On the Exchangeability Property in Causal Models

Adamo, Vincenzo

doi:10.21608/cjmss.2024.251550.1030

On the Exchangeability Property in Causal Models

Document Type : Original Article

Author

Vincenzo Adamo

Italian Revenue Agency, 00147 Roma, Italy

10.21608/cjmss.2024.251550.1030

Abstract

Exchangeability is one of the most important concepts in Bayesian probability theory [7], as well as in causal analysis, particularly within the theory based on the potential outcomes (see [18, 21, 23, 15]). In this paper, we propose a way to make explicit the link between the two concepts. We show they are almost coincident with the exchangeability property introduced by de Finetti [3], without making use of notions such as partial, conditional, or hierarchical exchangeability. To do this, we will start from the exchangeability property described in Greenland et al. [14], and assuming the use of a recursive linear Gaussian structural equation model, we will show how it is possible to exploit the properties of de Finetti's representation theorem, without performing any computation, to obtain an estimate of the average causal effect by calibrating a simple linear regression. This is achieved by showing the role of a specific subset of the latent variables in the data-generating process for the variable Y|X=x, linking the exchangeability property required for the identification of the causal coefficient, with the non-correlation between regressors and error term in linear regression, needed to obtain an unbiased coefficient estimation. The results here proposed are not restricted to the Gaussian family of random variables distributions.

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