Downloaded from https://www.everyspec.com AMCP 70410 3-9.2 SIMPLE ILLUSTRATION TABLE S.10 To use the Bayes procedure, we must first HONUIT COIN? state what our beliefs are about every pos- sibility in our conceptual model. Suppose Condition No. 1 No. 2 No. 3 we wish to take a stand on whether a coin (1) pbebllIty of husdd 0 1/2 1 Is honest or not, and 11 we can see of' it are " the results of lejtimate tlilps Row I in (2) dagrst*ofbalief ...... 11/2% 90% 1/2% Table 3-10 shows the 3 conditions we pro- (3) likelihood of test A results sume are possible In our conceptual model, 3-heds, .tell 0 1/A I the coin is either 2-tailed, fair, or 2-headed, (4) degree-of belief ......
Row 2 shows our dtgree-of-belief berore, any (5) likelihood of test 8 results tests are run; we are 99% sure the coin Is 3-heeds, 0-tells 0 1/8 1 fair, and we suspect !/2% each that the coin (6) degr4of-bellef . ..... 21 LO% 24 is 2-tailed or 2-headed, Let each test involve (7) likelihood of test C results t1-hseel, 2-talls 0 3/8 1 soeinS the results of 3 legitirmate flips of the 1S) dsr2.tell . 0 200 O coin. Run Test A and suppose the results are 3-heads, 0-tails, Now, separately for each condition, calculate the likelihood uf getting where the test result if that condition is true. Ob- viously, under condition #1 (2-tailed coin) ;ubevent i for E.
The event for example, the results of Test A would be space is partitioned (exhaustive impossible; so we put a zero there. By means and mutually exclusive sub- of Bayes formula, our "after test A" degree- events) into subovents, of-belief Is calculated, Reasonably enough, we now suspect more strongly that the coin H - event: new test results is not a fair one, but we no longer suspcict that It Is 2-tafled, After the results of Test h' (E'l, - prior probabilities assigned to B are In, we are even more suspicious about the E 1 the coin's being 2-headed.
But Test C clears things up, since we have observed at least 1 Pr I{tEtJ - likelihood of getting the now tall and I head, the coin can be neither 2. results, given that Ej were in headed nor 2-tailed. Therefore it must be fact true. fair, according to the conceptual modl we set up. The "flter test" degree-of-belief Pr I, 1H now probabilities assigned to often serves as the "before test" degree-of- the El, after seeing the test belief for a subsequent test, as in this illustra- results. tion. 2;j - implies sum over all I 3.9.3 BAYES FORMULA$, DISCRETE RANDOM VARIABLES In the illustiation in par. 3-9,2 (see Table 3-10), rows 4, 6, 8 were obtained from Eqs. 3-38 and 3-39 as shown in Example No.
29. 1V1rf I H) o: Pr{Ee}p i{H I El} (3-38) 3.9.4 PRINCIPLES FOR APPLICATION zoP'r{ I hl 1 (3-39) Apply the fbllowing principles: - '- ,','.r . . .. .....