6.16 Fraud detection. To prevent fraud, a credit-card company decides to contact Professor Villebanque and provides him with a random list of several thousand fraudulent and non-fraudulent events. There are many dierent types of events, e.g., transactions of various amounts, changes of address or card-holder information, or requests for a new card. Professor Villebanque decides to use SVMs with an appropriate kernel to help predict fraudulent events accurately.
It is dicult for Professor Villebanque to dene relevant features for such a diverse set of events. However, the risk department of his company has created a complicated method to estimate a probability P[U] for any event U. Thus, Professor Villebanque decides to make use of that information and comes up with the following kernel dened over all pairs of events (U; V ):
K(U; V ) = P[U ^ V ] ???? P[U] P[V ]: (6.31)
Help Professor Villebanque show that his kernel is positive denite symmetric.