We consider a problem of classifying a three-dimensional vector, where a value of each element is either 0 or 1, into either the class 1 or the class 2. Let be a vector. Assume that each element of of the class () independently follows a Bernoulli distribution, and let () be the probability of . Let be parameters of the class .
Let be a data set consisting of data. Let be the vector of the -th data, and be the class of . We assume that of the class is independently observed from the aforementioned distribution whose parameter is .
(1) Let be a prior probability for the class . We determine an estimated class of by comparing the posterior probabilities. Namely, we set if ; otherwise we set . Show a rule that assigns to an estimated class by using and .
(2) Let be a subset of the data set . By using , derive the maximum likelihood estimate of from the data set .
(3) Assume that a data set is given in Table 1. Compute the values of the maximum likelihood estimates from Table 1.
Table 1: A data set
1
1
2
1
3
2
4
1
5
2
(4) Let prior probabilities be and . Compute the estimated class of by substituting computed in (3) for of the rule shown in (1).
(5) Let a prior probability be . We classify by substituting computed in (3) for of the rule shown in (1). Explain the relation between and an estimated class .