Milan Studeny wrote:
> Simply, one can
> always compute conditionals from a joint probability measure but not
> conversely in general although this is possible in usual situations.
There are obvious cases that can't be represented by a belief network
(Bayesian network). These are when there are uncountably many variables
(a belief network assumes an enumeration of variables). For example,
think of my position at time T as a variable for each time T. It is not
unreasonable to model T as the reals (which are not enumerable). This
cannot be modelled as a belief network. Can it also not be modelled as a
joint? If not then we need some new concepts, as continuous time is
important to model.
[Even if we follow Jaynes' advice, it doesn't seem to get us out of
this. First the reals are a well defined limit of rationals which can be
defined as the limit of integers. Secondly, even if you don't think that
times form a continuum, you may want to have infinitely many time points
between two other points. You also probably want my position at some
time to be dependent on the previous time, not on the time that is
defined by the enumeration. If you insist on enumerating forward in time
you will soon get stuck in Zeno's paradox. If you insist on emumerating
with respect to a well defined enumeration (e.g., like the enumeration
of the rationals, but we only need infinitely many time points between
two points for this to hold) you won't get many sensible
independencies.]
David
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