# Invariance of causal prediction

I am reading Causal inference by using invariant prediction: identification and confidence intervals by Jonas Peters (link to the resource is here: https://rss.onlinelibrary.wiley.com/doi/full/10.1111/rssb.12167)

In part 2: Assumed invariance of causal prediction

Why and how can we reject that the invariance assumption holds for the empty set S = ∅?

Assumption 1.(invariant prediction). There is a vector of coefficients $\gamma^* = {({\gamma _1}^*,...,{\gamma _p}^*)^T}$ with support ${S^*} = \{ k:y_k^* \ne 0\} \subseteq \{ 1,...,p\}$ that satisfies, for all $e \in \varepsilon, X^e$ has an arbitrary distribution and $Y^e=\mu+X^e\gamma^* +\varepsilon^e, \varepsilon^e~F_e$ and $\varepsilon^e \!\perp\!\!\!\perp X_{S^*}^e$

"Take the example of Fig. 2(c). The variance of the activity of gene YMR321C is clearly higher for interventional than observational data, so we can reject that the invariance assumption holds for the empty set S = ∅. However, if conditioning on the activity X of gene YPL273W, the conditional distribution of the activity Y of gene YMR321C is not signiﬁcantly different between interventional and observational data, so the set S={YPL273W} fulfills the invariance assumption (3), at least approximately.)" • That's a seriously awesome paper! – usεr11852 May 29 '18 at 22:28
• Yes, could you please help me with my problem? – user2842390 May 30 '18 at 11:21
• I am still in part 1! :) – usεr11852 May 30 '18 at 18:18

## 1 Answer

How: If the distribution of Y varies with the environment of the observation, then you can reject the invariance assumption for the empty set. As the caption to Fig 2 states, in Figure 2c, the distribution of Y (=YMR321C) has larger variance in the interventional environment than the observational environment, so the empty set is not invariant. Formally this could be determined with a statistical test comparing the variances of observational and experimental environments.

Why: In Figure 2c, clearly some of the interventions are affecting Y (leading to the larger variance), therefore the empty set (no variables are causally effecting Y) is not plausible.