Improtance of posterior probabilities

There are many powerful reasons for wanting to compute the posterior probabilities, even if we subsequently use them to make decisions. 

These include: 

Minimizing risk. 

Consider a problem in which the elements of the loss matrix are subjected to revision from time to time .If we know the posterior probabilities, we can trivially revise the minimum risk decision criterion by modifying(1.81)appropriately. If we have only a discriminant function, then any change to the loss matrix would require that we return to the training data and solve the classification problem afresh.

 Reject option.

 Posterior probabilities allow us to determine a rejection criterion that will minimize the mis-classification rate, or more generally the expected loss, for a given fraction of rejected data points.

 Compensating for class priors. 

Consider our medical X-ray problem again, and suppose that we have collected a large number of X-ray images from the general population for use as training data in order to build an automated screening system. Because cancer is rare amongst the general population, we might find that, say, only 1 in every 1,000 examples corresponds to the presence of cancer. If we used such a data set to train an adaptive model, we could run into severe difficulties due to the small proportion of the cancer class. For instance, a classifier that assigned every point to the normal class would already achieve 99.9% accuracy and it would be difficult to avoid this trivial solution. Also, even a large data set will contain very few examples of X-ray images corresponding to cancer, and so the learning algorithm will not be exposed to a broad range of examples of such images and hence is not likely to generalize well. A balanced data set in which we have selected equal numbers of examples from each of the classes would allow us to find a more accurate model. However, we then have to compensate for the effects of our modifications to the training data. Suppose we have used such a modified data set and found models for the posterior probabilities. 

From Bayes’theorem(1.82),we see that the posterior probabilities are proportional to the prior probabilities, which we can interpret as the fractions of points in each class. We can therefore simply take the posterior probabilities obtained from our artificially balanced data set and first divide by the class fractions in that data set and then multiply by the class fractions in the population to which we wish to apply the model. Finally, we need to normalize to ensure that the new posterior probabilities sum to one. Note that this procedure cannot be applied if we have learned a discriminant function directly instead of determining posterior probabilities. 

Combining models. 

For complex applications, we may wish to break the problem into a number of smaller sub-problems each of which can be tackled by a separate module.