Model Reliance, Causal Effects and Conditional Model Reliance

Summary

MR studies how a model’s loss changes under an intervention on the data. Proposition 19: for a binary treatment $T=X_1$ under ignorability and positivity, the MR of the true regression function $f_0=E(Y\mid T,C)$ equals $1+\frac{\mathrm{Var}(T)}{\mathbb{E}\mathrm{Var}(Y\mid T,C)}{\sum }_t\{\mathbb{E}(Y_1-Y_0\mid T=t{)}^2+\mathrm{Var}(\mathrm{CATE}(C)\mid T=t)\}$. So reliance grows with treatment prevalence, average effect size and effect heterogeneity, and falls with residual noise. Conditional (“core”) model reliance $CMR=e_{\text{cond}}/e_{\text{orig}}$ permutes $X_1$ only among observations with the same $X_2$. It measures reliance on information unique to $X_1$ and avoids evaluating impossible $(x_1,x_2)$ combinations. Estimators: importance weighting, exact matching on discrete $X_2$, or residual imputation $X_1-{\mu }_1(X_2)$.

Overview

Section 8 frames MR causally (p. 30). The goal mirrors causal inference, with the model’s performance as the outcome and permuting $X_1$ as the intervention. Two results follow. When the model is nature’s own conditional-mean function, MR reduces to familiar causal quantities (§8.1). And an alternative perturbation that respects the dependence between $X_1$ and $X_2$ gives conditional importance (§8.2).

Main Content

MR and conditional average treatment effects (§8.1)

Rename $(Y,X_1,X_2)$ as $(Y,T,C)$: binary treatment $T$, baseline covariates $C$. Let $f_0(t,c):=E(Y\mid C=c,T=t)$, with no restriction on its form. Let $Y_1$ and $Y_0$ be potential outcomes, $Y=Y_0(1-T)+Y_{1}T$, $\mathrm{CATE}(c):=E(Y_1-Y_0\mid C=c)$ (Potential Outcomes Framework).

Proposition 19: Causal interpretation of MR (p. 31, Eq 8.1)

Let $e_{\text{orig}}$ and $e_{\text{switch}}$ use squared-error loss $L(f,(y,t,c))=(y-f(t,c){)}^2$. If conditional ignorability $(Y_1,Y_0)\perp T\mid C$ and positivity $0<P(T=1\mid C=c)<1$ for all $c$ hold, then

$$MR(f_0)=1+\frac{\mathrm{Var}(T)}{{\mathbb{E}}_{T,C}\mathrm{Var}(Y\mid T,C)}\sum _{t\in \{0,1\}}\{\mathbb{E}(Y_1-Y_0\mid T=t{)}^2+\mathrm{Var}(\mathrm{CATE}(C)\mid T=t)\},(8.1)$$

where $\mathrm{Var}(T)$ is the marginal variance of treatment assignment.

Reading.

• If everyone is treated, $\mathrm{Var}(T)=0$ and $MR(f_0)=1$: scrambling $T$ changes nothing.
• A larger average effect magnitude $\mathbb{E}(Y_1-Y_0\mid T=t{)}^2$ means more reliance.
• More heterogeneity across subgroups, $\mathrm{Var}(\mathrm{CATE}(C)\mid T=t)$, also means more reliance. “A treatment may be important for predicting $Y$ even if the average treatment effect is zero, so long as the treatment helps some subgroups more than others” (p. 31).
• More unexplained outcome noise, $\mathbb{E}\mathrm{Var}(Y\mid T,C)$, means less reliance.

Proof of Proposition 19 (App B.9, pp. 65-67)

Let $p=P(T=1)$ and $q=1-p$. Ignorability gives $f_0(t,c)=E(Y_t\mid C=c)$, so $e_{\text{orig}}(f_0)=q{\mathbb{E}}_{C\mid T=0}\mathrm{Var}(Y_0\mid C)+p{\mathbb{E}}_{C\mid T=1}\mathrm{Var}(Y_1\mid C)$ (B.50). For $e_{\text{switch}}$, independence of $T^{(a)}$ and $T^{(b)}$ gives four terms $A_{ij}=p^{i+j}{q}^{2-i-j}{\mathbb{E}}_{C\mid T=i}\mathbb{E}[\{Y_i-E(Y_j\mid C){\}}^2]$:

• $A_{00}=q^2{\mathbb{E}}_{C\mid T=0}\mathrm{Var}(Y_0\mid C)$ and $A_{11}=p^2{\mathbb{E}}_{C\mid T=1}\mathrm{Var}(Y_1\mid C)$;
• $A_{01}=pq{\mathbb{E}}_{C\mid T=0}[\mathrm{Var}(Y_0\mid C)+\mathrm{CATE}(C{)}^2]$ and $A_{10}=pq{\mathbb{E}}_{C\mid T=1}[\mathrm{Var}(Y_1\mid C)+\mathrm{CATE}(C{)}^2]$.

Summing, and using $p+q=1$, gives $e_{\text{switch}}(f_0)=e_{\text{orig}}(f_0)+\mathrm{Var}(T)\{{\mathbb{E}}_{C\mid T=0}{\mathrm{CATE}}^2+{\mathbb{E}}_{C\mid T=1}{\mathrm{CATE}}^2\}$ (B.52-B.54), with $pq=\mathrm{Var}(T)$. Then $\mathbb{E}[{\mathrm{CATE}}^2\mid T=t]=\mathbb{E}(Y_1-Y_0\mid T=t{)}^2+\mathrm{Var}(\mathrm{CATE}(C)\mid T=t)$. Divide by $e_{\text{orig}}(f_0)={\mathbb{E}}_{T,C}\mathrm{Var}(Y\mid T,C)$.

Combining with the bounds. Corollary 22 bounds $MR(f^{\star })$. When $f^{\star }=E(Y\mid X)$, Proposition 19 turns that bound into a statement about effect size and heterogeneity (App A.4).

Conditional (core) model reliance (§8.2)

Motivation (p. 31). When $X_1$ and $X_2$ are highly correlated or redundant, good models can rely on either, and MCR correctly reports a wide range. Often, though, the question is how much a model relies on information in $X_1$ “that cannot alternatively be gleaned from $X_2$“. The paper’s example: age and accumulated wealth are correlated and both predict future promotion. How much does a promotion model rely on information available only from wealth?

Definition: Conditional switched loss, CMR and CMCR (Eq 8.2, p. 32)

With $Z^{(a)},Z^{(b)}$ independent copies of $Z$:

$$e_{\text{cond}}(f):={\mathbb{E}}_{X_2}\left[{\mathbb{E}}_{(Y^{(b)},X_1^{(a)},X_2^{(b)})}\right[L\{f,(Y^{(b)},X_1^{(a)},X_2^{(b)})\}|X_2^{(a)}=X_2^{(b)}=X_2]],(8.2)$$

the expected loss over pairs whose $X_1$ values are switched given that the pair matches on $X_2$. Equivalently, noise is added to $X_1$ so that it carries no information about $Y$ given $X_2$, while the joint distribution of $(X_1,X_2)$ is preserved.

$$CMR(f):=\frac{e_{\text{cond}}(f)}{e_{\text{orig}}(f)}.$$

Conditional MCR (CMCR) is defined like Eq 2.2 with CMR in place of MR.

• If $X_1\perp X_2$, then CMR = MR.
• Otherwise CMR falls as $X_2$ becomes more predictive of $X_1$, and CMCR is generally closer to 1 than MCR.

Advantage (p. 32). CMR keeps perturbed inputs inside the support $\mathcal{X}$ rather than the product ${\mathcal{X}}_1\times {\mathcal{X}}_2$. So it is not influenced by impossible combinations of $x_1$ and $x_2$, while MR can be. This echoes Hooker (2007): evaluate models where the data actually live, which is the theme of Hooker Mentch Zhou 2021 - Permute-and-Predict Extrapolation - Overview. Challenge: exact matched pairs may be rare, so CMR is hard to estimate nonparametrically.

Estimating CMR (§8.2.1)

Estimators of $e_{\text{cond}}$ (§8.2.1, pp. 32-33; Eq 8.3)

Weighting (discrete, low-dimensional covariates). Use the importance weight $w(x_1,x_2):=P(X_1=x_1\mid X_2=x_2)/P(X_1=x_1)$ (see also Hooker 2007):

$${\hat{e}}_{\text{weight}}(f):=\frac{1}{n(n-1)}\sum _i\sum _{j\ne i}w({\mathbf{X}}_{1[i,\cdot ]},{\mathbf{X}}_{2[j,\cdot ]})L\{f,({\mathbf{y}}_{[j]},{\mathbf{X}}_{1[i,\cdot ]},{\mathbf{X}}_{2[j,\cdot ]})\}.$$

Unlikely combinations are down-weighted and impossible ones are discarded. It is unbiased if the probabilities are known.

Matching (discrete, low-dimensional $X_2$). This needs no model of $X_1\mid X_2$:

$${\hat{e}}_{\text{match}}(f):=\frac{1}{n(n-1)}\sum _i\sum _{j\ne i}\frac{1({\mathbf{X}}_{2[j,\cdot ]}={\mathbf{X}}_{2[i,\cdot ]})}{P(X_2={\mathbf{X}}_{2[i,\cdot ]})}L\{f,({\mathbf{y}}_{[j]},{\mathbf{X}}_{1[i,\cdot ]},{\mathbf{X}}_{2[j,\cdot ]})\}.(8.3)$$

It is unbiased if $P(X_2=x_2{)}^{-1}$ is known. The inverse-probability weight is needed because the share of matched pairs at a value $x_2$ differs from the share of observations there; summing matched pairs naively is biased. In practice use $P(X_2={\mathbf{X}}_{2[i,\cdot ]})\approx \frac{1}{n-1}{\sum }_{j^{\prime }\ne i}1({\mathbf{X}}_{2[i,\cdot ]}={\mathbf{X}}_{2[j^{\prime },\cdot ]})$, with minor adjustments to avoid dividing by zero. This is analogous to exact matching in causal inference (Stuart 2010), which works best when exact matches are common.

Imputation under homogeneous residuals (continuous or high-dimensional covariates). Let ${\mu }_1(x_2)=E(X_1\mid X_2=x_2)$ and assume the residual $X_1-{\mu }_1(X_2)$ is independent of $X_2$. Then

$$e_{\text{cond}}(f)=\mathbb{E}L\{f,(Y^{(b)},X_1^{(a)}-{\mu }_1(X_2^{(a)})+{\mu }_1(X_2^{(b)}),X_2^{(b)})\},$$

i.e. swap the residuals between random pairs. With ${\mu }_1$ known, the U-statistic is

$${\hat{e}}_{\text{impute}}(f):=\frac{1}{n(n-1)}\sum _i\sum _{j\ne i}L\{f,({\mathbf{y}}_{[j]},{\mathbf{X}}_{1[i,\cdot ]}-{\mu }_1({\mathbf{X}}_{2[i,\cdot ]})+{\mu }_1({\mathbf{X}}_{2[j,\cdot ]}),{\mathbf{X}}_{2[j,\cdot ]})\}.$$

In practice, plug in a regression or ML estimate of ${\mu }_1$. No explicit model of the joint distribution of $(X_1,X_2)$ is needed.

Unbiasedness of ${\hat{e}}_{\text{match}}$ and ${\hat{e}}_{\text{weight}}$ (App A.7, pp. 47-48)

Matching. Each summand has the same expectation. Write $p_{x_2}(x)=P(X_2=x)$ and condition on $X_2^{(a)}$:

$$\mathbb{E}[\frac{1(X_2^{(a)}=X_2^{(b)})}{p_{x_2}(X_2^{(a)})}L\{f,(Y^{(b)},X_1^{(a)},X_2^{(b)})\}]={\mathbb{E}}_{X_2^{(a)}}\{p_{x_2}(X_2^{(a)})\frac{1}{p_{x_2}(X_2^{(a)})}\mathbb{E}[L\mid X_2^{(b)}=X_2^{(a)},X_2^{(a)}]+0\}=e_{\text{cond}}(f).$$

Weighting (shown for discrete variables). Summing over $x_1^{(a)},x_2^{(b)},y^{(b)}$, the weight converts $P(X_1=x_1^{(a)})$ into $P(X_1=x_1^{(a)}\mid X_2=x_2^{(b)})$. The expectation then becomes ${\mathbb{E}}_{X_2^{(b)}}\mathbb{E}[L\mid X_2^{(a)}=X_2^{(b)},X_2^{(b)}]=e_{\text{cond}}(f)$.

Related approaches (pp. 33-34).

• Strobl et al. (2008) give a similar conditional permutation procedure for trees. A formal comparison is hard because they “do not define a specific estimand” and their method is tree-specific (Strobl et al 2008 - Conditional Variable Importance - Overview).
• Reduce the covariates to induce approximate independence before computing an MR-like measure (Gregorutti et al. 2017).
• Regroup predictors of $X_1$ into $X_1$ itself (Toloşi & Lengauer 2011; the Kirk, Lewin & Stumpf discussion of Meinshausen & Bühlmann 2010, see Meinshausen Buhlmann 2010 - Stability Selection - Overview).
• Extension: condition only on the part of $X_2$ believed to cause $X_1$, by changing the outer expectation in (8.2). The paper otherwise focuses on MR and notes that similar results “could potentially be carried over” to CMR.

Examples

Prop 19 with numbers (illustrative)

Take a 50/50 randomized promotion ($\mathrm{Var}(T)=0.25$) with residual variance $\mathbb{E}\mathrm{Var}(Y\mid T,C)=4$. Suppose the effect is $+1$ in half the customers and $-1$ in the other half, so ATE $=0$ and $\mathrm{Var}(\mathrm{CATE})=1$ in each arm. Then $MR(f_0)=1+\frac{0.25}{4}(0+1+0+1)=1.125$. The treatment matters for prediction (a 12.5% loss increase when scrambled) despite a zero average effect. A homogeneous effect of $+1$ gives the same 1.125: magnitude and heterogeneity enter symmetrically.

CMR for correlated media (marketing analogue)

TV and branded-search spend are strongly correlated across weeks. MR for search permutes it independently of TV, creating impossible weeks with heavy search and no TV, and can overstate reliance. CMR via ${\hat{e}}_{\text{impute}}$ regresses search on TV and other drivers, then swaps residuals across weeks. That isolates reliance on the part of search not predictable from TV, which is closer to the incremental question a budget planner asks. This is still predictive reliance, not causal incrementality; see Dillon et al - SHAP and Causal Insights - Overview.

Connections

• Causal vs predictive relevance. Prop 19 holds only for $f_0$ under ignorability. For an arbitrary fitted $f$, MR is a property of the model, not of nature. Compare the interventional/observational distinction in Janzing et al 2020 - Feature Relevance as a Causal Problem - Overview.
• Heterogeneous effects. The $\mathrm{Var}(\mathrm{CATE})$ term ties MR to the CATE literature: Metalearners for CATE.
• Matching and weighting estimators of CMR borrow directly from causal-inference adjustment: Matching Methods and Distance Measures, Common Support and Overlap.

See Also

• Strobl et al 2008 - Conditional Variable Importance - Overview: tree-based conditional permutation
• Hooker Mentch Zhou 2021 - Permute-and-Predict Extrapolation - Overview: why unconditional permutation extrapolates
• Fisher Rudin Dominici 2019 - Model Class Reliance - Overview