Model checking: in Exercise 2.13(below), the counts of airline fatalities in 19761985 were fitted to four different Poisson models.

Question: For each of the models, use simulations from the posterior predictive distributions to measure the discrepancies. Display the discrepancies graphically and give p-values.

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Exercise 2.13 is below:

(a) Assume that the numbers of fatal accidents in each year are independent with a Poisson() distribution. Set a prior distribution for and determine the posterior distribution based on the data from 1976 through 1985. Under this model, give a 95% predictive interval for the number of fatal accidents in 1986. You can use the normal approximation to the gamma and Poisson or compute using simulation.

(b) Assume that the numbers of fatal accidents in each year follow independent Poisson distributions with a constant rate and an exposure in each year proportional to the number of passenger miles flown. Set a prior distribution for and determine the posterior distribution based on the data for 19761985. (Estimate the number of passenger miles flown in each year by dividing the appropriate columns of Table 2.2 and ignoring round-off errors.) Give a 95% predictive interval for the number of fatal accidents in 1986 under the assumption that 8 1011 passenger miles are flown that year.

(c) Repeat (a) above, replacing fatal accidents with passenger deaths.

(d) Repeat (b) above, replacing fatal accidents with passenger deaths.

Incidentally, in 1986, there were 22 fatal accidents, 546 passenger deaths, and a death rate of 0.06 per 100 million miles flown. We return to this example in Exercises 3.12, 6.2, 6.3, and 8.14.

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 Fatal accidents 24 25 31 31 22 21 26 20 16 22 Passenger deaths 734 516 754 877 814 362 764 809 223 1066 Death rate 0.19 0.12 0.15 0.16 0.14 0.06 0.13 0.13 0.03 0.15 Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 Fatal accidents 24 25 31 31 22 21 26 20 16 22 Passenger deaths 734 516 754 877 814 362 764 809 223 1066 Death rate 0.19 0.12 0.15 0.16 0.14 0.06 0.13 0.13 0.03 0.15