Which practical interventions does the do-operator refer to in causal inference? Illustration on the example of obesity and cancer

For exposures $X$ like obesity, no precise and unambiguous definition exists for the hypothetical intervention $do(X = x_0)$. This has raised concerns about the relevance of causal effects estimated from observational studies for such exposures. Under the framework of structural causal models, we study how the effect of $do(X = x_0)$ relates to the effect of interventions on causes of $X$. We show that for interventions focusing on causes of $X$ that affect the outcome through $X$ only, the effect of $do(X = x_0)$ equals the effect of the considered intervention. On the other hand, for interventions on causes $W$ of $X$ that affect the outcome not only through $X$, we show that the effect of $do(X = x_0)$ only partly captures the effect of the intervention. In particular, under simple causal models (e.g., linear models with no interaction), the effect of $do(X = x_0)$ can be seen as an indirect effect of the intervention on $W$.