Subjective And Objective Bayesian Statistics 2nd Edition S. James Press - Solutions

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Exercise 2.6 Prove relation (4).(Y∗−A∗Y∗)V−1 ∗ (Y∗−A∗Y∗)+(γˆ−γ )(Z∗V−1 ∗ Z∗)(γˆ−γ )= (Y −Zγˆ)(Y −Z ˆ γ )+(˜Y − ˜Zγˆ)D−1( ˜ w)(˜Y − ˜Zγˆ)+(γ −γˆ)(ZZ+ ˜Z D−1( ˜ w)˜Z )(γ −γˆ).
Exercise 2.10.1 Consider a regression model Y = Xβ +e, e ∼ N(0,σ 2I) and suppose that we want to predict the value of a future observation, say y0, that will be independent of Y and be distributed N(x0β ,σ 2).(a) Find the distribution of y0−x0ˆβMSE 1+x0 (XX)−1x0.(b) Find a 95%
Exercise 2.10.2 Consider the model Y = Xβ +b+e, E(e) = 0, Cov(e) =σ 2I, where b is a known vector. Show that Proposition 2.1.3 is not valid for this model by producing a linear unbiased estimate of ρXβ , say a0+aY, for which a0 = 0.Hint: Modify ρMY.
Exercise 2.10.3 Consider the model yi = β1xi1 +β2xi2 +ei, eis i.i.d. N(0,σ 2).Use the data given below to answer (a) through (d). Show your work, i.e., do not use a regression or general linear models computer program.(a) Estimate β1, β2, and σ 2.(b) Give 95% confidence intervals for β1 and
Exercise 2.10.4 Consider the model yi = β1xi1 +β2xi2 +ei, eis i.i.d. N(0,σ 2).There are 15 observations and the sum of the squared observations is YY = 3.03.Use the normal equations given below to answer parts (a) through (c).(a) Estimate β1, β2, and σ 2.(b) Give 98% confidence intervals for
Exercise 2.10.5 Consider the model yi =β0+β1xi1+β2xi2+β11x2 i1+β22x2 i2+β12xi1xi2+ei, where the predictor variables take on the following values.i 1 2 3 4 5 6 7 xi1 1 1 −1 −1 0 0 0 xi2 1 −1 1 −1 0 0 0 Show that β0, β1, β2, β11 +β22, β12 are estimable and find (nonmatrix)
Exercise 3.1 (a) Show that the F test developed in the first part of this section is equivalent to the (generalized) likelihood ratio test for the reduced versus full models, cf. Casella and Berger (2002, Subsection 8.2.1). (b) Find an F test for H0 :Xβ = Xβ0 where β0 is known. (c) Construct a
Exercise 3.2 Redo the tests in Exercise 2.2 using the theory of Section 3.2. Write down the models and explain the procedure.
Exercise 3.3 Redo the tests in Exercise 2.3 using the procedures of Section 3.2.Write down the models and explain the procedure.Hints: (a) Let A be a matrix of zeros, the generalized inverse of A, A−, can be anything at all because AA−A = A for any choice of A−. (b) There is no reason why X0
Exercise 3.4 Show that β XMMPXβ = 0 if and only if Λβ = 0.Earlier, we found the reduced model matrix X0 = XU directly and then, forΛ = PX, we showed that C(MP) = C(X0)⊥C(X), which led to the numerator sum of squares. An alternative derivation of the test arrives at C(M−M0)
Exercise 3.5 Show that a necessary and sufficient condition for ρ1 Xβ = 0 andρ2Xβ = 0 to determine orthogonal constraints on the model is that ρ1Mρ2 = 0.
Exercise 3.6 In testing a reduced model Y = X0γ +e against a full model Y =Xβ +e, what linear parametric function of the parameters is being tested?
Exercise 3.7 Show that ρMY = ρ[Mρ(ρMρ)−ρM]Y so that to estimateρXβ , one only needs the perpendicular projection of Y onto C(Mρ).
Exercise 3.8 Show that Y(A−A0)V−1(A−A0)Y/[r(X)−r(X0)] is the appropriate numerator mean square for testing model (4) against model (2).
Exercise 3.9 Show that Y(A−A0)V−1(A−A0)Y equals the difference in the SSEs for models (3) and (1).
Exercise 3.9.1 Consider the model yi =β0+β1xi1+β2xi2+ei, eis i.i.d. N(0,σ 2).Use the data given below to answer (a) and (b).Obs. 1 2 3 4 5 6 y −2 7 2 5 8 −1 x1 4 −1 2 0 −2 3 x2 2 −3 0 −2 −4 1(a) Find SSR(X1,X2|J) = R(β1,β2|β0).(b) Are β0, β1, and β2 estimable?
Exercise 3.9.2 For a standard linear model, find the form of the generalized likelihood ratio test of H0 : σ 2 =σ 2 0 versus HA : σ 2 =σ 2 0 in terms of rejecting H0 when some function of SSE/σ 2 0 is small. Show that the test makes sense in that it rejects for both large and small values of
Exercise 3.9.3 Consider a set of seemingly unrelated regression equations Yi = Xiβi+ei, ei ∼ N 0,σ 2I, i = 1, . . . , r, where Xi is an ni × p matrix and the eis are independent. Find the test for H0 : β1 = · · · =βr.
Exercise 3.9.4 What happens to the test of Λβ = d if Λβ has no estimable part?
Exercise 3.9.5 Consider the model Y = Xβ +e, E(e) = 0, Cov(e) =σ 2I, (1)with the additional restrictionΛβ = d, where d =Λb for some (known) vector b and Λ = PX. Model (1) with the additional restriction is equivalent to the model(Y −Xb) = (M−MMP)γ +e. (2)If the parameterization of
Exercise 3.9.6 Discuss how the testing procedures from this chapter would change if you actually knew the variance σ 2.
Exercise 4.1 To develop some facility with this notation, let Tr = [ti j], where ti j =δir − Nr nfor r = 1, . . . ,t. Find Tr Tr, Tr Ts for s = r, and JTr.A very important application of this notation is in characterizing the vector MY.As discussed in Section 3.1, the vector MY is the base
Exercise 4.2 Show that MY = [ti j], where ti j = ¯ yi·and MαY = [ui j], where ui j = ¯ yi·− ¯ y·· .
Exercise 4.3 Verify that the estimate of μ +αi is ¯ yi· and that the algebraic formulas for the sums of squares in the ANOVA table are correct.Hint: To find, for example, Y (M−[1/n]Jn n )Y =YMαY, use Exercise 4.2 to get MαY and recall that YMαY = [MαY] [MαY].
Exercise 4.4 Verify that the formulas for expected mean squares in the ANOVA table are correct.
Exercise 4.5 Show that α1 =α2 = · · · =αt if and only if all contrasts are zero.
Exercise 4.7 Suppose N1 = N2 = · · · = Nt ≡ N. Rewrite the ANOVA table incorporating any simplifications due to this assumption.
Exercise 4.8 If N1 = N2 = · · · = Nt ≡ N, show that two contrasts λ 1β and λ 2βare orthogonal if and only if λ 1λ2 = 0.
Exercise 4.9 Find the least squares estimates of μ, α1, and αt using the side condition α1 = 0.
Exercise 4.10 Using ρ as defined by (1) and X as defined in Section 1, especially(4.1.1), show that(a) ρX =λ , where λ = (0,λ1, . . . ,λt ).(b) ρ ∈ C(X).
Exercise 4.3.1 An experiment was conducted to see which of four brands of blue jeans were most resistant to wearing out as a result of students kneeling before their linear models instructor begging for additional test points. In a class of 32 students, 8 students were randomly assigned to each
Exercise 4.3.2 After the final exam of spring quarter, 30 of the subjects of the previous experiment decided to test the sturdiness of 3 brands of sport coats and 2 brands of shirts. In this study, sturdiness was measured as the length of time before tearing when the instructor was hung by his
1.1 Give examples of subjectivity in medical diagnosis by a physician; in the introduction of “evidence” by an attorney in civil or criminal trials; in studies of the human genome by molecular biologists; in portfolio analysis of financial securities by certified financial analysts; in the
1.2” By going to the original reference sources, explain who Stigler believes actually wrote Bayes’ theorem, and also, explain what Stigler believes Bayes’ theorem was really saying.
1.3 In the observing of a desired experimental effect example of Section I .2 of this chapter, can you think of other interpretations of the same data?
1.4* Why is the book Ars Coqjectundi by James Bernoulli of particular interest to people interested in Bayesian statistical inference?
1.S In the observing of a desired experimental effect example of Section 1.2 of this chapter, which interpretation of the data seems most appropriate to you? Why?
1.6 Tn an experiment you were doing, or one that you were analysing, how would you decide whether certain observations were too large or too small to be considered as part of the basic data set?
t.7* Distinguish between prior and posterior distributions.
1.8* The methodological approaches to science of some of the most famous scientists in history (Aristotle, Marie Curie, Charles Darwin, Albert Einstcin, Michael Faraday, Sigmund Freud, Galileo Galilei, William Harvey, Antoine Lavoisier, Isaac Newton, Louis Pasteur, and Alexander von Humboldt) were
1.9* Explain the meaning of the statement, “Most scientific inference is already partly subjective, and it always has been.”
2.1 What is meant by “inverse probability”?
Distinguish between finite and countable additivity, and explain the impor- tance of the difference.
Explain the importance of the Rnyi axiom system of probability for Bayesian inference.
What is meant by "coherence"? Give an example (not from the text) of incoherence.
Explain what you mean by the statement that YOUR subjective probability, or YOUR degree of belief, that there will be measurable rain tomorrow in Riverside, CA, is 70 percent.
What is the meaning of the (von Mises) long-run frequency approach to probability?
By going to the original reference sources, explain who Stigler believes actually wrote Bayes' theorem, and also explain what Bayes' theorem was really saying about how to choose the prior distribution.
Explain why a frequency interpretation of probability cannot possibly interpret the statement in Exercise 2.5.
Why is the book Ars Conjectandi by James Bernoulli of particular interest to people interested in Bayesian statistical inference?
2.10* What is meant by "Savage's axioms," and what is the context to which this axiom system applies?
2.11* What is meant by utility theory?
2.12* Explain the assertion "Mathematical probability is merely a special case of subjective probability."
2.13* What is meant by a "normative theory of inference and decision-making behavior" as compared with an empirical theory?
3.1" Wiat is the likelihood principle and why is it important for Bayesian statistical science'?
3.2 How would you decide whether to use a normal likelihood function or a Student t likelihood fiinction for your data intended for a Bayesian analysis, assuming the data were likely 10 be roughly unimodal and symmetric? (Him see methods for comparing empirical distributions in other texts such as
3.3 Which computer program do you prefer for analysing data distributions'? Why'?
3.4 It is important to review your familiarity with histograms. The weights of40 male students at the Lhiversity of California, Riverside, were recorded to the nearest pound as shown below.(a) Using a statistical computer program such as SAS, SPSS, STATA, or MMTAB, construct an appropriate
3.5 Give three examples of classical statistical procedures that depend upon data that are not taken in a given experiment, but might have been taken. Such procedures violate the likelihood principle. (Hint: often, such procedures involve averaging over all possible values of the observables.)
3.6 Explain with three examples, the proposition: ”The likelihood function need not be a probability.“
3.7 Consider the average value of the function G(B) with respect to the posterior distribution of the unobservable 8, based upon a set of continuous data:x , , . . . ,x,. The average G is given by:where h denotes the posterior pdf, L denotes the likelihood function, and g Cenotes the prior pdf.
3.8* Explain why unbiasedness is a property that violates the spirit of the likelihood principle.
4.1” Distinguish between prior and posterior distributions.
4.2 You go to the doctor for a regular medical checkup and the doctor suggests a laboratory test for a particular form of cancer. Assume (unrealistically) that the doctor has available only the one test. We assume that she has not yet discovered any reason to believe, a priori, that you have the
4.3 In Exercise 4.1, suppose that when the doctor had asked you about your medical history, you did not know that there had been a strong incidence of cancer in your family. On further investigation, suppose that you found that three out of four of your grandparents had died early of the same type
4.4 In Exercise 4.2, find the value of your prior probability that yields a posterior probability of SO percent for having the disease.
4.5 The value of a second opinion: Consider the context of Exercise 4.1. Now suppose that because you feel that cancer is such a serious disease, you decide to seek a second opinion about your problem. You look for another physician, perhaps an oncologist, who will suggest either that a new
4.6 Tom Lee has been accused by the police of fathering Mary Jones’ child.Samples are taken of Tom Lee’s DNA and that of the child, and compared.Suppose that the probability of a DNA match given that Mr. Lee is not the true father is I percent, and the probability of a match given that Mr. Lee
4.7 As a particle physicist, you carry out an experiment involving a comparison between the physics of matter and the physics of antimatter. Is there a particular kind of symmetry or not? Suppose that your prior probability for finding this asymmetry is 50 percent. Moreover, suppose the experiment
4.8 Example 4.2 concerned quality control in manufacturing. Suppose instead of the sample selected in that example, the RAMM Company had actually selected a sample for testing with n = 100, and that five defectives were found in the sample. Calculate your posterior distribution for assuming the
4.9 Example 4.3 involved making an inference about which section of a large course a student came from. Adopt the same probabilities as in that example, but now suppose that 10 students were selected from a particular section (all from the same section) and that their average score was 72 percent.
4.10 In Example 4.5 we were concerned with the efficacy of a salesmanship course. Suppose in that example that your prior for followed the distribution N (4, 16) instead of the prior adopted in the example. For the same data as in that example, evaluate your posterior probability distribution for
4.11 Find the Bayes factor in Exercise 4.10.
4.12 Find the posterior odds against Tom Lee being the father of Mary Jones' child in Exercise 4.6.
4.13 Find the mean and variance of the posterior distribution in Example 4.2.
4.14 Find the means and variances of the posterior distributions given in Equation (4.15) for the cases of normal data with unknown mean and unknown variance, for: (a) A vague prior family of distributions; (b) A natural conjugate prior family of distributions.
4.15 Let XX, denote independent and identically distributed observations from N(0, 2). Suppose o is known, but you can assess a mixture prior for of the form: g(0) = ag(0) + (1 a)g(0), where for 0 1, 8,(0) = == 1 ex i = 1,2,
Suppose X denotes an observation that follows the exponential distribution with density given by: ,f(x 1 tl) = 8 exp(-6x). 0 c x, tl c 00. Give a vague prior density for 0.
Suppose that x , , . . . . .r, denotes i.i.d. observations h m the gamma distribution with density: f(x } 0, T) o( ,I?-’ exp(-rx), 00 i 0 < 00. 0 < t < 00.Give a vague prior density for ( 8 , ~ ) .
For Exercise 5. I , give a natural conjugate prior density for tl.
Suppose XI, . . . , X,, denotes an i.i.d. sample of p-vectors h m a N(B, A)distribution, where A denotes a knownp x p covariance matrix. Give a vague prior density for (8. A).
Give a natural conjugate prior density for the parameters in Exercise 5.4,
Give the Jeffreys’ prior density for the parameter in Exercise 5.1.
Give the Jefieys’ prior density for the parameters in Exercise 5.2.
Distinguish between objective and subjective prior distributions.
5.9* What i s meant by a public policy prior distribution?
5.10 What is a “reference prior” distribution?
5.11 How would you handle a problem in which the posterior distribution for an unknown quantity turned out to be improper?
5.12 Suppose the probability mass fimction for a random variable r is given by:(a) Find a vague prior density for p.@) Find the Jefkeys’ prior density for p.
5.13 For the distribution in Exercise 5.12, find a natural conjugate prior density for P .
5.14 What is a g-prior chstribution and when might you want to use one?
5.15 What is a mixture prior distribution and when might you want to use one?
5.16* What i s Laplace’s Principle of Insuficient Reason? When might you want to use such a principle?
5.17 (a) What are the advantages of adopting objective prior distributions?(b) What are the advantages of adopting subjective prior distributions?
5.18 What is an exponential power distribution family of prior densities?
5.19 What is stable estimution?
5.20 (a) What is a dafa-based prior distribution‘?5.21(b) What is meant by sample splitting in developing prior distributions‘?
Assume I am a geneticist. What sequence of questions would you ask me to assess tiactiles of my prior distribution for 8, the number of years from now when I believe it will be possible through genetic engineering to affect the intelligence of an unborn child?
Assume that I am a male sociologist. What sequence of questions would you ask me to asses fractiles of my prior distribution for 8. the proportion of time that husbands of working couples spend delivering and picking up their children from day care?
Assume that I am a female sociologist. What sequence of questions would you ask me to assess fractiles of my prior distribution for 0, the proportion of time that wives of worIung couples spend delivering and picking up their children from day care?

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Subjective And Objective Bayesian Statistics 2nd Edition S. James Press - Solutions

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