1. Show that if the prior probability of a hypothesis is close to unity, then the posterior probability p0 satisfies and more exactly .

2. Watkins (1986, Section 13.3) reports that theory predicted the existence of a Z particle of mass GeV, while first experimental results showed its mass to be GeV. Find the prior and posterior odds and the Bayes ratio for the hypothesis that its mass is less than 93.0 GeV.

3. An experimental station wishes to test whether a growth hormone will increase the yield of wheat above the average value of 100 units per plot produced under currently standard conditions. Twelve plots treated with the hormone give the yields:

Find the P-value for the hypothesis under consideration.

4. In a genetic experiment, theory predicts that if two genes are on different chromosomes, then the probability of a certain event will be 3/16. In an actual trial, the event occurs 56 times in 300. Use Lindley’s method to decide whether there is enough evidence to reject the hypothesis that the genes are on the same chromosome.

5. With the data in the example in Section 3.4 on ‘The Poisson distribution’, would it be appropriate to reject the hypothesis that the true mean equalled the prior mean (i.e. that )? [Use Lindley’s method.]

6. Suppose that the standard test statistic takes the value z=2.5 and that the sample size is n = 100. How close to does a value of θ have to be for the value of the normal likelihood function at to be within 10% of its value at ?

7. Show that the Bayes factor for a test of a point null hypothesis for the normal distribution (where the prior under the alternative hypothesis is also normal) can be expanded in a power series in as

8. Suppose that x1, x2, . Show over the interval the likelihood varies by a factor of approximately

9. At the beginning of Section 4.5, we saw that under the alternative hypothesis that the predictive density for was , so that

Show that a maximum of this density considered as a function of ψ occurs when , which gives a possible value for ψ if . Hence, show that if then for any such alternative hypothesis, the Bayes factor satisfies

and deduce a bound for p0 (depending on the value of ).

10. In the situation discussed in Section 4.5, for a given P-value (so equivalently for a given z) and assuming that , at what value of n is the posterior probability of the null hypothesis a minimum.

11. Mendel (1865) reported finding 1850 angular wrinkled seeds to 5474 round or roundish in an experiment in which his theory predicted a ratio of 1:3. Use the method employed for Weldon’s dice data in Section 4.5 to test whether his theory is confirmed by the data. [However, Fisher (1936) cast some doubt on the genuineness of the data.]

12. A window is broken in forcing entry to a house. The refractive index of a piece of glass found at the scene of the crime is x, which is supposed . The refractive index of a piece of glass found on a suspect is y, which is supposed . In the process of establishing the guilt or innocence of the suspect, we are interested in investigating whether is true or not. The prior distributions of and are both where . Write

Show that, if H0 is true and , then and are independent and

By writing and , go on to show that u has an distribution and that z has an , so approximately an , distribution. Conversely, show that if H0 is false and and are assumed independent, then and are all independent and

By writing

and

show that in this case u has an , so approximately an , distribution, while z has an , so approximately an , distribution. Conclude that the Bayes factor is approximately

Suppose that the ratio of the standard deviations is 100 and that , so that the difference between x and y represents two standard deviations, and that , so that both specimens are of commonly occurring glass. Show that a classical test would reject H0 at the 5% level, but that B=9.57, so that the odds in favour of H0 are multiplied by a factor just below 10.
[This problem is due to Lindley (1977); see also Shafer (1982). Lindley comments that, ‘What the [classical] test fails to take into account is the extraordinary coincidence of x and y being so close together were the two pieces of glass truly different’.]

13. Lindley (1957) originally discussed his paradox under slightly different assumptions from those made in this book. Follow through the reasoning used in Section 4.5 with representing a uniform distribution on the interval to find the corresponding Bayes factor assuming that , so that an variable lies in this interval with very high probability. Check that your answers are unlikely to disagree with those found in Section 4.5 under the assumption that represents a normal density.

14. Express in your own words the arguments given by Jeffreys (1961, Section 5.2) in favour of a Cauchy distribution

in the problem discussed in the previous question.

15. Suppose that x has a binomial distribution of index n and parameter θ, and that it is desired to test against the alternative hypothesis :

16. Twelve observations from a normal distribution of mean θ and variance are available, of which the sample mean is 1.2 and the sample variance is 1.1. Compare the Bayes factors in favour of the null hypothesis that assuming that (a) is unknown and (b) it is known that .

17. Suppose that in testing a point null hypothesis you find a value of the usual Student’s statistic of 2.4 on 8 degrees of freedom. Would the methodology of Section 4.6 require you to ‘think again’?

18. Which entries in the table in Section 4.5 on ‘Point null hypotheses for the normal distribution’ would, according to the methodology of Section 4.6, cause you to ‘think again’?