continuous distributions

[joost]

A part of my network consists of 2 continuous parent nodes "mu" and "sigma^2", with continuous child node A, with distribution A~N(mu,sigma^2). The mean of A "mu" is not known, but follows some distribution. Also the variance of A "sigma^2" is not known, but follows some distribution. Two questions:

1. How to set the prior distributions on "mu" and "sigma^2", if they are different from a Normal distribution. (In the table of these nodes I only have to specify the mean and sigma2. This means assumption of a normal distribution?) For example, I want to give "mu" and "sigma^2" a beta-pert-distribution.

2. How to fill in the table for A? Opening the table will me make fill in Mean, mu, sigma^2, Variance. Filling in 1 and 1 for mu resp. sigma^2, does this mean that the parent distributions only influence the distribution of A(Mean,Variance) by means of a "disturbance" (like in the modelling example Temperature.net from the course the quality of the thermometer does)? Thus does it only add mu to Mean and sigma^2 to Variance?

I hope someone can help me.
Joost

[Anders L Madsen]

There are two different approaches to modeling with continuous variables in the HUGIN tools.

One approach is to discretize the nodes representing continuous entities using Interval nodes. In your example the "mu", "sigma^2", and "A" would all be discretized into Interval nodes. The intervals of each Interval node would have to be determined by the user. The conditional probability distribution of each discrete node can subsequently be constructed using the Table Generator functionality.

The continuous nodes of a network in HUGIN are assumed to have a linear Gaussian distribution. This implies that a mean and a variance is specified for a continuous node without continuous parents for each configuration of the discrete parents, if any. For a continuous node with continuous parents a regression factor is specified for each continuous parent for each configuration of the discrete parents.

Thus, the alternative approach is the approximate the continuous distributions using mixtures of Gaussian distributions. Any continuous distribution can be approximated using a mixture of Gaussian distribution. Unfortunately, it may be difficult to identify the number of components and the weights of the mixture. Also, the use of mixtures of Gaussian distributions may lead to heavy computations.

I suggest to use the first approach.

Hope this is useful
-Anders