5.1 Patterns of probable inference  (Page 4/6)

In terms of the posterior probability, we have

We note that the odds favoring H , given positive indications from both survey companies, is 10.2 as compared with the odds favoring H , given a favorable test market, of 13.5. Thedifference reflects the residual uncertainty about the test market after the market surveys. Nevertheless, theresults of the market surveys increase the odds favoring a satisfactory market from the prior 3 to 1 to a posterior 10.2 to 1. In terms of probabilities,the market surveys increase the likelihood of a favorable market from the original $P\left(H\right)=0.75$ to the posterior $P\left(H\right|E_1{E}_2)=0.91$ . The conditional independence of the results of the survey makes possible direct use of the data.

A classification problem

A population consists of members of two subgroups. It is desired to formulate a battery of questions to aid in identifying the subclass membership ofrandomly selected individuals in the population. The questions are designed so that for each individual the answers are independent, in the sense thatthe answers to any subset of these questions are not affected by and do not affect the answers to any other subset of the questions. The answers are,however, affected by the subgroup membership. Thus, our treatment of conditional idependence suggests that it is reasonable to supose theanswers are conditionally independent, given the subgroup membership. Consider the following numerical example.

A classification problem

A sample of 125 subjects is taken from a population which has two subgroups. The subgroup membership of each subject in the sample is known. Each individual is asked a battery often questions designed to be independent, in the sense that the answer to any one is not affected by the answer to any other. The subjects answer independently. Data on the resultsare summarized in the following table:

Assume the data represent the general population consisting of these two groups, so that the data may be used to calculate probabilities and conditionalprobabilities.

Several persons are interviewed. The result of each interview is a “profile” of answers to the questions. The goal is to classify the person in one of the two subgroupson the basis of the profile of answers.

The following profiles were taken.

• Y, N, Y, N, Y, U, N, U, Y. U
• N, N, U, N, Y, Y, U, N, N, Y
• Y, Y, N, Y, U, U, N, N, Y, Y

Classify each individual in one of the subgroups.

Solution

          $A_i=$ the event the answer to the i th question is “Yes”

          $B_i=$ the event the answer to the i th question is “No”

          $C_i=$ the event the answer to the i th question is “Uncertain”

The data are taken to mean $P\left(A_1\right|G_1)=42/69,P\left(B_3\right|G_2)=19/56$ , etc. The profile

Y, N, Y, N, Y, U, N, U, Y. U corresponds to the event $E=A_1{B}_2{A}_3{B}_4{A}_5{C}_6{B}_7{C}_8{A}_9{C}_{10}$