Causal Bayes Nets and the Do-Operator

When the Arrows Mean Causes

So far, the arrows in our Bayes nets have meant one specific thing: “carries information about.” An arrow $A \to B$ said the two were probabilistically linked, and Chapter 9 taught us to read exactly which variables inform which off the graph’s shape. But the arrows never claimed anything about causation. A fork $A \leftarrow B \to C$ makes $A$ and $C$ dependent without either one causing the other.

This chapter asks what changes when we insist the arrows mean causes — and the answer is one of the most important ideas in all of statistics. It’s the difference between seeing and doing.

Chibany is at the campus health center for a checkup.

Doctor: “Just so you know, people with yellow teeth have a higher rate of lung cancer.”

Chibany: “Huh. So, I should paint my teeth white, then? To lower my risk?”

Doctor: (laughing) “That’s not how it works.”

But why isn’t that how it works? Chibany can draw the Bayes net: yellow teeth and lung cancer really are statistically associated — $P(\text{cancer} \mid \text{yellow teeth})$ really is higher than $P(\text{cancer})$. So why shouldn’t acting on one change the other? Answering that is the whole chapter.

The Same Statistics, Three Different Stories

Here’s the catch that makes causation subtle. The single observation “yellow teeth and lung cancer occur together more often than chance” is compatible with three completely different causal structures:

graph LR
    subgraph Story1["1. teeth cause cancer"]
        T1(("yellow<br/>teeth")) --> C1(("cancer"))
    end
    subgraph Story2["2. cancer causes teeth"]
        C2(("cancer")) --> T2(("yellow<br/>teeth"))
    end
    subgraph Story3["3. common cause (true)"]
        S3(("smoking")) --> T3(("yellow<br/>teeth"))
        S3 --> C3(("cancer"))
    end
    classDef node fill:none,stroke:#9bbcff,stroke-width:2px,color:#fff
    class T1,C1,T2,C2,S3,T3,C3 node

Story 1 says whitening would help. Story 3 — the true one — says it wouldn’t: smoking stains the teeth and causes the cancer, so the teeth are just a marker, a symptom riding alongside the real culprit. Whitening the teeth paints over the marker without touching the cause.

Here is the crucial fact: these three graphs are indistinguishable from observation alone. They predict the exact same correlations. No amount of watching people’s teeth and lungs can tell you which story is true. (This is the structure-learning difficulty we flagged back in Chapter 8 — and the reason causal claims need more than data.) To break the tie, you need to do something, not just see.

Why observation can’t tell them apart: the factorizations agree

This isn’t hand-waving — it falls right out of the Markov factorization from Chapter 8. Write $T$ for yellow teeth and $C$ for cancer. Each graph turns into a product of one factor per node (each node given its parents):

• Story 1, $T \to C$: $;P(T, C) = P(T),P(C \mid T).$
• Story 2, $C \to T$: $;P(T, C) = P(C),P(T \mid C).$

Now — are these two different distributions? No. Both are just two ways of writing the same joint, because the definition of conditional probability says $P(T),P(C \mid T)$ and $P(C),P(T \mid C)$ both equal $P(T, C)$ exactly (it’s the chain rule, applied in the two possible orders). Reverse the arrow and you haven’t changed the distribution one bit — you’ve only re-ordered the factorization. And the only independence statement either graph could make over two variables is “$T \perp C$” — which neither one asserts (an arrow between them means dependent). So stories 1 and 2 make identical claims about every observable probability: same joint, same marginals, same correlation, same (empty) set of independencies.

Two graphs that encode the exact same set of conditional independencies are called Markov equivalent, and the collection of all graphs sharing those independencies is a Markov equivalence class (we add this term to the glossary). Stories 1 and 2 sit in the same class — so observation, which only ever sees the joint distribution, is blind to which arrow direction is the true one. The data literally does not contain the answer.

What about Story 3, the common cause? It lives over three variables ($S$ = smoking, plus $T$ and $C$), and factorizes as

$$P(S, T, C) = P(S),P(T \mid S),P(C \mid S).$$

This fork does make an independence claim — $T \perp C \mid S$ (teeth and cancer are independent once you know whether the person smokes; all their correlation ran through smoking). So it is not Markov equivalent to stories 1 and 2 as written. But here’s the catch: we never observe $S$. Marginalize smoking out — sum it away, the way an observer who only sees teeth and lungs must — and what’s left is a plain dependence between $T$ and $C$:

$$P(T, C) = \sum_{s} P(s),P(T \mid s),P(C \mid s),$$

a distribution in which $T$ and $C$ are correlated — exactly the observable pattern stories 1 and 2 produce. So all three reproduce the same teeth–cancer correlation: stories 1 and 2 because they are Markov equivalent, and story 3 because its hidden common cause manufactures the same marginal dependence. Observation sees only that correlation, and three different mechanisms can paint it.

Intervention as Graph Surgery

What does “doing” mean, formally? Suppose Chibany actually whitens their teeth. They reach in and set the teeth-color variable to “white,” by an act of will — not because of anything upstream. Smoking didn’t decide it; Chibany’s wallet did.

That act severs the teeth variable from its usual cause. In graph terms, intervening on a variable $X$ deletes all the arrows pointing into $X$ — its parents no longer have any say, because you’ve overridden them. Everything else in the graph stays exactly as it was.

graph TD
    subgraph Before["Observe (see)"]
        S1(("smoking")) --> T1(("teeth"))
        S1 --> C1(("cancer"))
    end
    subgraph After["Intervene: do(teeth = white)"]
        S2(("smoking")) --> C2(("cancer"))
        T2["teeth = white<br/>(set by hand)"]
    end
    classDef node fill:none,stroke:#9bbcff,stroke-width:2px,color:#fff
    classDef fixed fill:#cfd6e6,stroke:#9bbcff,stroke-width:2px,color:#111
    class S1,T1,C1,S2,C2 node
    class T2 fixed

On the left, the natural network: smoking causes both teeth and cancer. On the right, after $do(\text{teeth} = \text{white})$: the arrow from smoking into teeth is cut. The intervened variable is drawn as a square rather than a circle, to signal it is set by hand and is no longer a free random variable — it carries no information about smoking. Crucially, the smoking→cancer arrow is untouched. This operation has a name and a notation, Pearl’s do-operator, written $do(X = x)$.

$P(Y \mid X)$ vs. $P(Y \mid do(X))$: the Numbers

Let’s make the smoking/teeth/cancer network concrete and compute both quantities. The structure is the fork (common cause) from above:

Note that teeth and cancer are conditionally independent given smoking — there’s no arrow between them, only a shared parent. Any association between them is entirely the confounder’s doing.

Observation: $P(\text{cancer} \mid \text{yellow teeth})$. Seeing yellow teeth makes smoking more likely (Bayes’ rule: $P(\text{smoking} \mid \text{yellow}) = \frac{0.8 \times 0.3}{0.8 \times 0.3 + 0.2 \times 0.7} \approx 0.63$), and smokers get cancer more often. So:

$$P(\text{cancer} \mid \text{yellow teeth}) \approx 0.63 \times 0.15 + 0.37 \times 0.01 \approx 0.098.$$

Yellow teeth are a real warning sign — about $9.8%$ cancer risk, versus the $5.2%$ base rate.

Intervention: $P(\text{cancer} \mid do(\text{yellow teeth}))$. Now we paint the teeth yellow (or whiten them — same logic). The do-operator cuts the smoking→teeth arrow, so the teeth color no longer tells us anything about smoking. Cancer still depends only on smoking, which is untouched at its base rate. So the answer collapses to the plain marginal:

$$P(\text{cancer} \mid do(\text{yellow teeth})) = P(\text{cancer}) = 0.30 \times 0.15 + 0.70 \times 0.01 = 0.052.$$

Two different answers from the “same” question. Observing yellow teeth: $9.8%$. Setting them yellow: $5.2%$ — exactly the base rate, no change. Whitening Chibany’s teeth would do nothing to their cancer risk, because the teeth were never a cause. The doctor was right.

Pearl’s Ladder of Causation

Judea Pearl organizes these distinctions into three rungs of a ladder of causation, each requiring strictly more than the last:

1. Association — $P(Y \mid X)$. Seeing. “What does observing $X$ tell me about $Y$?” Everything in Chapters 8–9 lived here.
2. Intervention — $P(Y \mid do(X))$. Doing. “What happens to $Y$ if I make $X$ take a value?” This chapter.
3. Counterfactuals — $P(Y_{x} \mid X = x’, Y = y’)$. Imagining. “Given what actually happened, what would have happened if $X$ had been different?”

We stay on rungs 1 and 2. Rung 3 — counterfactuals, the logic of regret and responsibility (“would the patient have survived if they’d been treated?”) — needs still more structure and is a topic for a later course.

A Cognitive-Science Aside: the Blicket Detector

Where do humans sit on this ladder? Strikingly high, and early. In a classic experiment, Gopnik and Sobel gave young children a “blicket detector” — a box that lights up when certain blocks (“blickets”) are placed on it. By watching which blocks made it activate, and crucially by intervening — placing and removing blocks themselves — children as young as three inferred which blocks were blickets, and did so in a way that matches Bayes-net learning with interventions, not mere association. Toddlers, it turns out, are already climbing to rung 2. (Gopnik & Sobel, 2000; the broader program is reviewed in Gopnik et al., 2004.) Causal reasoning isn’t a late, fragile achievement — it looks like a core part of how minds are built.

GenJAX Implementation

The do-operator has a beautifully simple implementation: write two generative functions. The observational one samples every node from its parents as usual; the interventional one drops the intervened node’s parents and treats it as fixed. Comparing Monte Carlo estimates from the two makes the see/do gap concrete.

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import jax
import jax.numpy as jnp
import jax.random as jr
from genjax import gen, flip, ChoiceMap

# Shared CPT numbers for the smoking / teeth / cancer network.
P_SMOKE = 0.3
P_YELLOW_SMOKE, P_YELLOW_NOSMOKE = 0.8, 0.2
P_CANCER_SMOKE, P_CANCER_NOSMOKE = 0.15, 0.01

@gen
def observational():
    """The natural network: smoking causes both teeth and cancer."""
    smoking = flip(P_SMOKE) @ "smoking"
    teeth = flip(jnp.where(smoking, P_YELLOW_SMOKE, P_YELLOW_NOSMOKE)) @ "teeth"
    cancer = flip(jnp.where(smoking, P_CANCER_SMOKE, P_CANCER_NOSMOKE)) @ "cancer"
    return cancer

@gen
def intervened_yellow():
    """do(teeth = yellow): teeth is set by hand, so its arrow from smoking is
    CUT — we simply don't sample it from smoking anymore. Because nothing
    downstream of teeth depends on it (cancer's only parent is smoking), the
    value we'd set teeth to is irrelevant to cancer — so we just leave it out
    and sample the rest of the graph forward. No conditioning needed."""
    smoking = flip(P_SMOKE) @ "smoking"
    # teeth would go here, but the intervention removes it from the graph.
    cancer = flip(jnp.where(smoking, P_CANCER_SMOKE, P_CANCER_NOSMOKE)) @ "cancer"
    return cancer

N = 60000

# OBSERVE yellow teeth: condition the observational model on teeth = yellow.
obs = ChoiceMap.d({"teeth": 1})
keys = jr.split(jr.key(0), N)
def observe_one(k):
    trace, log_weight = observational.generate(k, obs, ())
    return trace.get_choices()["cancer"].astype(float), log_weight
cancer_obs, log_w = jax.vmap(observe_one)(keys)
w = jnp.exp(log_w - jnp.max(log_w)); w = w / jnp.sum(w)
p_see = jnp.sum(cancer_obs * w)

# DO yellow teeth: just sample the intervened model forward (no conditioning).
cancer_do = jax.vmap(lambda k: intervened_yellow.simulate(k, ()).get_retval())(keys)
p_do = jnp.mean(cancer_do.astype(float))

print(f"P(cancer | teeth = yellow)      = {float(p_see):.3f}   (observe — confounded)")
print(f"P(cancer | do(teeth = yellow))  = {float(p_do):.3f}   (intervene — base rate)")

Output:

P(cancer | teeth = yellow)      = 0.098   (observe — confounded)
P(cancer | do(teeth = yellow))  = 0.052   (intervene — base rate)

The code makes the abstract idea tangible: the only difference between the two models is that intervened_yellow deletes the teeth line. That one deletion — cutting teeth off from smoking — is graph surgery, and it changes the answer from $9.8%$ to the $5.2%$ base rate. Whitening does nothing; the doctor was right; and you can see exactly why in five lines of difference between two models.

Exercises

A companion notebook works through these interactively:

📓 Open in Colab: 10_causal_bayes_nets.ipynb

Special thanks to JPPCA for their generous support of this tutorial series.