Questions and Answers of Bayesian Statistics An Introduction

Suppose there is a loss function ℒ (0, a) defined byOn the basis of an observation x you have to take action a0, a1 or a2. For what values of the posterior probabilities p0 and p1 of the hypotheses
A child is given an intelligence test. We assume that the test result x is N(θ, 100) where θ is the true intelligence quotient of the child, as measured by the test (in other words, if the child
In Section 7.8, a point estimator δn, for the current value λ of the parameter of a Poisson distribution was found. Adapt the argument to deal with the case where the underlying distribution is
The sample correlation coefficient between length and weight of a species of frog was determined at each of a number of sites. The results were as follows: Find an interval in which you are 95% sure
Three groups of children were given two tests. The numbers of children and the sample correlation coefficients between the two test scores in each group were as follows:Is there any evidence that the
Suppose you have sample correlation coefficients r1, r2, ...,rk on the basis of sample sizes n1, n2, ...,nk, Give a 95% posterior confidence interval for ζ = tanh-1 p.
Show that in the special case where the sample correlation coefficient r = 0 and the prior takes the special form p(p) ∝ (1 - p2)k/2 the variablehas a Student's t distribution on k + n + 1 degrees
By writing and using repeated integration by parts, show that the posterior distribution of p can be expressed as a finite series involving powers ofand Student's t integrals.
By substitutingin the formfor the posterior density of the correlation coefficient and then expandingas a power series in u, show that the integral can be expressed as a series of beta functions.
Fill in the details of the derivation of the prior from Jeffreys' rule as outlined at the end of Section 6.1. p(0, V, p) x (v)-¹ (1-²)-3/2
The following data consist of the estimated gestational ages (in weeks) and weights (in grammes) of 12 female babies: Give an interval in which you are 90% sure that the gestational age of a
Show directly from the definitions that, in the notation of Section 6.3, See = {yi - a - b(x¡ — x)}².
Observations y; for i = -m, -m + 1, ... , m are available which satisfy the regression model yi~ N(a + Bu; + yvi, d), where u = i and v₁ = i² — m(m + 1). Adopting the reference prior p(a, b,
Fisher (1925b, Section 41) quotes an experiment on the accuracy of counting soil bacteria. In it, a soil sample was divided into four parallel samples, and from each of theses after dilution seven
In the case of the data on scab disease quoted in Section 6.5, find a contrast measuring the effect of the season in which sulphur is applied and give an appropriate HDR for this contrast. 6.5.5
The data below [from Wishart and Sanders (1955, Table 5.6)] represent the weight of green produce in pounds made on an old pasture. There were three main treatments, including a control (O)
Express the two-way layout as a particular case of the general linear model.
Show that the matrix A+ = (ATA)-1 AT which arises in the theory of the general linear model is a generalized inverse of the (usually non-square) matrix A in that (a) AAA = A (b) A+ AA+ A+ (c)
Express the bivariate linear regression model in terms of the original parameters n = (n0, n1)T and the matrix A0 and use the general linear model to find the posterior distribution of n.
Two analysts measure the percentage of ammonia in a chemical process over 9 days and find the following discrepancies between their results:Investigate the mean discrepancy θ between their results
With the same data as in the previous question, test the hypothesis that there is no discrepancy between the two analysts.Previous questionTwo analysts measure the percentage of ammonia in a chemical
Suppose that you have grounds for believing that observations xi, yi for i = 1, 2 ... , n are such that xi ~ N(θ, ϕi ) and also yi ~ N(θ,ϕi ) but that you are not prepared to assume that the
How much difference would it make to the analysis of the data in Section 5.1 on rat diet if we took ω = ½ (ϕ + ψ) instead of ω = ϕ + ψ.
Two analysts in the same laboratory made repeated determinations of the percentage of fibre in soya cotton cake, the results being as shown:Investigate the mean discrepancy θ between their mean
A random sample x = (x1,x2,...,xm) is available from an N(λ,ϕ) distribution and a second independent random sample y (y1, y2,..,Yn) is available from an N(µ,2ϕ) distribution. Obtain, under the
Verify the formula for S1 given towards the end of Section 5.2.Formula 8- (x - y) ' (m₁²¹ + n₁¹' ) ² ² $1
The following data consists of the lengths in mm of cuckoo's eggs found in nests belonging to the dunnock and to the reed warbler:Investigate the difference θ between these lengths without making
Show that if m = n then the expression f12/f2 h in Patil's approximation reduces to 4(m – 5) 3 + cos 40
Suppose that Tx Ty and θ are defined as in Section 5.3 and thatShow that the transformation from (Tx, Ty) to (T, U) has unit Jacobian and hence show that the density of T satisfies T = T, sin 0 Ty
Show that if x ~ Fvi,v2 then VIX V₂ + V₁x ~ Be (v₁, 1₂).
Measurement errors when using two different instruments are more or less symmetrically distributed and are believed to be reasonably well approximated by a normal distribution. Ten measurements with
Repeat the analysis of Di Raimondo's data in Section 5.6 on the effects of penicillin of mice, this time assuming that you have prior knowledge worth about six observations in each case suggesting
The undermentioned table [quoted from Jeffreys (1961, Section 5.1)] gives the relationship between grammatical gender in Welsh and psychoanalytical symbolism according to Freud:Find the posterior
Show that if π ≌ p then the log odds-ratio is such that A-A'=(-p)/{л(1-л)}.
Suppose that x ~ P(8.5), that is x is Poisson of mean 8.5, and y ~ P(11.0). What is the approximate distribution of x - y?
Show that if the prior probability π0 of a hypothesis is close to unity, then the posterior probability p0 satisfies 1- p0 ≅ (1 - π0 )B-1 and more exactly 1- Po ≅ (1 - π0)B-1 + (1 -
Watkins (1986, Section 13.3) reports that theory predicted the existence of a Z particle of mass 93.3 ≠ 0.9 GeV, while first experimental results showed its mass to be 93.0 ≠ 1.8 GeV. Find the
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
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.
Suppose that the standard test statistic z = (X̅- θ0)/√(ϕ/n) takes the value z = 2.5 and that the sample size is n = 100. How close to θ0 does a value of θhave to be for the value of the
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 λ
At the beginning of Section 4.5, we saw that under the alternative hypothesis that θ ~ N(θ,ψ) the predictive density for X̅ was N(θ0, ψ + ϕ), so thatShow that a maximum of this density
In the situation discussed in Section 4.5, for a given P-value (so equivalently for a given z) and assuming that ϕ = ψ, at what value of n is the posterior probability of the null hypothesis a
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
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 N(θ1,ϕ). The refractive index of a piece of glass
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 P1 (θ) representing a uniform
Express in your own words the arguments given by Jeffreys (1961, Section 5.2) in favour of a Cauchy distributionin the problem discussed in the previous question.Previous questionLindley (1957)
Suppose that x has a binomial distribution B(n, θ) of index n and parameter 0, and that it is desired to test H0 : θ = θ0 against the alternative hypothesis H1: θ ≠ θ0:(a) Find lower bounds on
Twelve observations from a normal distribution of mean θ and variance ϕ 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
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'?
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'?Table in Section 4.5
Laplace claimed that the probability that an event which has occurred n times, and has not hitherto failed, will occur again is (n + 1)/(n + 2) [see Laplace (1774)], which is sometimes known as
Find a suitable interval of 90% posterior probability to quote in a case when your posterior distribution for an unknown parameter π is Be(20, 12), and compare this interval with similar intervals
Suppose that your prior beliefs about the probability π of success in Bernoulli trials have mean 1/3 and variance 1/32. Give a 95% posterior HDR for π given that you have observed 8 successes in 20
Suppose that you have a prior distribution for the probability π of success in a certain kind of gambling game which has mean 0.4, and that you regard your prior information as equivalent to 12
Suppose that you are interested in the proportion of females in a certain organization and that as a first step in your investigation you intend to find out the sex of the first 11 members on the
Show that if g(x) = sinh-1 √(x) thenDeduce that if x ~ NB(n,π) has a negative binomial distribution of index n and parameter π and z = g(x) then Ez ≅ sinh-1 √(x) and Vz ≅ 1/4n. What does
The following data were collected by von Bortkiewicz (1898) on the number of men killed by horses in certain Prussian army corps in twenty years, the unit being one army corps for one year:Give an
Recalculate the answer to the previous question assuming that you had a prior distribution for λ of mean 0.66 and standard deviation 0.115.Previous questionThe following data were collected by von
Find the Jeffreys prior for the parameter a of the Maxwell distribution and find a transformation of this parameter in which the corresponding prior is uniform. p(x|a) = 2 √/²=7a²³/²x²
Use the two-dimensional version of Jeffreys' rule to determine a prior for the trinomial distribution(cf. Exercise 15 on Chapter 2).Exercise 15Suppose that the vector x = (x, y, z) has a trinomial
Suppose that x has a pareto distributionPa(ξ, y ), where ξ is known but y is unknown, that isUse Jeffreys' rule to find a suitable reference prior for y. p(x|y) = yx-¹(5,0)(x).
Consider a uniform distribution on the interval (a, β), where the values of a and β are unknown, and suppose that the joint distribution of a and β is a bilateral bivariate Pareto distribution
Suppose that observations x1, x2, ..., Xn are available from a densityExplain how you would make inferences about the parameter θ using a conjugate prior. p(x|0) = (c + 1)0−(c+¹) xc (0 < x < 0).
What could you conclude if you observed two tramcars numbered, say, 71 and 100?
We sometimes investigate distributions on a circle. Find a Haar prior for a location parameter on the circle (such as µ, in the case of von Mises' distribution).
Suppose that the prior distribution p(µ,σ) for the parameters µ, and σ of a Cauchy distributionis uniform in µ and σ, and that two observations x1 = 2 and x2 = 6 are available from this
Show that if the log-likelihood L(θ I x) is a concave function of θ for each scalar x (that is, L"(θ l x) ≤ ( 0 for all θ), then the likelihood function L(θ l x) fore given an n-sample x =
Show that if an experiment consists of two observations, then the total information it provides is the information provided by one observation plus the mean amount provided by the second given the
Find the entropy H{p(θ)} of a (negative) exponential distribution with density p(θ) β-1 exp(-θ/β).
Prove the theorem quoted without proof in Section 2.4. Theorem 2.1. A random sample x = (x₁,x2,...,xn) of size n is taken from N(0, 0) where is known. Suppose that there exist positive constants a,
Suppose that k ~ B(n, π). Find the standardized likelihood as a function of π for given k. Which of the distributions listed in Appendix A does this represent?Appendix A.Some facts are given about
Suppose we are given the 12 observations from a normal distribution: and we are told that the variance ϕ = 1. Find a 90% HDR for the posterior distribution of the mean assuming the usual reference
With the same data as in the previous question, what is the predictive distribution for a possible future observation x?Previous questionSuppose we are given the 12 observations from a normal
A random sample of size n is to be taken from an N(θ,ϕ) distribution where ϕ is known. How large must n be to reduce the posterior variance of ϕ to the fraction ϕ/k of its original value (where
Your prior beliefs about a quantity θ are such thatA random sample of size 25 is taken from an N(θ, 1) distribution and the mean of the observations is observed to be 0.33. Find a 95% HDR for θ.
Suppose that you have prior beliefs about an unknown quantity θ which can be approximated by an N(λ,ϕ) distribution, while my beliefs can be approximated by an N(μ,ψ) distribution. Suppose
Under what circumstances can a likelihood arising from a distribution in the exponential family be expressed in data translated form?
Suppose that you are interested in investigating how variable the performance of schoolchildren on a new mathematics test, and that you begin by trying this test out on children in 12 similar
The following are the dried weights of a number of plants (in g) from a batch of seeds:Give 90% HDRs for the mean and variance of the population from which they come. 4.17, 5.58, 5.18, 6.11, 4.50,
Find a sufficient statistic for μ, given an n-sample x = (x1, x2 ,. . . , xn) from the exponential distributionwhere the parameter μ, can take any value in p(x|u) = μ΄' exp(-x/μ) (0 < x < )
Find a (two-dimensional) sufficient statistic for (a, β) given an n-sample x = (x1, x2,...,xn) from the two-parameter gamma distributionwhere the parameters a and β can take any values in p(x|a,
Find a family of conjugate priors for the likelihood 1(β|x) = p(x|a,β), where p(x |a, β) is as in the previous question, but a is known.Previous questionFind a (two-dimensional) sufficient
Show that the tangent of a random angle (i.e. one which is uniformly distributed on [0, 2π)) has a Cauchy distribution C(0,1).
Suppose that the vector x = (x, y, z) has a trinomial distribution depending on the index n and the parameter π = (π , p, σ), where π + p + σ = 1, that isShow that this distribution is in the
Suppose that the results of a certain test are known, on the basis of general theory, to be normally distributed about the same mean μ with the same variance ϕ, neither of which is known. Suppose
Suppose that your prior for θ is a 2/3 : 1/3 mixture of N(0, 1) and N(1, 1) and that a single observation x ~ N(θ, 1) turns out to equal 2. What is your posterior probability that θ > 1?
A random variable X is said to have a chi-squared distribution on v degrees of freedom if it has the same distribution as whcre Z1, Z2 , ... , Zv are independent standard normal variates. Use the
A card came is played with 52 cards divided equally between four players, North, South, East and West, all arrangements being equally likely. Thirteen of the cards are referred to as trumps. If you
(a) Under what circumstances is an event A independent of itself? (b) By considering events concerned with independent tosses of a r ~ d die and a blue die, or otherwise. give examples of events A,
Whether certain mice are black or brown depends on a pair of genes, each of which is either B or b. If both members of the pair are alike, the mouse is said to be homozygous, and if they are
The example on Bayes' Theorem in Section 1.2 concerning the biology of twins was based on the assumption that births of boys arid girls occur equally frequently, and yet it has been known for a very
Suppose a red and a blue die are tossed. Let x be the sum of the number showing on the red die and twice the number showing on the blue die. Find the density function and the distribution function of
Suppose that k ~ B(n,π) where n is large and π is small but nπ = λ has an intermediate value. Use the exponential limit ( 1 + x )n → ex to show that P(k = 0) ≌ e-λ and P(k = I) ≌
Suppose that m and n have independent Poisson distributions of means λ and µ, respectively (see question 6) and that k = m + n. (b) Generalize by showing that k has a Poisson distribution of mean
Modify the formula for the density of a one-to-one function g(x) of a random variable x to find an expression for the density of x2 in terms of that of x, in both the continuous and discrete case.
Suppose that x1, x2 , ... , Xn are independently and all have the same continuous distribution, with density f(x) and distribution function F(x). Find the distribution functions ofin terms of F(x),
Suppose that u and v are independently uniformly distributed on the interval [0, 1], so that the divide the interval into three sub-intervals. Find the joint density function of the lengths of the
Show that two continuous random variables x and y are independent (i.e. p(x, y) = p(x )p(y) for all x and y) if and only if their joint distribution function F(x, y) satisfies F(x, y) = F(x) F(y) for
Suppose that the random variable x has a negative binomial distribution NB(n, π) of index n and parameter π , so thatFind the mean and variance of x and check that your answer agrees with that

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