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statistics principles and methods

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

Assume I am a geologist. What sequence of questions would you ask me to assess fractiles of my prior distribution for (r, f?), the polar coordinates(r = radial distance from here, 8 = angular direction from here) of the location that I believe is most likely to contain large reserves of natural gas?
5.25 Assume that I am a macro economist. What sequence of questions would you ask me to assess fiactiles of my prior distribution for 0, my probability that the gross domestic product will decline in both of the next two quarters?
5.26 Suppose that a data distribution for a random variable X follows the law N(0, 2), and that X,..., X, denotes a random sample from this distribution. Suppose further that for a prior distribution family you assume that (802) follows the natural conjugate distribution N(0, Ko) for some
5.27* In assessing a multivariate prior distribution by assessing a p-dimensional vector from N subjects and merging them by density estimation, we used a kernel, K(0), which we took to be normal. Suppose instead that we choose K(8) to be multivariate Student 7, so that K() {n+60) 0.5(n+p) for some
5.28* How should an expert be defined for the purposes of group prior distribution assessment?
5.29 Suppose you are trying to decide whether to buy shares in the XYZ corporation. You seek an opinion from several of the well-known stock brokerage firms in the closest major financial center. One decision rule is to follow the advice of the majority of opinions. Another might involve finding
5.30* Compare the mean assessed probability with the median assessed probability in the nuclear war application in Section 5.4. How do you interpret the substantial discrepancy?
5.31* By using the availability heuristic, explain why many people do not perceive the advent of a nuclear war in the near future as a likely event.
5.32* Let 0 denote the expected number of live chickens in California on a given day. Explain how you would formulate your prior distribution for 0.
5.33* Explain the "closure property" for natural conjugate prior distributions.
5.34* If X given follows a Poisson distribution, give a natural conjugate prior for A.
5.35* Give another method that has been proposed for assessing multivariate prior distributions. (Hint: see Kadane et al., 1980.)
5.36* Suppose X: px 1 and (X10) ~N(0.1). Give a natural conjugate prior family for 6.
5.37* Suppose X: px 1 and L(X|A) = N(0, 2), for A=-1. Give a natural conjugate prior family for A. (Assume is a positive definite matrix.)
5.38* Suppose X: p x 1, and (X | 0, A) = N(0, A'); E = A' denotes a posi- tive definite matrix. Find a natural conjugate prior family for (0, A) by noting that g(0, A) = g(1A)2(A), and assuming that a priori, var (0|) o .
6.1 Suppose that the posterior distribution of (3 is N(0, I ) with density h(0ly) o( exp(-0.5Bz).(a) Use a standard normal generator to obtain M = 5000 draws from the posterior density. Use these draws to estimate the posterior mean and posterior standard deviation of 8.@) Set up a
6.2 Repeat the preceding exercise with the tailored proposal generator 01~9 - N(0, 1.5).
6.3 Suppose that the target distribution is N(0.5, I ) truncated to the interval (0, m)with density(a) What is the normalizing constant of this target density?@) Use the expression given in Equation (6.18) to obtain M = 5000 draws from the target density. Use this sample to find the posterior mean
6.4 Suppose that the posterior density Ct1y is N(0. I), as in Problem 6.1, and that z ly, 8 - N((3, 0.5).The goal is to sample the m e t density zly.(a) %rive the target density h(z 1,~) = Ih(z Iy, fl)h(8l y) db, by direct calculation.(Hint: the marginalized density is normal).(b) Use your result
6.6 Montgomery, et al. (2000, p. 75) give n = 25 observations on the amount of time taken to service vending rnacliines by the route driver. The amount of time y (measured in minutes) is associated with the number of cases of product stocked (xl) and the distance in feet walked by the route driver
7.1* Let XI, . . . , &, denote i.i.d. observations from the distribution Adopt a natural conjugate prior distribution for /3 and find(a) the posterior density kernel for ,ll given x I , . . . , x , ~ ;(b) the large-sample posterior normal distribution for p. given xI , . . . , xN
7.2* Use the Lindley approximation of Theorem 7.2 to approximate the posterior mean in Exercise 7.1, part (a).
7.3* Give the Tiemey-Kadane approximation for the posterior variance in Exercise
7.1, part (a).
7.4* Explain the method of Gauss-Hermite quadrature, and explain how you would use it to evaluate using a three-point quadrature grid. (Hint: take PI = 3 in Table 7.2.)
7.5” Explain what is meant by “importance sampling.” When would you usc it?
7.6* Explain how you would use simulation of the posterior distribution to evaluare the posterior distribution of 03,w here y f ) I I x , = u + b x , + ~ j , i = l , . . , , n, E, - N(0, 02),th e E, are uncorrelated, (a,b ) are unknown coefficients, and the prior distribution is
7.7 If you did not use a program you wrote yourself, which commercial computer program would you use if you intended to carry out a Bayesian analysis of the coefficients in a univariate multiple regression? Why‘?
Describe how you would go about finding a Bayesian estimator for a data distribution not discussed in this chapter, such as a gamma-data distribution with known scale factor.
Find a Bayes' estimator for a univariate normal distribution with mean zero but unknown variance, under a vague prior.
Explain the use of a "highest posterior density" (HPD) credibility interval.
Explain how to find an HPD credibility interval for a bimodal posterior distribution.
What is an empirical Bayes' estimator?
What are the differences between a confidence interval and a credibility interval.
Why is admissibility a questionable property of an estimator for a Bayesian?
How does the expected loss of an empirical Bayes' estimator compare with that of a Bayesian estimator? With that of an MLE?
Give a Bayesian estimator for the parameter of an exponential distribution under a natural conjugate prior.
Suppose you have a sample of size n from N(0, 1), and 0=5(..... ). Adopt a vague prior distribution for , and give a Bayes' estimator for .
Explain the form of the Bayesian estimator for the sample mean in the case of a univariate normal data distribution with known variance, under a natural conjugate prior distribution for the sample mean.
8.12* Suppose X ~N(0, 1), and we take three independent observations, X = 2, X = 3. x3 = 4. If we adopt a vague prior for 6, then: (a) Find the posterior distribution for 8 given the data; (b) Find a two-tailed 95 percent credibility interval for 0.
Suppose that XI . . . , X I , are independent and identically distributed as N(O, 4). Test the hypothesis that tf” : 0 = 3 versus the alternative hypothesis Ho : U # 3. Assume that you have observed 2 = 5, and that your prior probabilities are P(FI,} = 0.6, and P ( H , ) = 0.4. .4ssume that your
Give the Bayes’ factor for Exercise 9.1.
Suppose the probability mass function for an obsewable variable X is given by: f ( x I A) = (e-AL’)/x!, x = 0, 1,. . . . I . =- 0. Your prior density for i. is given by: g(2} = 2e-2”. You observe X = 3. Test the hypothesis /f(, : i = I , versus the alternative hypothesis H , : i, # 1. Assume
Explain the use of the intrinsic Bayrs’factor.
Explain the difference between the Lindley and Jeffreys methods of Bayesian hypothesis testing.
What is meant by “Lindlcy’s paradox”‘?
Explain some of the problems associated with the use of p-values and significance testing.
Explain how frequentist hypothesis testing violates the likelihood principle.
Suppose XI , ,. . .&,, n = 50, are i.i.d. observations from R(0.7). You observe .t = 2. Suppose your prior distribution for H is vague. Use the Lindley hypothesis twting procedure to test HL, : 0 = 5 , versus HI : 8 f 5.
Suppose that X,.. . . ,A’,, are i.i.d. following the law h‘(0,d ).W e assunie that o2 is unknown. Form the sample mean and variance: k =:I /n X, and 2 = I / n c;(T - 2)’.Y ou observe X = 5, s2 = 37, for n = 50. You adopt the prior distributions: 0 - hr( I , I ) and g( l/02)rx ($)4e-Zui,w ith
Find the Bayes’ factor for the hypothesis testing problem in Exercise 9.10.
Suppose r denotes the number of successes in n trials, and r follows a binomial distribution with parameter p. Carry out a Bayesian test of thc hypothesis H : p = 0.2, versus the alternative A : p = 0.8, where these are the only two possibilities. Assume that r = 3, and n = 10, and that the prior
Suppose that X denotes the number of successes in n repetitions of an experiment carried each time independently of previous trials. Let 8 denote the probability of success of the experiment in a single mal. Suppose that your prior beliefs about 8 are well represented by a beta distribution with
Suppose X I 0 - N(0. I), and the prior density for 0 is N ( 0 , 1). Give the predictive density for a new observation.What is de Finetti’s theorem?
Suppose that for given 0, X, , . . . , X, are i.i.d. observations from N ( 0 , 7 ) , and that your prior density for 0 is N ( 5 , 3 ) . Find your predictive distribution for a new data point, y.
Provide conditions under which a set of exchangeable random variables might be treated approximately as if they were independent. (Hint:th ink about finite versus infinite exchangeability,)
Explain what a de Finerti transform is.
Suppose the sampling distribution for some experimental data is given by N ( N , o’), where both parameters are unknown. If the prior distribution for these parametes is in the naturai conjugate prior family (normal-inverted gamma), find the de Finetti transform.
Suppose you are carrying out an experiment in which the response data follow a binomial distribution with probability of success on a single trial=p. You search the literature and find that another researcher has carried out an analogous experiment in which there were Itr replications of her
Go to the original sources and explain the notion of “partial exchangeability.”(Hint:s ee Diaconis and Freedman, 198I ).
Among all distributions that have a finite first moment, what is the maxent distribution?
Among all distributions that have a finite second moment, what is the maxent distribution?
What can be said about de Finetti’s theorem for finitely exchangeable events? (Hint:s ee haconis, 1977.)
Suppose A@) and h(x) denote two pdfs for two continuous random variables. Give the Kullback-Leibier divergence for the first density function relative to the second. Why might this divergence hnction be a useful construct?
IIow are the mtmpy in a distribution and the iplfornzation in a distnbution related?
Find the entropy in the nonnal distribution, N ( 0 , l), relative to a uniform distribution.
Following the suggestion at the end of Section 12.4.9, assume that ,u and xi are independent a priori, assume that the ais are exchangeable, and that at - ,V(
What is the role of exchangeability in de Finetti’s theorem?
A simple regression has been carried out on a set of response variables,(I!, , . . . , yJ, and a corresponding set of explanatory variables, (xl, . . . . &).There is just one explanatory variable. The Gaussian linear regression model has been adopted. Adopt a vague prior density for the regression
10.19 Asme that X,. . . . , X, denotes an i.i.d. sample from N ( O , I), so that.k I 13 - N(13, l/n). Let Y denote a new observation from the same distribution as the Xs. Adopt a vague prior density for 6, and show that the resulting predictive density for 0,12) is N ( k 1 + l/n).
Let 6, denote the proportion of defectives in a large population of cell phone parts. A sample of s i x 15 is taken from the population and three defectives a x found. Suppose the prior distribution for I ) is beta, with density:g(0) 0: I P ( 1 - tr)9 ’ $ 0 5 tr 5 1.Adopt the loss function:Give a
Suppose you are considering the purchase of a certain stock on the New York Stock Exchange, tomorrow. You will take either action a t : buy thc stock tomorrow, or action aZ:n ot to buy the stock tomorrow. For these two actions, you have the following loss function:L(a,, 0) = 500 + 0.20, L(a,, 0) =
Explain the importance of Bayesian estimators with respect to proper prior distributions in decision theory.
You have data from a multivariate normal distribution N(O,Z), with C = 51. From- your da’a, you find that % = (1,3,4.4)’. Adopt the loss function L(f?,I)) = (I) - O)’A(O - 0). with A = 41. Your prior distribution for 0 is U ,.,A ’(+, 21), with 4 = (2,2,3,7)’. Find a Bayesian estimator for
Your data (xl, . . . , x,) arise from a distribution with probability density f’(x I 0) = 6’e-&,x > 0; Adopt a na$rral conjugate prior density for 8, and the loss function, L(8, 0) = 4 I 6, - 8 1, and iind a Bayesian estimator for 8.
Why is admissibility not usually an important criterion for a Bayesian scientist for choosing estimators which have good decision-theorctic properties?
11.8 Suppose you have data (xl, . . . , xn), allp-dimensional, from a multivariate normal distribution N(B, S), with C unknown. Adopt the loss function:where S denotes the sample covariance matrix. Adopt a natural conjugate prior distribution for ( 0 , s ) and develop the appropriate Bayes’
Adopt a zero/one loss function and estimate the mean in the context in which we have a sample ( x I , ,. . , x n ) from N(O, 5) , with X = 8, and a normal prior density 8 - N(4,9).
For the problem in Exercise 1 1.9, suppose there were no time to collect data that would bear on the problem (ignore the observation that X = 8), and a decision had to be made about 8. Find a Bayes’ estimate of 0.
11.11* For the model in Seetion 12.2.1, give:(a) The modal Bayesian estimator for p2;(b) The minimum risk Bayesian estimator with respect to a quadratic loss function for pz;(c) The minimum risk Bayesian estimator with respect to an absolute value loss function for p2.
Suppose there are 4 possible actions, ai, i = 1,2,3,4, and three possible states of nature, Oi, i = 1.2,3, in a decision problem, and we have the loss matrix:Loss a 02 03 5 04 e 5 2 6 1 02 4 3 1 2 0, 0 8 2 9. Suppose the prior probabilities for are as shown in the following table: Prior
11.13 [Research Exercise] It may be noted from Section 11.2.1 that quadratic loss 11.14 functions are unbounded in the parameter 6. This may lead to gambling gains in utility that can be infinite. This is known as the St. Petersburg Paradox. Explain. (Hint: see, for example, Shafer, 1988.)
11.14 [Research Exercise] What if there were more than one rational decision maker involved in the decision-making process. How might the decision be made? (Hint: see, for example, Kadane et al., 1999.)
Consider the univariate multiple regression model y | x = Bo + B x + + Bpxip + 24;. i = 1,....n. where y, denotes yield for a crop in year i, x, denotes the observed value of explanatory variable j during year i, u, denotes a disturbance term in year i,=(B): (p+1) x 1 is a vector of coefficients to
Consider the simple univariate regression model of Section 12.2.1. Introduce a natural conjugate prior for (B. B. ). (a) Find the joint posterior density for (B1, B2, ). (b) Find the marginal posterior density for (B1, B2). (c) Find the marginal posterior density for (o). (d) Find the marginal
For the simple univariate regression model of Section 12.2.1, suppose that instead of assuming the ups are independent N(0, 2), we assume that they are independent and that u, ~N(0, 0), where of ox, (heteroscedasti- city). Answer parts (a)-(e) of Exercise 12.2 using the interpretation of of in this
For the simple univariate regression model of Section 12.2. I suppose that instead of asuming the u,s are independent, we assume serially correlated errors with corr(u,u,) = p , i # j , var(u,) = cr2, 0 < p c 1. Adopt a vague prior for (p,, &, &), assume p is u priori independent of (PI, /I2, d),
For the model in Exercise 12.2 give a highest posterior density credibility interval for p2. of 95 percent probability.
For the model in Exercise 12.2 give:(a) The model Bayesian estimator for &;(b) The minimum-risk Bayesian estimator with respect to a quadratic loss(c) The minimum-risk Bayesian estimator with respect to an absolute error
Suppose X : p x I and L(X I 0) = N(8. I ) . Give a natural conjugate prior family for 8.
Suppose X : p x I and C(X I 8) = N ( 8 , I ) , for A E Z-'. Give a natural conjugate prior family for 2 is a positive definite matrix.
Suppose X : p x 1 and C(X I 6,Z) = N ( 8 , Z) for Z a positive definite matrix. Find a natural conjugate prior family for (d, by noting that p(B, A) =p,(8 I -4)p2(A) and A = 2-l and by assuming that a priori var(tl I Z) o( Z.
Use the example in Section 12.2.6 involving average microprocessor speeds in year i,y,, and average RAM memory in year i, x,, and test the hypothesis (using Bayesian procedures) that H: f12 = 3, versus the alternative liypothesis that A: & # 3 at the 5 percent level of credibility.(a) Use the
Do Exercise 12.1 assuming a g-prior for (B, 02) instead of the vague prior.
Provide conditions under which a set of exchangeable random variables might be treated approximately as if they were independent.
What can be said about de Finetti's theorem for finitely exchangeable events? ( H i m see Chapter 10.)
.&=me that p,a, are independent u priori, and assume that the a,s are exchangeable and thata, - N(c*, @*) for all i.(a) Find the prior density for 0,.&) Find the prior density for B.(c) Find the joint posterior density for (B, 2).
Adopt the two-way layout where i = 1 ,..., I , j = 1 ,..., J , k = 1 ,..., K, and the main effects, interaction effects, and disturbances are interpreted as in the conventional ANOVA model. Explain how you would generalize the results of this chapter for the one-way layout. (Hint: See Press and
How would you generalize the results of this chapter for the one-way layout to a one-way layout with covariates (MANOCOVA)?
Explain why the Bayesian estimators in the example involving test scores in Section 12.4.8 are so close to the MLEs.

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