tool. * Goal: Discover the architecture of the causal graph.

Causal Discovery via Shrinkage

When we apply “sparsity priors” (like Horseshoe or Spike-and-Slab) to a system of equations, the model can discriminate between:

1. Instruments: Variables that drive Treatment (\(T\)) but not Outcome (\(Y\)).
2. Confounders: Variables that drive both.

The model effectively “learns” exclusion restrictions. It separates the levers that move the treatment from the confounders that muddy the waters.

Non-Parametric Approaches

When Parameters Fail

Sometimes, the shape of the “tank” is too complex for linear equations.

If we force a linear line on a non-linear world, our structural parameter (\(\alpha\)) collapses.

The Solution: Bayesian Additive Regression Trees (BART).

Flexible Imputation

With BART, we replace rigid parameters with flexible function approximation.

1. Fit a flexible model for \(E[Y | X, T]\).
2. Impute \(Y(1)\): Set everyone to treated, predict outcomes.
3. Impute \(Y(0)\): Set everyone to control, predict outcomes.

Even if the “coefficient” is uninterpretable, if the model learns the shape of the tank, the imputed difference recovers the causal effect.

Conclusion

Epistemic Modesty

“Every causal model, like every fish tank, is a ‘small world’ whose regularities we can nurture but never universalize.”

Bayesian structural causal inference unites epistemic modesty with computational rigor.

Each model is not a final map of the world. It is a provisional machine for generating causal understanding.

Final Thought

Our task is not to master the ocean.

Our task is to build clear tanks, understand their physics, and know when to change the water.