Pearl 2014 - Appendix Derivations of Transport Formulas

Summary

The Appendix derives, line by line, the transport formulas (5.9) for Fig. 6(d) and (5.10) for Fig. 7, justifying each step by a condition of Theorem 3 or a do-calculus rule. The pattern is always: condition on the S-admissible set, peel $do$-terms off $S$ using Rule 1 or 3, and finally replace $S$-conditioned terms with starred (target) distributions.

Overview

Both derivations start from $P^{\ast }(y\mid do(x))=P(y\mid do(x),s)$ (for Fig. 7 with two selection variables, $s$ and $s^{\prime }$). Read the result as a recipe for what to measure where: factors containing a $do$ are estimated in the source experiment; factors containing only observed variables and the $S$-node become starred factors estimated from the target’s observational data.

The PDF text of the appendix is two-column and garbled in places; the justification annotations below follow the paper’s printed order. The exact graph structure of Figs. 6(d) and 7 is drawn in the original; the roles of the variables below are inferred from the final formulas (so treat the role descriptions as a reading aid, not as a transcription of the figure).

Main Content

Derivation of (5.9), Fig. 6(d)

Variables: $X$ treatment, $Y$ outcome, $Z$ an S-admissible set, $W$ a set that separates $X$ from $Z$. Selection node $S$.

$$\begin{array}{rlrl}{P}^{\ast }(y\mid do(x))& =P(y\mid do(x),s)& & \\ & =\sum _{z}P(y\mid do(x),s,z)P(z\mid do(x),s)& & \\ & =\sum _{z}P(y\mid do(x),z)P(z\mid do(x),s)& & \text{(Thm 3 condition 2: S-admissibility of }Z\text{ for the effect of }X\text{ on }Y)\\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid do(x),w,s)P(w\mid do(x),s)& & \\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid w,s)P(w\mid do(x),s)& & \text{(Thm 3 condition 3: }(X\perp Z\mid W,S)\text{ in the }W\text{-modified graph)}(A.1)\\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid w,s)P(w\mid do(x))& & \text{(Thm 3 condition 2: empty set S-admissible for }X\to W)\\ & =\sum _{z}P(y\mid do(x),z)\sum _w{P}^{\ast }(z\mid w)P(w\mid do(x)).& & \end{array}$$

Resulting formula (5.9)

$$P^{\ast }(y\mid do(x))=\sum _{z}P(y\mid do(x),z)\sum _{w}P(w\mid do(x))P^{\ast }(z\mid w)$$

Estimable pieces: $P(y\mid do(x),z)$ and $P(w\mid do(x))$ from the experiment on $\Pi$; $P^{\ast }(z\mid w)$ from observations on ${\Pi }^{\ast }$.

Derivation of (5.10), Fig. 7 (two selection nodes $S,S^{\prime }$)

Reading from the final formula: $Z$ is S-admissible, $W$ separates $X$ from $Z$, and $T$ is a set that is S’-affected and S-admissible for the effect on $W$.

$$\begin{array}{rlrl}{P}^{\ast }(y\mid do(x))& =P(y\mid do(x),s,s^{\prime })& & \\ & =\sum _{z}P(y\mid do(x),s,s^{\prime },z)P(z\mid do(x),s,s^{\prime })& & \\ & =\sum _{z}P(y\mid do(x),z)P(z\mid do(x),s,s^{\prime })& & \text{(Thm 3 cond. 2, S-admissibility of }Z\text{)}\\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid do(x),s,s^{\prime },w)P(w\mid do(x),s,s^{\prime })& & \\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid s,s^{\prime },w)P(w\mid do(x),s,s^{\prime })& & \text{(Thm 3 cond. 3, }(X\perp Z\mid W,S,S^{\prime })\text{)}(A.2)\\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid s,s^{\prime },w)\sum _{t}P(w\mid do(x),s,s^{\prime },t)P(t\mid do(x),s,s^{\prime })& & \\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid s,s^{\prime },w)\sum _{t}P(w\mid do(x),t)P(t\mid do(x),s,s^{\prime })& & \text{(Thm 3 cond. 2, S-admissibility of }T\text{ for }X\to W\text{)}\\ & =\sum _{z}P(y\mid do(x),z)\sum _{w}P(z\mid s,s^{\prime },w)\sum _{t}P(w\mid do(x),t)P(t\mid s,s^{\prime })& & \text{(Thm 3 cond. 1 / Rule 3, }(X\perp T\mid S,S^{\prime })\text{)}\\ & =\sum _{z}P(y\mid do(x),z)\sum _w{P}^{\ast }(z\mid w)\sum _{t}P(w\mid do(x),t)P^{\ast }(t).& & \end{array}$$

Resulting formula (5.10)

$$P^{\ast }(y\mid do(x))=\sum _{z}P(y\mid do(x),z)\sum _w{P}^{\ast }(z\mid w)\sum _{t}P(w\mid do(x),t)P^{\ast }(t)$$

Measurement plan stated by the authors: variables $U$ and $V$ (present in Fig. 7) need not be measured; $W$-specific causal effects need not be estimated in the experiment; only $P^{\ast }(z\mid w)$ and $P^{\ast }(t)$ are needed from the target.

Reading the derivation pattern

Step type	Tool	Effect on $S$
Condition on $Z$ (or $W$, $T$)	law of total probability	none
Drop $s$ from a $do$-term	Rule 1 + S-admissibility	removes $S$ from the experimental factor so it can come from $\Pi$
Drop $do(x)$ from a non-$do$ term	Rule 3 or "$X\perp Z\mid W$"	converts the factor to observational form: $S$ stays as conditioning
Final replacement	definition of the $S$-node ($P(\cdot \mid s^{\ast })=P^{\ast }(\cdot )$)	the factor becomes a target distribution

This is exactly Theorem 1’s criterion: $S$ ends up only as a conditioning variable in $do$-free terms.

Python: Monte Carlo check of (5.9) on a toy chain

Model: latent $U\to X$, $U\to Y$; $X\to W\to Z\to Y$; $S\to Z$, so only the mechanism $W\to Z$ differs in the target. Experimental pieces $P(y\mid do(x),z)$ and $P(w\mid do(x))$ are estimated from simulated source experiments; $P^{\ast }(z\mid w)$ from simulated target observational data; the answer is compared with the target ground truth (simulating $do(x)$ in the target) and with naively reusing the source effect.

import numpy as np
rng = np.random.default_rng(0)
N = 400_000
def draw(n, tgt, do_x=None):
    u = rng.random(n) < .5                                  # latent confounder of X and W,Y? (here U->X, U->Y)
    x = (rng.random(n) < np.where(u, .7, .3)).astype(int) if do_x is None else np.full(n, do_x)
    w = (rng.random(n) < .2 + .5*x).astype(int)              # X -> W (shared mechanism)
    pz = (.5 + .3*w) if tgt else (.1 + .2*w)                 # W -> Z ; S -> Z changes this mechanism
    z = (rng.random(n) < pz).astype(int)
    y = (rng.random(n) < .2 + .5*z + .1*u).astype(int)       # Z -> Y (+ confounding by U), shared
    return x, w, z, y
# --- source experiment: estimate P(y|do(x),z) and P(w|do(x))
est_y = {}; est_w = {}
for x in (0, 1):
    _, w, z, y = draw(N, tgt=False, do_x=x)
    est_w[x] = w.mean()
    for zz in (0, 1): est_y[(x, zz)] = y[z == zz].mean()
# --- target observational data: estimate P*(z|w)
x, w, z, y = draw(N, tgt=True)
est_z = {ww: z[w == ww].mean() for ww in (0, 1)}             # P*(Z=1|w)
def eq59(x):
    tot = 0
    for zz in (0, 1):
        for ww in (0, 1):
            pw = est_w[x] if ww else 1-est_w[x]
            pz = est_z[ww] if zz else 1-est_z[ww]
            tot += est_y[(x, zz)] * pz * pw
    return tot
# --- target ground truth by simulating do(x) in the target
truth = {x: draw(N, tgt=True, do_x=x)[3].mean() for x in (0, 1)}
naive = {x: draw(N, tgt=False, do_x=x)[3].mean() for x in (0, 1)}
print("effect truth  :", round(truth[1]-truth[0], 3))
print("effect Eq 5.9 :", round(eq59(1)-eq59(0), 3))
print("effect naive  :", round(naive[1]-naive[0], 3))

Output (seed 0, $N=400,000$): truth $0.075$, Eq. 5.9 $0.074$, naive source effect $0.051$. The formula recovers the target effect from source experiments plus target observations; carrying the source effect over understates it.

Examples

Measurement plan in a marketing rollout

Suppose a retailer ran a geo test on paid social ($X$) with intermediate metric $W$ = site visits and $Z$ = add-to-cart rate, outcome $Y$ = purchases. If the new market differs only in how visits convert to add-to-cart (the $S\to Z$ story of Fig. 6d), (5.9) says: take $P(w\mid do(x))$ and $P(y\mid do(x),z)$ from the test, but measure $P^{\ast }(z\mid w)$ from the new market’s observational funnel data. No new experiment needed. See Pearl 2014 - Marketing Application Transporting Geo and Lift Results.

Connections

• Proof patterns reused in the multi-source derivations of Bareinboim 2016 - Transportability and Data Fusion.
• Conceptually the same decomposition as mediation formulas in Instrumental Variables and Principal Stratification (post-treatment variables).
• Rules used: Pearl 2014 - Threats vs Licenses and SCM Preliminaries (Rules 1-3); criterion: Pearl 2014 - S-Admissibility and the Graphical Criterion.

See Also

• Pearl Bareinboim 2014 - External Validity and Transportability - Overview