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# Bayesian Networks

## Evidential Reasoning

From the product rule (or chain rule), one can express the probability of any desired proposition in terms of the conditional probabilities specified in the network. For example, the probability that the Sprinkler is on given that the Pavement is slippery is:

These expressions can often be simplified in ways that reflect the structure of the network itself. The first algorithms proposed for probabilistic calculations in Bayesian networks used a local distributed message-passing architecture, typical of many cognitive activities. Initially, this approach was limited to tree-structured networks but was later extended to general networks in Lauritzen and Spiegelhalter’s (1988) method of junction tree propagation. A number of other exact methods have been developed and can be found in recent textbooks.

It is easy to show that reasoning in Bayesian networks subsumes the satisfiability problem in propositional logic and, therefore, exact inference is NP-hard. Monte Carlo simulation methods can be used for approximate inference (Pearl, 1988) giving gradually improving estimates as sampling proceeds. These methods use local message propagation on the original network structure, unlike junction-tree methods. Alternatively, variational methods provide bounds on the true probability.

## Causal Reasoning

Most probabilistic models, including general Bayesian networks, describe a joint probability distribution (JPD) over possible observed events, but say nothing about what will happen if a certain intervention occurs. For example, what if I turn the Sprinkler on instead of just observing that it is turned on? What effect does that have on the Season, or on the connection between Wet and Slippery? A causal network, intuitively speaking, is a Bayesian network with the added property that the parents of each node are its direct causes, as in Figure 2.4. In such a network, the result of an intervention is obvious: the Sprinkler node is set to X3=on and the causal link between the Season X1 and the Sprinkler X3 is removed (Figure 2.5). All other causal links and conditional probabilities remain intact, so the new model is: