However, I then noticed another problem. We end up calling X a variable and
then saying things as if it actually were. Consider manipulation. We say we
manipulate X (or the state of X). That makes no sense if taken literally.
We cannot manipulate a function or its state. We can manipulate the values
of its arguments, which in turn determines the value it returns. Indeed,
this is what we are doing. We are really manipulating x so the value of the
Cartesian product lies in the subset where x has that value. In general, we
usually speak of X as if it were x.
I'd greatly appreciate any additional comments on the this. I feel there
are two obtions:
1) Abandon tradition and write P(X). Indeed this random variable usage may
not be all that traditional anyway. It seems it may be something started by
us fairly recently. Looking in other (I admit only the few I own) texts, it
is used only for real-valued functions in Fisherian texts. The other
Bayesian texts I found discuss applications of Bayes' Theorem without
random variables. When we write P(Cancer=present), Bayesians are conceiving
the event `Cancer is present' for an individual patient without reference
to some other event in another space. The random variable notation is
really being used as notation for a plain old event; not one determined by
some function. This is fundamentally different than writing P(height=72)in
a Fisherian application. There we are speaking of a population of
individuals, with the uniform distribution, and we mean the set of those
who are 72 inches tall.
2) Stick with formally defining X as a random variable, but then note that
in Bayesian applications that this is only true implicitly, that actually
we indentify X and its values directly. So in practice X becomes more of an
actual variable, and we say things like `manipulating X'.
Sincerely,
Rich