# Measuring Causal Invariance Formally

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## Abstract

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## 1. Introduction

Minimal Criterion: C is a cause of E if there is at least one ideal intervention on C that changes the value of E

## 2. Mutual Causal Information and Range of Influence

## 3. Measuring Invariance

## 4. Example

## 5. Invariance and Non-Nominal Variables

## 6. Conclusions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Causal diagrams between C and E with different degrees of invariance. In each diagram, each value of C (e.g., ${c}_{1}$, ${c}_{2}$) has the same probability. If there are four possible values of C, then each value has a probability of $\frac{1}{4}$. Furthermore, when more than one arrow leaves from a particular value of C, each arrow has the same probability conditioned on the value of C it leaves from. For instance two arrows leaving a particular value of C implies a conditional probability of $\frac{1}{2}$ for each arrow. In a context where there are six possible values of C, the overall probability of an arrow leaving a value of C from which overall two arrows are leaving is thus $\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}$.

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Bourrat, P. Measuring Causal Invariance Formally. *Entropy* **2021**, *23*, 690.
https://doi.org/10.3390/e23060690

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Bourrat P. Measuring Causal Invariance Formally. *Entropy*. 2021; 23(6):690.
https://doi.org/10.3390/e23060690

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Bourrat, Pierrick. 2021. "Measuring Causal Invariance Formally" *Entropy* 23, no. 6: 690.
https://doi.org/10.3390/e23060690