2 Spam filtering We would like to test a method to filter spam e-mails. We consider that a sender can be sorted according to 3 states : legitimate : The majority of his e-mails are not spam. suspicious : Some e-mails are spam and a label is flagged on the correspondence from this sender. spam : The majority of his e-mails are spam. All the new e-mails are put in a dedicated spam folder. We model the state evolution of the system as a Markov chain with the following state transition matrix to apply when a new mail form the sender is received: Currently the sender is considered as... 0.7 Next mail will potentially be... legitimate suspicious spam 0.2 0.1 0.3 0.4 0.3 0.3 0.7 legitimate suspicious spam 0 Questions We suppose that initially the sender is legitimate. Compute the probability that he is still legitimate after 3 iterations. Give a simplified formula in order to compute the evolution of the distribution without measure- ment update. We would like now to evaluate new mails with an estimation of the quality. The estimation is faulty such that: The e-mail is estimated as... legitimate suspicious spam legitimate 0.6 0.4 0 E-mail is actually... suspicious 0.3 0.7 0 spam 1 0 0 Questions Make the computation with a measurement update for each step, considering that the first is evaluated as suspicious, the second one and third one as legitimate