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I'm working with some folks to improve a benchmarking database for MCMC samplers. So there are canonical models, data, and posteriors. It would be useful to be able to automatically rank models according to their complexity, which I feel should involve more than simply counting the number of model parameters but instead convey the degree of complexity in how the model is structured with respect to those parameters. I'm personally pretty math-naive but read a lot of pop-sci stuff so have a vague feeling that there might be graph-theoretic and/or information-theoretic tools for this kind of complexity quantification.

If it helps, we're focusing at the moment on models expressed in Stan, which translates to C++, so maybe there's a way to take the compiled binaries (which would include all the math for the distributions used) and perform some kind of analysis on it?

Alternatively, in the graph-theory world, someone recently developed tools to translate models into a graph representation, but I don't know how robust/mature that is.

Or maybe I'm wrong and there's no general solution to quantifying algorithmic complexity in this way?

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I'm way out of my depth here, but after some more searching I think this paper might address what I'm looking for.

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