AFAICT, this is the question ‘how much worse do your predictions get as you discard information in some orderly fashion?’, as framed by physicists.
Do “renormalisation groups”, whatever they are, fit in here?
Petri, G., Expert, P., Turkheimer, F., Carhart-Harris, R., Nutt, D., Hellyer, P. J., & Vaccarino, F. (2014). Homological scaffolds of brain functional networks. in Journal of The Royal Society Interface, 11(101), 20140873. DOI
Talks about a fun-sounding “persistent homology” idea, which sounds a little like some kind of topological measure theory to my analytics-biassed perspective:
Persistent homology is a recent technique in computational topology developed for shape recognition and the analysis of high dimensional datasets [36,37]. It has been used in very diverse fields, ranging from biology [38,39] and sensor network coverage  to cosmology . Similar approaches to brain data [42,43], collaboration data  and network structure  also exist. The central idea is the construction of a sequence of successive approximations of the original dataset seen as a topological space X. This sequence of topological spaces is such that whenever and is called the filtration.
- Bar-Yam, Y. (2003). Dynamics Of Complex Systems. Westview Press.
- Castiglione, P., & Falcioni, M. (2008). Chaos and Coarse Graining in Statistical Mechanics. Cambridge, UK ; New York: Cambridge University Press.
- NESCI's multiscale methods page