Heuristics for relative frequencies of grammatical features?

Question

Mathematician here, very interested in linguistics but no formal training. Apologies if the question is absurd, ambiguous, or unanswerable.

One thing I've found interesting in the process of learning languages and studying linguistics is the diversity of ways languages evolve, and how they hit upon radically different systems for organizing communication. A secondary surprise, however, is that language is not "anything goes," and there are some properties, or combinations of properties that are never seen in real languages. Even among properties that are observed, some seem to be rare and some common, and this is what I mean by "relative frequencies".

A curious example I've read about recently is that some languages have a "tripartite alignment" system which is finer than the nominative/accusative or ergative/absolutive alignments. Still, this system seems to be very rare.

This brings me to my question: what heuristic best explain the existence of such rare features? I'm vaguely aware of the "parametric" model of generative linguistics, which I'd consider fairly "combinatorial," although I'm not sure how popular it is in current academia. Let me outline three, not-necessarily-mutually-exclusive, possibilities:

[Uniform, Independent Binary Combinatorics] This model assumes that fundamental grammatical principles are (approximately) mutually independent and are "on" or "off" with equal probability. Such a process can still produce features with relative frequencies, due to the interaction between different "switches". For example, a rare feature may, due to the geometry of linguistic space, require a very particular set of switches to be turned on.

[Probabilistic] The process is (approximately) governed by simple probabilistic rules, but they are either not uniform, not mutually independent, or not binary (or some combination thereof!). A feature is then rare if its set of generating parameters has low probability with this joint distribution.

[Noise] The generative process has substantive noise, and this largely explains rare features.

To be extra clear, this is not the sort of question that admits a tidy answer, and I'm not expecting one. I'm interested in answers of the form "nobody really believes in the binary model anymore due to..." or "parametric models actually never assume independence or uniformity, and this would explain such rare features..." or "no parametric model has yet to explain away noise in generative linguistics". It's more of an invitation to ruminate on what is known in this space, so feel free to take the question statement as an open-ended prompt to share relevant thoughts or research.

Answer 1

Unfortunately, little is known about the big questions touched here.

At first sight, there seem to be enough languages in the world (approx. 6000) to use some statistical methods, but this is not easy because

• Not all of the 6000 languages are described in sufficient detail to include them in the sample (some of them are just known by their name without much of further information), reducing the usable sample to a few hundred languages
• The language of the world are not independent of each other, they form some very large and member-rich families like Niger-Congo, Sino-Tibetan or Austronesian, others are isolates or members of very small families

One could also try evolutionary approaches, studying the change of languages over time. Unfortunately, only very few languages are known with a big time depth, and the sample of ancient languages is very biased with respect to language families.

Typologists (e.g. Annemarie Verkerk) try nowadays to combine language trees and typological classification to estimate the probability of certain typological changes by inferring the linguistic type of ancestor nodes in the tree. This is a very interesting kind of research and it questions earlier simple typological models (e.g., linguistic types changing in circles with a given direction).