I've been wrestling with understanding how to properly use the I(,)
interval censoring construct for a while. I've finally figured out a
point of view that lets me understand it in a declarative fashion (what
it means) rather than operationally (how it affects the simulation).
The crux of the problem is, I believe, that the notation used in the
BUGS language is misleading. One writes
v ~ distr(params) I(lower, upper)
as if the I(,) construct were part of the model, i.e., the prior joint
distribution for the model variables. This leads one to want to think
of this as a truncated distribution, even though the documentation
emphasizes that it is not. But the information given by that I(lower,
upper) annotation is not part of the model -- it is part of the DATA!
That is, the posterior distribution that WinBUGS simulates is
- the prior distribution specified by the model
- conditioned on the data values specified in the data file
- and also conditioned on the set of propositions (lower <= v <= upper),
for every occurrence of some v ~ distr(params) I(lower, upper) in the
model file.
This implies that the interval constraints belong in the DATA file, and
not the MODEL file... or, at least, if they are in the model file they
should be separated out from the model itself. This also opens up the
possibility of more general constraints of the form e_1 <= e_2, where
both e_1 and e_2 can involve zero, one, or more than one model variable.
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