Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0003/ efr-0003 /efr-0003/ Structural Causal Model Definition Let be a (finite) directed acyclic graph (DAG). Then a structural causal model (SCM) on consists of For each vertex , measurable spaces A family of independent random variables ---these are called the exogenous variables For each , a measurable function (note that if has no parents, the product is just a singleton) References Context Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0002/ efr-0002 /efr-0002/ Bidirectionality in graphical models Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0003/ efr-0003 /efr-0003/ Structural Causal Model Definition Let be a (finite) directed acyclic graph (DAG). Then a structural causal model (SCM) on consists of For each vertex , measurable spaces A family of independent random variables ---these are called the exogenous variables For each , a measurable function (note that if has no parents, the product is just a singleton) There is a significant existing literature (see Abstracting causal models, Reference , Reference . For a survey see Reference ) studying transformations between structural causal models. These have generally been called something like abstractions, the typical examples being the relationship between a high-level and a low-level model of the same system. However, the precise properties that we should ask for in such a transformation have turned out to be somewhat subtle. The most obvious condition to impose is to ask, for any possible variable we could intervene on, say , and any other variable (or collection of variables) , that the diagram of kernels commutes---in other words, given an intervention in the low-level model, the two possible distributions in the high-level model agree. The main problem with this condition is that it is simply too strict. We generally can't ask that every low-level intervention is well-modeled by the corresponding high-level intervention. Rather, each high-level intervention corresponds to some distribution of the corresponding low-level variables, and we must ask that a diagram of this form: commutes. For example, if the low-level variables are the velocities of all the gas molecules in a container, and the high-level variables are thermodynamic quantities like temperature and pressure, we can't expect that every set of velocities corresponding to a given temperature will lead to a given distribution of outcomes---rather, intervening on the temperature leads to some particular distribution of possible sets of velocities, and this leads to the observed distribution of outcomes. The key here is that control information flows in the opposite direction from measurement information. This phenomenon repeats in many examples. Backlinks Related Contributions