Dave:
In the simulation and reliability communities there are attempts to
reconcile continuous time "physics" with discrete events like component
failures, alarms, etc. Rather than have a Bayesian network at each real
valued time index, we might be better to focus on Bayesian networks that
occur at random times, like the discrete events and integrate the
transitions between these Bayesian networks. One would need a method for a
"Bayesian " inversion of the integration in order to determine when and what
root cause events occurred that result in current alarms or component
failures.
I recommend the book, C. Smidts, J. Devooght, and P.E. Labeau, "Dynamic
reliability: future directions: International Workshop Series on Advanced
Topics in Reliability and Risk Analysis", University of Maryland, ISBN
0-9652669-3-1 for an introduction. Jacques Devooght's article may give some
insight to a methodology that could be adapted. (My own contribution to the
book discusses some of these issues, but is a long way from a full
development).
Bob.
----- Original Message -----
From: "David Poole" <poole@cs.ubc.ca>
To: <uai@cs.orst.edu>
Sent: Sunday, July 29, 2001 2:15 PM
Subject: Re: [UAI] Definition of Bayesian network
> 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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