Abstract
Recent progress in the field of machine learning suggests that the joint distribution of two variables X, Y sometimes contains information about the underlying causal structure, e.g., whether X is the case of Y or Y the cause of X, given that exactly one of the alternatives is true. To provide an idea about these statistical asymmetries I show some intuitive examples, both hypothetical toy scenarios as well as scatter plots from real world data. I sketch some recent approaches to infer the causal direction based on these asymmetries and give some pointers to physics literature that relate them to the thermodynamic arrow of time.
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Notes
- 1.
If X and Y refer to quantum measurements at two particles coming from a common source, there may not be such a variable Z that screens off the dependences in the sense of (1) due to quantum entanglement and the violation of Bell’s inequality. Nevertheless, one may visualize the scenario as causal relation of type (b) and replace Z with some joint quantum state | ψ〉 to indicate that the common cause is no longer a random variable.
- 2.
This is different, however, in quantum theory: a joint operator describing two systems that are correlated by a common cause is positive, while an operator describing correlations of cause and effect has positive partial transpose [4]. Therefore, one can sometimes tell apart cases (a) and (c) versus case (b) in Fig. 1, see [5].
- 3.
Here we have implicitly assumed that the joint probability distribution has a joint density to simplify explanations.
- 4.
Note that causal faithfulness has also been discussed in the context of Bell’s inequality: Wood and Spekkens [8] argue that classical explanations of the EPR-scenario (with superluminal communication, for instance) require similar fine-tuning of parameters.
- 5.
Note that vanishing correlation is not enough because linear least square regression automatically yields uncorrelated residuals. Instead, one needs a statistical dependence test that is able to detect higher order dependences.
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Janzing, D. (2017). Statistical Asymmetries Between Cause and Effect. In: Renner, R., Stupar, S. (eds) Time in Physics. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-68655-4_8
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