I'll add Minimum Description Length to the mix. Under certain definitions and conditions, it equals the Bayesian Information Criterion plus an extra term, which I consider a very interesting result in this "two faces of the same coin" perspective.
This is exactly why I think they are one and the same. It's relatively trivial to just plop a (lossy) machine learned markov chain instead of one learned (perfectly) from the data into PPM. With zero changes to the rest of the algorithm.
There is the Kolmogorov Complexity [1], Normalized Information Distance [2] and Normalized compression distance [3] that correlates those.
Finally, there's the Pre-Big Bang Informational Compression and the Delayed Release of Antimatter [4]
All big {rabbit/black} holes to lose some time, if you have any.
[0] https://en.wikipedia.org/wiki/Prediction_by_partial_matching
[1] https://en.wikipedia.org/wiki/Kolmogorov_complexity
[2] https://homepages.cwi.nl/~paulv/papers/chapter08.pdf
[3] https://en.wikipedia.org/wiki/Normalized_compression_distanc...
[4] https://philarchive.org/rec/GREPBI