This is really useful information because it can help you identify what information is truly relevant for the estimation of certain parameters (so sufficient statistics) or help you crystallize your understanding of the implications of the model you’ve created. In other words, it helps show you the ways in which your model says different aspects of your data should influence others.
This creates testable implications of the model. If your model says that two variables should be conditionally independent given a third, but they’re not, you have an avenue for refinement. You can also clearly identify your assumptions or the implications of your assumptions.
Another great thing about them is that exact inference for certain (most) structures is known to be computationally infeasible. There are a lot of different inference schemes available that can help you with different approximations with various drawbacks/advantages, heuristics that sort of work, or even ways of drawing samples from the true distribution if you can identify the structures. See belief propagation, loopy belief propagation, sequential Monte Carlo, and Markov chain Monte Carlo methods.
On top of this it helps you see everything in a general framework. Lots of the fundamental pieces of ML models are really just slight tweaks to other things. For instance, SVMs are linear models on kernel spaces with a specific structural prior. Same with splines; it’s just a different basis function. All of this helps you see the pieces of different methods that are actually identical. This helps you make connections and learn more effectively, in my opinion.
