Rens, although my characterization of a "probabilistic grammar"
assumed the most commonly held interpretation -- the basic idea of
which goes back to Booth (1969), and is still active in more recent
innovations such as stochastic tree-adjoining grammars -- you are
quite right in pointing out that other interpretations of
"probabilistic grammar" are possible.
However, it is not clear to me how the competence/performance
distinction has "actually disappeared" in the framework you describe.
Crucial to your approach is the "large corpus of analyzed utterances,"
and implicit in that phrase is some set of rules -- discrete,
nonprobabilistic rules -- for what constitutes an allowable analysis.
Admittedly those rules may be incredibly simple, for example, you can
allow all possible binary trees covering a sentence as the set of
possible analyses for that sentence; but that is still different from
an underlying set of rules that allows 3-way branching. And either of
those is different from an underlying set of rules that talks about
dependency structures rather than constituency. And in any of those
cases, you still have to talk about the possible combinatory
operations on analyses (e.g. some time ago you and I had a discussion
about the fact that your framework allowed tree substitution but not
TAG-style tree adjunction). These seem to me to be more substantive
issues than simple conventions on "formal notation."
I realize that "rules" may be a loaded term here; feel free to
substitute "structures" or "analyses" if you prefer. Whatever you
choose to call them, the principles (another loaded term; use its
unloaded sense) that determine what makes an analysis possible are a
critical part of your theory, even if you choose to keep them
incredibly simple or to leave them implicit rather than explicit. If
your theory is intended to be a cognitive model, then you have no
choice but to interpret those underlying principles as something that
people know about the possible structures of their language, as
distinguished from the actual structures that get observed or
produced. And that's competence.
Best,
Philip