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WHY MAKING BAYESIAN NETWORKS BAYESIAN MAKES SENSE.

WHY MAKING BAYESIAN NETWORKS BAYESIAN MAKES SENSE. Dawn E. Holmes Department of Statistics and Applied Probability University of California, Santa Barbara CA 93106, USA. Subjective Probability. Rational degrees of belief. Keynes ‘s consensual rational degrees of belief.

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WHY MAKING BAYESIAN NETWORKS BAYESIAN MAKES SENSE.

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  1. WHY MAKING BAYESIAN NETWORKS BAYESIAN MAKES SENSE. Dawn E. HolmesDepartment of Statistics and Applied ProbabilityUniversity of California, Santa BarbaraCA 93106, USA

  2. Subjective Probability • Rational degrees of belief. • Keynes‘s consensual rational degrees of belief.

  3. What is a Bayesian Network? • Directed acyclic graph • Nodes are variables (discrete or continuous) • Arcs indicate dependence between variables. • Conditional Probabilities (local distributions) • Missing arcs implies conditional independence • Independencies + local distributions => specification of a joint distribution

  4. Bayesian Networks • In classical Bayesian network theory a prior distribution must be specified. • When information is missing, we are able to find a minimally prejudiced prior distribution using MaxEnt

  5. A Simple Bayesian Network

  6. Priors for Bayesian Networks • Using frequentist probabilities results in a rigid network. • The results obtained using Bayesian networks are only as good as their prior distribution. • The maximum entropy formalism.

  7. Maximum Entropy and the Principle of Insufficient Reason. • The principle of maximum entropy is a generalization of the principle of insufficient reason. • It is capable of determining a probability distribution for any combination of partial knowledge and partial ignorance.

  8. Two Results • An iterative algorithm for updating probabilities in a multivalued multiway tree in given. • A Lagrange multiplier technique is used to find the probability of an arbitrary state in a Bayesian tree using only MaxEnt.

  9. FRAGMENT OF STATE TABLE

  10. A Simple Bayesian Network

  11. A Simple BN with Maximum Entropy

  12. Maximum Entropy in Bayesian Networks • Maximum entropy provides a technique for eliciting knowledge from incomplete information. • We use the maximum entropy formalism to optimally estimate the prior distribution of a Bayesian network. • All and only the information provided by expert knowledge is used.

  13. What use are Subjective Bayesian Prior Distributions • Why determine the prior distribution for a Bayesian network using maximum entropy? • Any problem involving probabilities can be represented by a Bayesian network.

  14. Independence • Proofs must not use techniques outside of MaxEnt. • Proofs have already been given elsewhere.

  15. Thank You!

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