Hi, I am a bit confused on whether bayesian networks can model non-linear dependence between nodes/random variables, since I have read some controversial information in papers.
Specifically,Motomura et al.claim that:
" However, most of conventional Bayesian networks can handle discrete variables. Well known Bayesian network
for continuous variables exist but it can handle only linear dependency between child and parent nodes"
( Y.Motomura and I.Hara. Bayesian network learning system based on neural
networks. AFSS2000,International Symposium on Theory and Applications
of Soft Computing, 2000.)
However, taken from this link: http://www.bioss.ac.uk/people/dirk/essays/GeneExpression/bayes_net.html
Multinomial distribution for the possible states of a child variable given the state of the parents: ... Can model nonlinear dependencies
Linear Gaussian. Learn a linear regression model for the child variable given its parents Disadvantage: Can only model linear dependencies.
--
So I suppose it depends on the chosen representation of the random variables.
On the other hand, Bayesian Neural Network can be seen as a non-linear Gaussian regression (with the assumption of gaussian distributions). Am I correct? Can someone clear this a bit further for me?
For example, it can be possible fuzzy Bayesian networks ( see my papers, J.Fusion Energy , 2015; J. Of Physics& Astronomy, 2013;2016)
Yes they can. Linearity only reduces the computation cost of inference over network. In BN dependencies are represented with conditional density functions which doesn't impose any linearity constraint.
Y ~ P(Y|X)
Dependencies between categorical variables are simplest possible example of nonlinear relationship. When people refer to linear BN they usually mean that the random variables are linearly translated version of each other with an independent additive random variable (w).
Y = F*X+w
When w is Gaussian, the inference tasks on BN are simplified to linear algebraic equations.
Please have a look at GeNIe (http://www.bayesfusion.com/, free for academic teaching and research use). It allows for creating Bayesian networks with continuous nodes/distributions with any relationships between them, not restricted to linear relationships.
I am also confused by this question, I don't know how this question evolves over time. In some papers, it reads that it is possible using some commecial package (e.g. analytica). But from none of them I can find the open access code. Does anyone know how to model nonlinear relationship in R? I don't think bnlearn has this function.
Thanks!
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