I've recently been reading a lot of papers from the more traditional Generativist literature on computational models for language acquisition, so I'm going to write a post discussing how these traditional approaches relate to the kind of natural language processing (NLP)-style, data-driven, structured probabilistic models I work with (and hinted at in the previous post). Specifically, I'm going to outline the Universal Grammar and Principles & Parameters-based approach to language acquisition. We will see that it looks pretty different from the approaches adopted for grammar induction in natural language processing, which typically involve estimating probability distributions over structural analyses given sentences (and possibly sentence meanings). I'll argue next that they are actually related in a deep way. Specifically, they both propose that children simplify their exploration of the space of possible grammars by learning in a smaller space that is related to the space of possible grammars, and the space proposed by P&P-based approaches is potentially a special case of the spaces used by data-driven techniques.