Janzing 2020 - Inputs as Causes - Interventional Equals Marginal Expectation

Summary

The paper’s positive argument. Treat the algorithm as a causal system whose inputs $X_1,\dots ,X_n$ are the only causes of its output $Y=f(X)$, and keep them formally separate from the true features ${\tilde{X}}_1,\dots ,{\tilde{X}}_n$ of the real-world object. Intervening on $X_T$ then leaves $X_{\bar{T}}$ at its natural joint law, so $\mathbb{E}[Y\mid do(X_T=x_T)]=\mathbb{E}[f(x_T,X_{\bar{T}})]$ (eq. 14). This justifies the marginal value function “from the causal perspective”. The price of the construction is its scope: the causality is that of the “technical input / output system”, and “Y is the output of the system and not a property of the external world.”

Overview

Eq. (5) showed that an intervention on $X_1$ cuts its dependence on the remaining causes of $Y$ while keeping their mutual dependence. This note applies that idea to the trivially known causal graph of a prediction algorithm: inputs → output.

Main Content

The causal reading of an algorithm (§3, p.5)

The authors “consider the inputs of an algorithm as causes of the output”. They emphasise that this is not a claim about the world: “we are not talking about the causal relation between any features in the real world outside the computer (where the attribute predicted by Y may be the cause of the features), but only about causality of this technical input / output system”. Footnote 4 adds: “Accordingly, Y is the output of the system and not a property of the external world.”

Definition: True features vs inputs (§3, p.5, Fig. 2)

• ${\tilde{X}}_1,\dots ,{\tilde{X}}_n$: the true features obtained from an object (a customer, an image, a market-week).
• $X_1,\dots ,X_n$: the inputs plugged into the algorithm.
• $Y=f(X_1,\dots ,X_n)$: the algorithm’s output.

Fig. 2, top (usual learning scenario): ${\tilde{X}}_j\to X_j$ for each $j$, with $X_j={\tilde{X}}_j$, and $X_1,\dots ,X_n\to Y$. The ${\tilde{X}}_j$ may be arbitrarily dependent among themselves. Fig. 2, bottom (hypothetical intervention): the inputs $X_1,X_2$ are cut from ${\tilde{X}}_1,{\tilde{X}}_2$ and set to fixed values $x_1,x_2$. The remaining inputs $X_3,\dots ,X_n$ are still copied from the object’s true features and are therefore sampled from the usual joint distribution $P_{X_3,\dots ,X_n}$.

What is actually drawn (p.6, checked against the rendered figure). The figure uses exactly five true features and five inputs, not general $n$:

• Top panel: a box labelled “object with features” holds five white nodes ${\tilde{X}}_1,\dots ,{\tilde{X}}_5$. Each ${\tilde{X}}_j$ has one arrow to its own black input node $X_j$ (${\tilde{X}}_1\to X_1,\dots ,{\tilde{X}}_5\to X_5$), and all five inputs $X_1,\dots ,X_5$ point to the black output $Y$. No edges are drawn among the ${\tilde{X}}_j$.
• Bottom panel: the same box with ${\tilde{X}}_1,\dots ,{\tilde{X}}_5$. The input nodes for features 1 and 2 are relabelled with the fixed lower-case values $x_1,x_2$ and have no incoming arrows (the edges ${\tilde{X}}_1\to X_1$ and ${\tilde{X}}_2\to X_2$ are deleted). $X_3,X_4,X_5$ keep their arrows from ${\tilde{X}}_3,{\tilde{X}}_4,{\tilde{X}}_5$, and $x_1,x_2,X_3,X_4,X_5$ all still point to $Y$.

Caption (p.6): “Top: Causal structure of our prediction scenario: The output $Y$ is determined by the inputs $X_1,\dots ,X_n$. In the usual learning scenario these inputs coincide with features ${\tilde{X}}_1,\dots ,{\tilde{X}}_n$ ob some object, that is $X_j={\tilde{X}}_j$. Bottom: To evaluate the impact of some inputs, say $X_1,X_2$, for the output $Y$ we consider a hypothetical scenario where we adjust these inputs to some fixed values $x_1,x_2$ and sample the remaining inputs from the usual joint distribution $P_{X_3,\dots ,X_n}$.” (“ob some object” is the original typo for “of some object”.) So the caption speaks of general $n$ while the drawing shows $n=5$.

This separation lets us “talk about a hypothetical scenario where the inputs are changed compared to the true features” (p.5). We can feed the algorithm a TV value that this market-week did not have, without asking what would have happened to the market-week.

Proposition: Interventional = marginal expectation (§3, eq. 14, p.5)

Under the causal structure of Fig. 2, with the inputs $X_1,\dots ,X_n$ the only causes of $Y$ and an intervention that “leaves the remaining components unaffected”, the remaining inputs are “sampled from their natural joint distribution without conditioning”, and

$$\mathbb{E}[Y\mid do(X_T=x_T)]=\mathbb{E}[f(x_T,X_{\bar{T}})].\text{\hspace{1em}(14)}$$

Proof sketch. In the post-intervention graph, $X_T$ has no parents. $X_{\bar{T}}$ keeps its parents ${\tilde{X}}_{\bar{T}}$ and hence its marginal law $P_{X_{\bar{T}}}$, which does not depend on $x_T$ because the link ${\tilde{X}}_T\to X_T$ is cut. $Y$ is a deterministic function of $(X_T,X_{\bar{T}})$. So $\mathbb{E}[Y\mid do(x_T)]=\int f(x_T,x_{\bar{T}})d{P}_{X_{\bar{T}}}(x_{\bar{T}})$. Equivalently, apply the back-door/adjustment formula with $X_{\bar{T}}$ as the adjustment set. It blocks every back-door path from $X_T$ to $Y$, because all of them run $X_T\leftarrow {\tilde{X}}_T\cdots {\tilde{X}}_{\bar{T}}\to X_{\bar{T}}\to Y$. ∎

Corollary: Causal justification of marginal SHAP (§3, pp.5-6)

“Since the interventional expectations coincide with the marginal expectations, we have thus justified the use of marginal expectations for the Shapley values from the causal perspective.” Using $f_T(x)=\mathbb{E}[Y\mid do(X_T=x_T)]$ in eq. (2) yields interventional Shapley values, which coincide with marginal Shapley values (eq. 4).

Minimal causal knowledge. “the fact that the inputs $X_1,\dots ,X_n$ cause the output $Y$ is the only causal knowledge needed to compute (14)“. The separation of $\tilde{X}$ from $X$ “allows us to be agnostic about the causal relations between the true features in the real world” (p.5).

What the argument licenses, and what it does not

Question	Object	Does eq. (14) answer it?
How does the model’s output respond if I set input $X_j$?	$\mathbb{E}[f(X)\mid do(X_j=x_j)]$	Yes. Needs only “inputs cause outputs”.
Does the model use input $j$?	Sensitivity for $f$	Yes. Marginal ${\varphi }_j=0$ whenever $f$ ignores $x_j$ (Lemma 1 contrast).
How would the real outcome respond if the real feature were set?	$\mathbb{E}[\tilde{Y}\mid do({\tilde{X}}_j=x_j)]$	No. That needs a causal model of $\tilde{X}$ and $\tilde{Y}$. $f$ is just a regression fitted to observational data, so its coefficients carry confounding.
Is the model’s input-output response plausible at $(x_T,X_{\bar{T}})$ combinations never seen in training?	Extrapolation of $f$	No. Eq. (14) asks for $f$ at those points but says nothing about whether $f$ is reliable there.

The third row is the crux for applied use. If $\tilde{Y}$ (real sales) is a cause of some ${\tilde{X}}_j$ — the parenthetical in the quotation above, “where the attribute predicted by Y may be the cause of the features” — then the model’s dependence on $X_j$ can be anti-causal, and a large interventional SHAP value is pure prediction, not effect. The fourth row is the objection pressed by Kumar et al 2020 - Problems with Shapley-Value Explanations - Overview and Hooker Mentch Zhou 2021 - Permute-and-Predict Extrapolation - Overview.

Examples

Example: Media-mix regression with a demand confounder (marketing illustration)

Setup. True world: an unobserved demand index $D$ raises both the planned TV budget ${\tilde{X}}_{\text{TV}}$ and real sales $\tilde{Y}$. TV’s real effect is ${\beta }_{\text{TV}}=1$ per unit and $D$‘s effect is $2$. TV is planned as ${\tilde{X}}_{\text{TV}}=D+\epsilon$. A model trained without $D$ learns $f(x)=\hat{\beta }{x}_{\text{TV}}+\dots$ with $\hat{\beta }>1$, because it absorbs part of $D$.

Interventional SHAP for a week with TV $=x$ is $\hat{\beta }(x-\mathbb{E}{X}_{\text{TV}})$. Per eq. (14) this is exactly how the model’s output moves when the TV input is set to $x$, so the attribution is correct for the model.

Reading it as incremental sales would overstate TV’s effect by the factor $\hat{\beta }/{\beta }_{\text{TV}}$. Eq. (14) has nothing to say about that error. It is removed by identification: experiments or geo tests (Experimental Benchmarks for Observational Ad Measurement), controls for $D$ (Logic of Regression Adjustment), or orthogonalised estimators (DML Estimators for ATE and the Interactive Model). The practical cautions are developed in Dillon et al - SHAP and Causal Insights - Overview.

Connections

• Do-operator semantics and graph surgery: Do-Calculus in Summary Causal DAGs, DAGs and Causal Identification, Canonical Causal DAGs.
• Potential-outcomes reading of the world-level estimand: Potential Outcomes Framework.
• Same theme in a different literature: Zhao & Hastie (2019) on when partial dependence plots are causal (cited by the paper, p.4). They need the back-door criterion to hold for the real features.
• For marketing measurement, a model-level interventional attribution belongs with prediction diagnostics, not lift estimates. See Chan Perry 2017 - Challenges in Media Mix Modeling - Overview and Causal Machine Learning - Overview.

See Also

• Janzing 2020 - Conclusions, Limitations and Model-vs-World Attribution — synthesis and critique
• Janzing et al 2020 - Feature Relevance as a Causal Problem - Overview