a) Consider the cost matrix for a two-class problem. Let C(+, +) = C(−,−) = p, C(+,−) = C(−, +) = q, and q > p. Show that minimizing the cost function is equivalent to maximizing the classifier's accuracy.
(b) Show that a cost matrix is scale-invariant. For example, if the cost matrix is rescaled from C(i, j) → βC(i, j), where β is the scaling factor, the decision threshold (Equation 5.82) will remain unchanged.
(c) Show that a cost matrix is translation-invariant. In other words, adding a constant factor to all entries in the cost matrix will not affect the decision threshold (Equation 5.82).