I think the use of symmetries for causal inference is an important insight in physics. Without constraints, there are an enormous amount of possible causal theories that can explain any given set of observations. But adding symmetries can often pin down the theory uniquely, and in a testable way (because the symmetry would fail for almost all the ways the theory could be false).
It also turns out that there’s an obscure method in the causal inference literature that I think can be analogized to this. If you’ve got two variables A and B, then the causal relationship A→B or B→A can be hard to figure out, just from their distribution P(A,B). However, if you can embed them into a family Ai and Bi which have sufficiently different distributions, but which all share the mechanisms, then likely precisely one of P(Ai|Bi) and P(Bi|Ai) will be constant as a function of i, with the constant one being the one that encodes the direction of causality. I think this is a special-case of the physics method of requiring theories to respect symmetries, where in this case the symmetry group is the group of permutations on i.
I think the use of symmetries for causal inference is an important insight in physics. Without constraints, there are an enormous amount of possible causal theories that can explain any given set of observations. But adding symmetries can often pin down the theory uniquely, and in a testable way (because the symmetry would fail for almost all the ways the theory could be false).
It also turns out that there’s an obscure method in the causal inference literature that I think can be analogized to this. If you’ve got two variables A and B, then the causal relationship A→B or B→A can be hard to figure out, just from their distribution P(A,B). However, if you can embed them into a family Ai and Bi which have sufficiently different distributions, but which all share the mechanisms, then likely precisely one of P(Ai|Bi) and P(Bi|Ai) will be constant as a function of i, with the constant one being the one that encodes the direction of causality. I think this is a special-case of the physics method of requiring theories to respect symmetries, where in this case the symmetry group is the group of permutations on i.