I really like how straightforward Perfors et al's (2011) Bayesian model is - it's very easy to see how and why they get the results that they do from child-directed speech. They're very careful to say precisely what their model is doing: Assuming there are hierarchical representations in the child's hypothesis space already, these representations can be selected as the ones that best match the child-directed input. In addition, I think they highlight how previous approaches to this problem have tended to split along two distinct dimensions: domain-specific vs. domain-general, and structured vs. unstructured. It's always useful to figure out where the current approach is adding to the existing discussion.
The only real issue I see is the one Berwick et al. (2011) pointed out: The (infamous) poverty of the stimulus (PoS) problem relating to structure dependence is not the one Perfors et al. (2011) are addressing. In particular, the traditional PoS problem has to do with hypothesizing what kind of rules will relate a declarative utterance (e.g., "I can have an opinion") to its interrogative equivalent (e.g., "Can I have an opinion?"). This relationship isn't addressed in Perfors et al.'s model - all that model is concerned with is the ability to assign structure to these utterances. As far as it knows, there's no relationship between the two. And this is where we see the real divergence from the traditional PoS problem, where it was assumed that the child is trying to generate an interrogative using the same semantic content that would be used to make the declarative. This is why the "rules of transformation" were hypothesized in the first place (granted, with the assumption that the declarative version was more basic, and the interrogative version had to be created from that basic version). So, long story short, the Perfors et al. model is learning something that is different from the original PoS problem.
However, it's fair to assume that knowing there are hierarchical structures is a prerequisite for creating rules that use those hierarchical structures. In this sense, what Perfors et al. have shown is really great - it allows the building blocks of the rules (hierarchical structures) to be chosen from among other representations. However, as Berwick et al. point out, it still remains to be shown how having structures building blocks leads you to create structure-dependent rules. Perfors et al. assume that this is an automatic step: [end of section 1.2] "...any reasonable approach to inducing rules defined over constituent structure should result in appropriate structure-dependent rules". Phrased that way, it does sound plausible - and yet, I think there's a real distinction, especially if we're concerned about relating the declarative and interrogative versions of an utterance. Making a structure-dependent rule requires using the available structure as the context of the rule. So this means you could make a structure-independent rule just by not using structure in the context of the rule - even if your building blocks are structured.
Example of a structure-independent rule using structure building blocks:
Move the auxiliary verb after the first NP.
Building blocks: auxiliary verb, NP (structured)
Context: first (not structured)
So again, I think that what Perfors et al. have shown is great in terms of understanding the stages of learning - it's important to know that the preference for hierarchical structure in language doesn't have to be innate (even if the ability to consider hierarchical structure in the hypothesis space may be). However, I do think it falls short of addressing the PoS problem that linguists typically associate with structure dependence. This isn't a failing of Perfors et al. - it just means that people really have to be careful about how they interpret these results. It's very tempting to say that the structure-dependence PoS problem has been solved if you don't give this a very careful read and know what linguists think the problem actually is.
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