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Book review: Darwinia
Reviewed: Friday, August 11, 2006

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tinderbox

Causal Independence for Probability Assessment
David Heckerman and John S. Breese, 1995
Monday, June 28, 2004

A Bayesian Network (BN) encode probabilistic relationships into a directed acyclic graph where the vertexes are variables and the edges (can) represent causality. The network both represents expert knowledge and provides mechanisms for computation. At issue in this paper is how to efficiently represent certain types of causally independent knowledge. For example, there may be an effect that can be caused by n different things. In the general case, I would need to specify 2<sup>n</sup> different parameters to completely describe this. If these causes are independent, however, I may be able to describe the situation using only n parameters. This is a big win.

Details, details

Heckerman first describes the noisy-or model -- where each cause has some chance of turning on the effect, all the causes are independent and everything is binary (causes and the effect). The idea is to note that for each cause C<sub>i</sub> there is some probably q<sub>i</sub> that the effect won't happen even if the cause is true. (If the effect always happened, then it would be deterministic and you wouldn't need the noise!). Now suppose that two (and only two) causes i and j are true, then the odds that the effect will still be false is:

<center>1 - q<sub>i</sub> q<sub>j</sub></center>

In general, noisy-or works by taking the product of the q<sub>i</sub>'s.

In a 1989 paper, Max Henrion describes how to extend noisy-or to non-binary causes and effects and how to add a leaky node to account for that we know not what. In this paper, Heckerman shows how to generalize the framework so as to model noisy-max, noise-and, noisy-addition and so forth.

He then goes on to describe four specializations of the general model. These are: amechanistic, temporal, decomposable and multiply decomposable causal independence. Each has certain requirements and certain benefits. Heckerman stresses that "the preferred form will depend on the specific causes and effects being modeled as well as the expert providing the model." As usual, you cannot really use this stuff without thinking about it.


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Copyright -- Gary Warren King, 2004 - 2006