Section 20.2.Learning with Complete Data 723 pper pper3 Figure 20.6 A Bayesian network that corresponds to a Bayesian learning process.Poste- rior distributions for the paramete r variables日，θi,and can be inferred from their prior distributions and the evidence in the Flvor and Wrapper,variables. derstand how the structure of a Bayes net can be learned from data.At present,structural in thei ill giv nly a hrief sketch idea oach is to od model We n start wit inks ar h n g parer node,fitting the para ith the ely .we itia mg model ch to retuning cach ure ations car clude rev change add ng,or de leting a ring is given fo ve parents only g th odes tha er in the sear st able derings ive methods for deciding when a good st re has been found dependen ns implicit tin the structure ar in the P(Fri/Sat,Bar|WillWait)=P(Fri/Sat]WillWait)P(Bar|WillWait) and we can check in the data that the same equation holds between the corresponding condi- tional frequencies.Now.even if the structure describes the true causal nature of the domain. statistical fuctuations in the data set mean that the equation will never he satisfied eractl so we need to perform a suitable statistical test to see if there is sufficient evidence that the independence hypothesis is violated.The complexity of the resulting network will dependSection 20.2. Learning with Complete Data 723 Flavor1 Wrapper1 Flavor2 Wrapper2 Flavor3 Wrapper3 Θ Θ1 Θ2 Figure 20.6 A Bayesian network that corresponds to a Bayesian learning process. Poste￾rior distributions for the parameter variables Θ, Θ1, and Θ2 can be inferred from their prior distributions and the evidence in the Flavor i and Wrapper i variables. understand how the structure of a Bayes net can be learned from data. At present, structural learning algorithms are in their infancy, so we will give only a brief sketch of the main ideas. The most obvious approach is to search for a good model. We can start with a model containing no links and begin adding parents for each node, fitting the parameters with the methods we have just covered and measuring the accuracy of the resulting model. Alterna￾tively, we can start with an initial guess at the structure and use hill-climbing or simulated annealing search to make modifications, retuning the parameters after each change in the structure. Modifications can include reversing, adding, or deleting arcs. We must not in￾troduce cycles in the process, so many algorithms assume that an ordering is given for the variables, and that a node can have parents only among those nodes that come earlier in the ordering (just as in the construction process Chapter 14). For full generality, we also need to search over possible orderings. There are two alternative methods for deciding when a good structure has been found. The first is to test whether the conditional independence assertionsimplicit in the structure are actually satisfied in the data. For example, the use of a naive Bayes model for the restaurant problem assumes that P(Fri/Sat, Bar|WillWait) = P(Fri/Sat|WillWait)P(Bar|WillWait) and we can check in the data that the same equation holds between the corresponding condi￾tional frequencies. Now, even if the structure describes the true causal nature of the domain, statistical fluctuations in the data set mean that the equation will never be satisfied exactly, so we need to perform a suitable statistical test to see if there is sufficient evidence that the independence hypothesis is violated. The complexity of the resulting network will depend