Chapter 3 showed how the binomial moments and factorization bounds, derived using the method of indicator functions, are endowed with mathematical elegance (simple, symmetric, and uniform treatment on upper and lower bounds). However, the indicator function bounding approach is unable to include optimality. If the derived bound is optimal, there is no space for improvement in terms of the given information. On the other hand, if the derived bound is not optimal, the bound can be improved to increase the efficiency of statistical inference. To overcome the drawback of being unable to measure optimality when using the indicator function approach, this chapter discusses another bounding technique that takes optimality into consideration (see, for example, Boros and Pre´kopa (1989), and Chen (1998)).