We can view the graph as encoding a generative sampling process executed by nature, where the value for each variable is selected by nature using a distribution that depends only on its parents. In other words, each variable is a stochastic function of its parents. Based on this intuition, perhaps the most natural network structure for the distribution in this example is the one presented in the figure left.

The edges encode our intuition about the way the world works. Intuitively, each variable in the model depends directly only on its parents in the network. We formalize this intuition later. One such model, P I , represents the distribution in the population of intelligent versus less intelligent student.

For example, we might believe that a smart student in an easy class is 90 percent likely to get an A, 8 percent likely to get a B, and 2 percent likely to get a C. Conversely, a smart student in a hard class may only be 50 percent likely to get an A. In general, each variable X in the model is associated with a conditional probability distribution CPD that specifies a distribution over the values of X given each possible joint assignment of values to its parents in the model. For a node with no parents, the CPD is conditioned on the empty set of variables. One possible choice of CPDs for this domain is shown in figure left.

The network structure together with its CPDs is a Bayesian network B ; we use B-student to refer to the Bayesian network for our student example.

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How do we use this data structure to specify the joint distribution? Consider some particular state in this space, for example, i1, d0, g2, s1, l0. Intuitively, the probability of this event can be computed from the probabilities of the basic events that comprise it: the probability that the student is intelligent; the probability that the course is easy; the probability that a smart student gets a B in an easy class; the probability that a smart student gets a high score on his SAT; and the probability that a student who got a B in the class gets a weak letter.

The total probability of this state is:. Clearly, we can use the same process for any state in the joint probability space.

## [] Learning Bayesian Networks with Low Rank Conditional Probability Tables

In general, we will have that. This equation is our first example of the chain rule for Bayesian networks which we will define in a general setting in section 3. We denote these random variables with their first letters I, D, G, S and L respectively in the discussion below. Each of the random variables in this example are discrete random variables and take values from a finite domain.

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Consequently, the CPD at each node v can be represented as tables, where rows are indexed by values of the random variables associated with Deps v and the columns are the probabilities associated with each value of Xv. For example, the CPD associated with the node for the random variable G has 2 columns associated with the 2 possible values of G namely g0, g1 , and 4 rows corresponding to various combinations of values possible for the direct dependencies of the node namely namely i0, d0, i0, d1, i1, d0 , and i1, d1. For instance. The probabilities in row g0 are left untouched.

The question is whether this is a slip or a fundamental change. We think that by this local swap the meaning of l0 and l1 should be reversed! Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide. European Workshop on Probabilistic Graphical Models. Bayesian Networks with Function Nodes.

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Bendtsen, M. Condamin, L. Wiley, New York Google Scholar.

Cowell, R. Springer Google Scholar. Dean, T. Ejsing, E. Geiger, D.

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## Bayesian Networks with Function Nodes

Jensen, F. Version 8. Koller, D. In: Proc.