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Consider the setting in which we wish to learn a nondeterministic (probabilistic) function f : X -+ {0, 11, which has two discrete output values For example, the instance space X might represent medical patients in terms of their symptoms, and the target function f (x) might be 1 if the patient survives the disease and 0 if not Alternatively, X might represent loan applicants in terms of their past credit history, and f (x) might be 1 if the applicant successfully repays their next loan and 0 if not In both of these cases we might well expect f to be probabilistic For example, among a collection of patients exhibiting the same set of observable symptoms, we might find that 92% survive, and 8% do not This unpredictability could arise from our inability to observe all the important distinguishing features of the patients, or from some genuinely probabilistic mechanism in the evolution of the disease Whatever the source of the problem, the effect is that we have a target function f (x) whose output is a probabilistic function of the input Given this problem setting, we might wish to learn a neural network (or other real-valued function approximator) whose output is the probability that f (x) = 1 In other words, we seek to learn the target function, f ' : X + [O, 11, such that f '(x) = P (f (x) = 1) In the above medical patient example, if x is one of those indistinguishable patients of which 92% survive, then f'(x) = 092 whereas the probabilistic function f (x) will be equal to 1 in 92% of cases and equal to 0 in the remaining 8% ' How can we learn f using, say, a neural network One obvious, bruteforce way would be to first collect the observed frequencies of 1's and 0's for each possible value of x and to then train the neural network to output the target frequency for each x As we shall see below, we can instead train a neural network directly from the observed training examples of f, yet still derive a maximum likelihood hypothesis for f ' What criterion should we optimize in order to find a maximum likelihood hypothesis for f' in this setting To answer this question we must first obtain an expression for P(D1h) Let us assume the training data D is of the form D = {(xl,dl) (x,, dm)},where di is the observed 0 or 1 value for f (xi) Recall that in the maximum likelihood, least-squared error analysis of the previous section, we made the simplifying assumption that the instances (xl x,) were fixed This enabled us to characterize the data by considering only the target values di Although we could make a similar simplifying assumption in this case, let us avoid it here in order to demonstrate that it has no impact on the final outcome Thus treating both xi and di as random variables, and assuming that each training example is drawn independently, we can write P(D1h) as.

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