Re: [UAI] Definition of Bayesian network

From: David Poole (poole@cs.ubc.ca)
Date: Sun Jul 29 2001 - 13:15:09 PDT

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    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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