[Thrun] So we’re now ready to define Bayes networks in a more general way. Bayes networks define probability distributions over graphs or random variables. Here is an example graph of 5 variables, and this Bayes network defines the distribution over those 5 random variables. Instead of enumerating all possibilities of combinations of these 5 random variables, the Bayes network is defined by probability distributions that are inherent to each individual node. For node A and B, we just have a distribution P of A and P of B because A and B have no incoming arcs. C is a conditional distribution conditioned on A and B. D and E are conditioned on C. The joint probability represented by a Bayes network is the product of various Bayes network probabilities that are defined over individual nodes where each node’s probability is only conditioned on the incoming arcs. So A has no incoming arc; therefore, we just want it P of A. C has 2 incoming arcs, so we define the probability of C conditioned on A and B. And D and E have 1 incoming arc that’s shown over here. The definition of this joint distribution by using the following factors has one really big advantage. Whereas the joint distribution over any 5 variables requires 2 to the 5 minus 1, which is 31 probability values, the Bayes network over here only requires 10 such values. P of A is one value, for which we can derive P of not A. Same for P of B. P of C given A B is derived by a distribution over C conditioned on any combination of A and B, of which there are 4 of A and B as binary. P of D given C is 2 parameters for P of D given C and P of D given not C. And the same is true for P of E given C. So if you add those up, you get 10 parameters in total. So the compactness of the Bayes network leads to a representation that scales significantly better to large networks than the common natorial approach which goes through all combinations of variable values. That is a key advantage of Bayes networks, and that is the reason why Bayes networks are being used so extensively for all kinds of problems. So here is a quiz. How many probability values are required to specify this Bayes network? Please put your answer in the following box.