Principles Of Uncertainty 2nd Edition Joseph B. Kadane - Solutions

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1. Vocabulary. State in your own words the meaning of:(a) Gamma function(b) Gamma distribution(c) Exponential distribution(d) Chi-square distribution
2. Find the constant for the distribution in (8.36).
3. Consider the density e−x, x > 0 of the exponential distribution.(a) Find its moment generating function.(b) Find its nth moment.(c) Conclude that Γ(n + 1) = n!
4. Suppose X has a standard normal distribution. Then E(X) = 0 and Var(X) = 1, so E(X2) = 1. So∞ 1 1 = √ x2 2 ∞−x 2 e 2/2dx = √ x e−x2/2dx.2π −∞ 2π 0 Substitute y = x2/2 in this integral to evaluate Γ(3/2).
1. Write down the constant omitted from (8.74) to make (8.74) the conditional density of m and R given X.
1. Write down the omitted constant in (8.78).
2. Suppose (p1, . . . , pk) have a Dirichlet distribution with parameters (α1, . . . , αk). Find the covariance between pi and pj .
1. For each of the following, display the likelihood in the form of (8.82) and the conjugate prior in the form of (8.83):(a) the multivariate normal case with known precision (section 8.2).(b) the normal linear model (section 8.3) with known precision.(c) the univariate normal with known mean and
2. Show that the density f(x|θ) = 1/θ, 0 < x < θ, and 0 otherwise, is not a member of the exponential family
1. Vocabulary. State in your own words the meaning of:(a) fixed vs. random effects(b) mixed effect model(c) empirical Bayes(d) fully hierarchical model
2. Think of your own example of a hierarchical structure to model some phenomenon of interest to you.
1. Vocabulary. Explain in your own words what missing data are.
2. Choose one of the examples in section 9.2.1. Choose a simple preliminary model for the problem.
1. Vocabulary. State in your own words the meaning of:(a) directed(b) acyclic(c) DAG(d) Bayesian network
2. Choose one of the examples in section 9.2. Draw a DAG for it. Explain the assumptions implicit in the DAG you drew.
1. State in your own words a definition of(a) trapezoid rule(b) simulation(c) Monte Carlo method(d) pseudo-random number generator(e) rejection sampling(f) importance sampling(g) control variate(h) antithetic variate(i) stratification(j) conditional means
2. Consider 1 x2dx 0(a) Compute�it analytically.(b) Using evaluation at 10 points, approximate it using the trapezoid rule, in R.
3. Suppose that m the stratification variance σ2 i=1 i /ni is to be minimized subject to the m constraint i=1 ni = n.(a) Let the minimization be taken over all real positive numbers ni, not just the integers.the m Show that optimal ni’s satisfy ni = nσi/i=1 σi , i = 1, . . . , m. [Hint: Use a
1 (a) What proposal distribution q(x, y) for a Metropolis-Hastings algorithm leads to this transition matrix?
2 (b) Does this specification satisfy rthe hypothesis after Lemma 10.3.3? Why or why not?(c) Show that both π 1 and 0
1 = π2 = are stationary probability vectors for this 0 1 transition matrix.
3 (d) What other assumption of the theorem rdoes this transition matrix fail to satisfy?
1. State in your own words the meaning of(a) Gibbs Step(b) Gibbs Sampler(c) random walk sampler?
2. Make a sampler in pseudo-code exemplifying each of the three special cases mentioned in problem 1. Give examples of when it would be useful and efficient to use each, and explain why.
1. State in your own words the meaning of(a) mixing of a Markov Chain(b) burn-in(c) equilibrium(d) adaptive algorithms(e) importance sampling reweighting of chains
2. Reconsider the algorithm you wrote to answer question 2(c) in section 10.3.3. Make up some data, and run your algorithm on a computer. How much burn-in do you allow for, and why? How do you decide whether the output of your algorithm has converged?
1. State in your own words the meaning of(a) reversible jump algorithm(b) variable dimensions in the parameter space
2. Give some examples of when variable dimensions would be important.
3. Explain why the Jacobian appears in the reversible jump algorithm.
1. State in your own words the principle ideas of variational inference.
2. What are the critical assumptions of variational inference? When are they more or less likely to be a practical issue?
3. Consider the model in Section 8.5 (univariate normal distribution with uncertain mean and precision). In this case the posterior is available analytically, and the posterior on the two parameters are not independent. Apply variational inference, and explore how much error is introduced through
1. Explain backward induction.
2. Try to find optimal strategies by considering first Jane’s first move, then Dick’s move, and finally Jane’s second move, the third move in the game. Is this simpler or more difficult?Why? You may assume that the loss functions LJ and LD represent the players’ losses, that they are both
3. Suppose x = 1, y = −1, s0 = 0, r = 2 and q = 3. Find the optimal strategies, again under the assumptions specified in problem 2.
4. Investigate the behavior of u∗, v∗ and w∗ as r → ∞.
5. Prove (11.3).
6. Prove (11.6).
7. Prove (11.9).
8. Choose what you consider to be a reasonable choice for PD(w | u, do(v)) other than the choice of w∗ with probability 1, and minimize (11.4) with respect to your choice.
9. Construct a contract that is better for both parties than they can do for themselves by playing the game. Make whatever assumptions you need, for example losses (11.1) and(11.2), and special values of q and r. Is a side-payment necessary to make your proposed contract better for both? If so,
1. Suppose someone offers to buy from you or sell to you for 40 cents a ticket that pays $1 if you snap your fingers in the next minute. Describe two ways in which you could make that person a sure loser.
2. Examine the behavior of the strategies (11.32), (11.38) and (11.43) as k → ∞.
3. Prove (11.18).
1. Suppose it happens that Dan and Edward have the same distribution, and in particularµd = µe and τd = τe.(a) What is r? What is r˜?(b) What is g(y)?(c) What is y(r˜), the largest root of g(y) = 0?(d) Apply the theorem. What is the optimal sample size?(e) Give an intuitive explanation of
2. Consider the case n = 0.(a) What will Edward’s estimate be?(b) What will Dan’s expected loss be? Find this by evaluating R(0).(c) Explain your answer to (b).
3. Consider the case in which τe → 0.(a) What is r?(b) How does the analysis compare to that found in exercise 1 above?
1. Vocabulary: State in your own words the meaning of(a) constant-sum game(b) minimax strategy(c) maximin strategy(d) value of a zero-sum, 2 person game(e) dominance(f) n-person game
2. Why might a good prescriptive theory of how to play a game require a good descriptive theory of the opponents’ play?
3. Suppose, in the single iteration Prisoner’s Dilemma (section 11.6.4), that the prisoner’s loss function is monotone but not necessarily linear in the amount of jail time they serve. This means that each prefers less jail time to more jail time. Show that under this assumption the same result
4. Consider the following modification of the iterated Prisoner’s Dilemma problem. Instead of punishments, years in jail, suppose the problem is phrased in terms of rewards. If both cooperate, they each get 3 points. If both defect, they each get 1 point. If one cooperates and the other defects,
5. Recall that a median of an uncertain quantity X is a number m such that P{X ≤ m} ≥1/2 and P{X ≥ m} ≥ 1/2.
1. State in your own words what is meant by(a) weak Pareto condition(b) strong Pareto condition(c) Bayesian group(d) autocratic compromise
2. Show that if one Bayesian is indifferent, say U1(d, θ) = b for all dED and θEΩ, then the Pareto condition is vacuous regardless of whether p1(θ) = p2(θ) for all θEΩ, or whether p1(θ) = p2(θ).
3. Show that, in Case 2(b), if p1(θ) = p2(θ) and UD(d, θ) = aUJ (d, θ) +b, where a > 0, then p(θ) = αpD(θ)+(1−α)pJ (θ)(0 ≤ α ≤ 1) and U(d, θ) = UD(d, θ) satisfy the Pareto condition.
4. Verify that (11.73) implies (11.90).
5. Verify that (11.76) implies (11.90).
6. Show that r = −1, s = −1 implies p(E) = pD(E) and U(c) = UD(c).
7. Show that r = 1, s = 1 implies p(E) = pJ (E) and U(c) = UJ (c).
2. Suppose your prior on µ (which is two-dimensional) is normal, with mean (2, 2) and precision matrix I, and suppose you observe a normal random variable with mean µ, and precision matrix ( 2 0 0 2 ). Suppose the observation is (3, 300).(a) Compute the posterior distribution on µ that results
1. Prove that the result derived in section 8.1 is a special case of the result derived in section 8.2.
4. Compare your answers to questions 2 and 3 above. Do you find them equally satisfactory?Why or why not?
3. Do the same problem, except that the observation now has the value 300.
2. Suppose your prior on µ is well represented by a normal distribution with mean 2 and precision 1. Also suppose you observe a normal random variable with mean µ and precision 2. Suppose that observation turns out to have the value 3. Compute the posterior distribution that results from these
1. Vocabulary. Explain in your own words the meaning of:(a) precision
2. An employer’s health plan offers to employees the opportunity to put money, before tax, into a health account the employee can draw upon to pay for health-related expenditures.Any funds not used in the account by the end of the year are forfeited. Suppose the employee’s probability
1. Suppose the result of a taxation audit using sampling is that the amount of tax owed,θ, has a normal distribution with mean $100,000 and a standard deviation of $10,000.Using the loss function (7.69), how much tax should be collected if:(a) b/a = 1(b) b/a = 2(c) b/a = 4(d) b/a = 19?
7. Suppose that your investment advisor informs you that she believes you face an infinite series of independent favorable bets, where your probability of success is 0.55. Suppose that she proposes that you use log (fortune) as your utility function, and that therefore at each opportunity, she
5. Suppose your utility is log f and you are offered the opportunity to buy as many tickets paying $1 if event A occurs and 0 otherwise. You have probability q that event A will occur. Tickets cost $ x each. How many tickets would you optimally buy?n 6. Reconsider the maximization of (7.60) subject
3. Suppose a person’s fortune is f = $1000, and his utility function is log(f). Suppose this person can buy tickets on the mutually exclusive events A1, A2 and A3 with prices x1 = 1/6, x2 = 1/3 and x3 = 1/2. Suppose this person’s probabilities on these three events are, respectively q1 = 1/2,
2. In your view, what is the significance of Kelly’s work?
1. Vocabulary. Explain in your own words:(a) Lagrange multipliers(b) Median
3. Suppose a decision-maker has absolute local risk aversion r(f).(a) Show that the risk of gain or loss of h with equal probability (±h, each with proba2 bility 1 ), is equivalent, asymptotically as h → 0, to the sure loss of h r(f). 2 2(b) Show that the gain of ±h with respective
2. Are you risk averse? If so, does absolute or relative risk aversion describe you better? Are you comfortable with constant risk aversion as describing the way you want to respond to financial risk? What constant k would you choose?
1. Vocabulary. Explain in your own words:(a) local absolute risk aversion(b) local relative risk aversion(c) concave function
5. Why does lexicographic utility violate the Archimedean condition?
4. Suppose utility is log-dollars. Find a random variable such that expected utility is infinite.
3. What’s wrong with infinite expected utility, anyway?
2. Is Pascal’s Wager an example of unbounded utility?
1. Vocabulary. Define in your own words:(a) St. Petersburg Paradox(b) Pascal’s Wager(c) Archimedean condition(d) Lexicographic utility
2. Prove that, if losses are defined in (7.7), minimizing expected loss is the same as maximizing expected utility.
1. Vocabulary. Define in your own words:(a) consequence(b) utility(c) loss(d) E-optimality(e) Pascal’s Wager
3. Suppose that in the example of section 7.2, your utilities are as follows:U(c4) = 1, U(c3) = 1/3, U(c2) = 0, U(c1) = 2/3.Suppose your probability of rain is 1/2. What is your optimal decision?
2. Assess your own utilities for the decision problem discussed in this section. Is there a probability for rain, r, above which maximization of expected utility suggests taking an umbrella, and below which not? If so, what is that probability? Would you, in fact, choose to take an umbrella if your
1. Vocabulary. State in your own words the meaning of:(a) consequence(b) utility of a consequence(c) utility of a decision
4 Did you ever have to make up your mind?”
3 It’s not often easy and not often kind.
2 And pick up on one and leave the other behind?
1 “Did you ever have to make up your mind?
3. Let Z,Z1, Z2, . . . be independent and identically distributed random variables satisfying E(Z) = 0 and V (Z) = 1. n Suppose i=1 Zi/√n has the same distribution as Z for all n.Prove that Z has a normal distribution.
2. How would you interpret the central limit theorem in terms of subjective probability and coherence?
1. Prove the statements (i), (ii), (iii) and (iv) found between (6.49) and (6.50).
4. Use the moment generating function of the Poisson distribution to verify:a) E(Z) = λb) V (Z) = λ
3. Use the moment generating function of the binomial distribution to verify:a) E(Y ) = npb) V (Y ) = np(1 − p)where Y has a binomial distribution (6.3).
2. Find the moment generating function of a negative binomial random variable.
1. Find the moment generating function of a geometric random variable (see section 3.7).
3. Is it important? Why or why not?
2. Is it a paradox?
1. What is the Borel-Kolmogorov Paradox?
2. From Corollary 5.7 1 we know that if a matrix A has two identical rows, say rows i and j, then | A |= 0. As in exercise 1, find x = 0 such that Ax = 0.

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Principles Of Uncertainty 2nd Edition Joseph B. Kadane - Solutions

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