I think of “gears-level model” and “causal DAG” as usually synonymous. There are some arguable exceptions—e.g. some non-DAG markov models are arguably gears-level—but DAGs are the typical use case.
The obvious objection to this idea is “what about feedback loops?”, and the answer is “it’s still a causal DAG when you expand over time”—and that’s exactly what gears-level understanding of a feedback loop requires. Same with undirected markov models: they typically arise from DAG models with some of the nodes unobserved; a gears-level model hypothesizes what those hidden factors are. The hospital example includes both of these: a feedback loop, with some nodes unobserved. But if you expand out the actual gears-level model, distinguishing between different people with different diseases at different times, then it all looks DAG-shaped; the observed data just doesn’t include most of those nodes.
This generalizes: the physical world is always DAG-shaped, on a fundamental level. Everything else is an abstraction on top of that, and it can always be grounded in DAGs if needed.
Instead of stopping at: “It has to do with gears.” keep going to get more specific, find subsets of things with gears: “gear AND oval-shape AND a sprocket is missing AND there is a cardan shaft AND …” But if indeed only things with gears are affected do not expand with “gears AND needs oil” because that already follows from gears.
The advantage of using causal DAGs for our model, even when most of the nodes are not observed, is that it tells us which things need to be included in the AND-clauses and which do not. For instance, “gear AND oval-shaped” vs “gear AND needs oil”—the idea that the second can be ignored “because that already follows from gears” is a fact which derives from DAG structure. For a large model, there’s an exponential number of logical clauses which we could form; a DAG gives formal rules for which clauses are relevant to our analysis.
I think of “gears-level model” and “causal DAG” as usually synonymous. There are some arguable exceptions—e.g. some non-DAG markov models are arguably gears-level—but DAGs are the typical use case.
The obvious objection to this idea is “what about feedback loops?”, and the answer is “it’s still a causal DAG when you expand over time”—and that’s exactly what gears-level understanding of a feedback loop requires. Same with undirected markov models: they typically arise from DAG models with some of the nodes unobserved; a gears-level model hypothesizes what those hidden factors are. The hospital example includes both of these: a feedback loop, with some nodes unobserved. But if you expand out the actual gears-level model, distinguishing between different people with different diseases at different times, then it all looks DAG-shaped; the observed data just doesn’t include most of those nodes.
This generalizes: the physical world is always DAG-shaped, on a fundamental level. Everything else is an abstraction on top of that, and it can always be grounded in DAGs if needed.
The advantage of using causal DAGs for our model, even when most of the nodes are not observed, is that it tells us which things need to be included in the AND-clauses and which do not. For instance, “gear AND oval-shaped” vs “gear AND needs oil”—the idea that the second can be ignored “because that already follows from gears” is a fact which derives from DAG structure. For a large model, there’s an exponential number of logical clauses which we could form; a DAG gives formal rules for which clauses are relevant to our analysis.