1
$\begingroup$

The mutual information between two random variables X and Y can be stated formally as follows:

I(X ; Y) = H(X) – H(X | Y)

Where I(X ; Y) is the mutual information for X and Y, H(X) is the entropy for X and H(X | Y) is the conditional entropy for X given Y. The result has the units of bits.

Is the above a realistic representation of the weights along the edge of a bayesian network? Or is a probabilistic representation more suitable? If so, what is the best representation?

How should the edge weights be view from probabilistic perspective in a bayesian network context for directed edges; The probability of the nodes I understand to be posterior or marginal probability, but the edges are slightly more ambiguous.

Update 2021/12/08:

enter image description here

Improve this question
$\endgroup$
2
  • $\begingroup$ I've never seen edge weights being used in the context of Bayesian networks, mainly because Bayesian networks are supposed to represent whether one random variable is conditionally independent from another. I've not seen the "degree of conditional independence" being used before. Could you give an example of this? Whether from a book, a paper, or otherwise? $\endgroup$
    – mhdadk
    Commented Nov 13, 2021 at 14:09
  • $\begingroup$ If weight should be introduced, it should be combined effect from entire network, not only single $I(X,Y)$ on two nodes. $\endgroup$
    – patagonicus
    Commented Dec 8, 2021 at 17:53

1 Answer 1

Reset to default
1
+50
$\begingroup$

If you could provide the community with literature resources or a graphic/sketch of your problem formulation and thoughts, then you will receive more specific answers.

And to answer your questions as it stands:

Is the above a realistic representation of the weights along the edge of a bayesian network?

No.

Or is a probabilistic representation more suitable? If so, what is the best representation?

Yes.

Bayesian networks do not have weights associated with their edges. An edge in a Bayesian network represents the conditional probability for a corresponding random variable, which is represented as a node/vertex in the graph.

Finally, the edges refer to conditional probability distributions, whereas the components of mutual information are based on joint probability distribution and marginal probability distribution.

Improve this answer
$\endgroup$
3
  • $\begingroup$ I have updated the question. Does weights still represent conditional probability of child given parent in this case? +1 so far $\endgroup$
    – StatsBio
    Commented Dec 8, 2021 at 20:46
  • $\begingroup$ Yes, according to this (arxiv.org/abs/1703.04025) paper, the weights are estimates for the values of the parameters associated with each conditional probability distribution in the Bayesian network. $\endgroup$
    – SolingerStuebchen
    Commented Dec 8, 2021 at 22:45
  • $\begingroup$ please could you update your answer with some more details of why conditional probability is the edge weights in this case (some mathematical explanation would also be helpful). I had some thoughts and wonder if instead parents nodes represent the prior, then the edge weights should represent the likelihood and then the nodes would represent the posterior. @SolingerMuc $\endgroup$
    – StatsBio
    Commented Dec 10, 2021 at 17:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.