4.16 () Consider a binary classification problem in which each observation xn is known to belong to one of two classes, corresponding to t = 0 and t = 1, and suppose that the procedure for collecting training data is imperfect, so that training points are sometimes mislabelled. For every data point xn, instead of having a value t for the class label, we have instead a value πn representing the probability that tn = 1.

Given a probabilistic model p(t = 1|φ), write down the log likelihood function appropriate to such a data set.