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# Bayes' Decision Theory [5+2* P]

a)
[2 P] In many pattern classification problems one has the option either to assign the pattern to one of classes, or to reject it as being unrecognizable. If the cost for rejects is not too high, rejection may be a desirable action. Suppose that, for a new value of , the true class is and that we assign to class . Assume, In doing so, we incur the loss

 (1)

where is the loss incurred for choosing the (+1)th action, rejection, and is the loss incurred for making a substitution error. Show that the minimum risk is obtained if we decide if for all and if , and reject otherwise. What happens if ? What happens if ?

b)
[3+2* P] Consider the classification problem with rejection option of a).

1. Use the results of a) to show that the following discriminant functions are optimal for such problems:

 (2)

2. Plot these discriminant functions and the decision regions for the two-class one-dimensional case having

• ,
• ,
• , and
• ,
where denotes the normal distribution with mean and variance .
3. Describe qualitatively what happens as is increased from 0 to 1.
4.  [2* P] Repeat for the case having

• ,
• ,
• , and

Next: Markov Blanket [4 P] Up: MLA_Exercises_2007 Previous: Curse of Dimensionality [2*
Haeusler Stefan 2007-12-03