Chapter 15

bayesian decision analysis

15-1.P( | x) = = x/n = 6/20

s / P(s) / P(x | s) / P(s) P(x | s) / P(s | x)
.1 / .2 / .0089 / .00178 / .0252
.2 / .3 / .1091 / .03273 / .4655
.3 / .1 / .1916 / .01916 / .2726
.4 / .1 / .1244 / .01244 / .1769
.5 / .1 / .0370 / .00370 / .0526
.6 / .1 / .0049 / .00049 / .0069
.7 / .1 / .0002 / .00002 / .0003
1.0 / .07032 / 1.00000

Credible set of posterior probability 0.9148 is [0.2, 0.4].

Bayesian Revision
Binomial
p / 0.1 / 0.2 / 0.3 / 0.4 / 0.5 / 0.6 / 0.7 / Total
Prior / 0.2 / 0.3 / 0.1 / 0.1 / 0.1 / 0.1 / 0.1 / 1
New Evidence
n / 20 / x / 6
Total
Joint Prob. / 0.0018 / 0.0327 / 0.0192 / 0.0124 / 0.0037 / 0.0005 / 0.0000 / 0.0703
Posterior / 0.0252 / 0.4655 / 0.2726 / 0.1769 / 0.0526 / 0.0069 / 0.0003

15-2. = x/n = 7/17

Credible set of posterior probability 0.9061 is [0.2, 0.4]

15-3.

Bayesian Revision
Binomial
p / 0.1 / 0.2 / 0.3 / 0.4 / 0.5 / 0.6 / 0.7 / Total
Prior / 0.0002 / 0.142208 / 0.374266 / 0.389638 / 0.089154 / 0.004505 / 0.000034 / 1
New Evidence
n / 10 / x / 3
Total
Joint Prob. / 0.000011 / 0.028630 / 0.099865 / 0.083769 / 0.010448 / 0.000191 / 0.000000 / 0.222914
Posterior / 0.00005 / 0.12844 / 0.44800 / 0.37579 / 0.04687 / 0.00086 / 0.00000 / 1.0000

Credible set of posterior probability 0.95223 is [0.2, 0.4]

15-4.

Bayesian Revision
Binomial
p / 0.05 / 0.15 / 0.2 / 0.25 / Total
Prior / 0.3 / 0.5 / 0.1 / 0.1 / 1
New Evidence
n / 18 / x / 1
Total
Joint Prob. / 0.1129 / 0.0852 / 0.0081 / 0.0034 / 0.2096
Posterior / 0.5386 / 0.4065 / 0.0387 / 0.0161

15-5. = x/n = 3/10

R / P(R) / P(X | R) / P(R) P(X | R) / P(R | X)
.25 / .1 / .2503 / .02503 / .1129
.30 / .2 / .2668 / .05336 / .2407
.35 / .2 / .2522 / .05044 / .2275
.40 / .3 / .2150 / .06450 / .2909
.45 / .1 / .1665 / .01665 / .0751
.50 / .1 / .1172 / .01172 / .0529
1.0 / .22170 / 1.0000

15-6. = x/n = 4/12

R / P(R) / P(X | R) / P(R) P(X | R) / P(R | X)
.25 / .1129 / .1936 / .02186 / .10291
.30 / .2407 / .2312 / .05336 / .26198
.35 / .2275 / .2366 / .05383 / .25341
.40 / .2909 / .2129 / .06193 / .29155
.45 / .0751 / .1699 / .01276 / .06007
.50 / .0529 / .1208 / .00639 / .03008
1.0000 / .22170 / 1.00000

A credible set of posterior probability 0.97 is [0.25, 0.45].

A credible set of posterior probability 0.91 is [0.25, 0.40].

15-7. / R / P(R) / P(X | R) / P(R X) / P(R | X)
.1 / .111 / .0574 / .0064 / .0633
.2 / .111 / .2013 / .0224 / .2216
.3 / .111 / .2668 / .0296 / .2928
.4 / .111 / .2150 / .0239 / .2364
.5 / .111 / .1172 / .0130 / .1286
.6 / .111 / .0425 / .0047 / .0465
.7 / .111 / .0090 / .0010 / .0099
.8 / .111 / .0008 / .00009 / .0009
.9 / .111 / .0000 / .0000 / .0000
.10109 / 1.0000

15-8.x/n = 17/20

S / P(S) / P(X | S) / P(S) P(X | S) / P(S | X)
.70 / .1 / .0716 / .00716 / .0423
.75 / .2 / .1339 / .02678 / .1584
.80 / .3 / .2054 / .06162 / .3644
.85 / .2 / .2428 / .04856 / .2872
.90 / .1 / .1901 / .01901 / .1124
.95 / .1 / .0596 / .00596 / .0352
1.0 / .16909 / 1.0000
Bayesian Revision
Binomial
p / 0.7 / 0.75 / 0.8 / 0.85 / 0.9 / 0.95 / Total
Prior / 0.1 / 0.2 / 0.3 / 0.2 / 0.1 / 0.1 / 1
New Evidence
n / 20 / x / 17
Total
Joint Prob. / 0.0072 / 0.0268 / 0.0616 / 0.0486 / 0.0190 / 0.0060 / 0.1691
Posterior / 0.0423 / 0.1584 / 0.3644 / 0.2872 / 0.1124 / 0.0352
15-9. / S / P(S) / P(X | S) / P(S) P(X | S) / P(S | X)
.70 / .0423 / .0278 / .00118 / .0071
.75 / .1584 / .0669 / .01060 / .0638
.80 / .3644 / .1369 / .04989 / .3001
.85 / .2872 / .2293 / .06585 / .3962
.90 / .1124 / .2852 / .03206 / .1929
.95 / .0352 / .1887 / .00664 / .0399
.16622 / 1.0000

15-10.= d/n = 2/15

x / P(x) / P(d | x) / P(x) P(d | x) / P(x | d)
.05 / .1 / .1348 / .01348 / .055503
.10 / .2 / .2669 / .05338 / .219788
.15 / .4 / .2856 / .11424 / .470375
.20 / .2 / .2309 / .04618 / .190143
.25 / .1 / .1559 / .01559 / .064191
1.0 / .24287 / 1.000000

A credible set of posterior probability 0.9445 is [0.10, 0.25].

A credible set of close to the same probability is [0.05, 0.20].

15-11.M’ = 8,500’= 1,000n = 35M = 9,210= 365

M” = = 9,207.3

”2 = = 3,792” = 61.58

The posterior distribution of the population mean is normal with mean 9,207.3 and standard deviation 61.58

Bayesian Revision
Normal
Prior
M / 8500 / (M) / 1000
New Evidence
n / 35 / x-bar / 9210 / s / 365
Posterior / Credible sets
M / 9207.31 / (M) / 61.5792 / 1
95% / 9207.308 / + or - / 120.6928

15-12.M’ = 15,000’= 4,000n = 12M = 9,867= 1,055

M” = = 9,896.58

”2 = = 92,217.498” =303.67

The posterior distribution of the population mean is normal with mean 9,896.58 and standard deviation 303.67. A 95% HPD credible set for average expected monthly sales is: 9,896.58 1.96(303.67) = [9,301.39, 10,491.77].

Bayesian Revision
Normal
Prior
M / 15000 / (M) / 4000
New Evidence
n / 12 / x-bar / 9867 / s / 1055
Posterior / Credible sets
M / 9896.58 / (M) / 303.673 / 1
95% / 9896.585 / + or - / 595.1879

15-13.M’ = 94’= 2n = 10M = 96Assume = 1

M” = = 95.95

”2 = = .0976” = 0.312

The posterior distribution of the population mean is normal with mean 95.95 and standard deviation .312.

15-14.n = 15M = 95= 1

M” = = 95.386

”2 = = .03961” = 0.199

The posterior distribution of the population mean is normal with mean 95.386 and standard deviation 0.199. A 95% HPD credible set for the mean rating of the foie gras is: 95.386

1.96 (0.199) = [94.996, 95.776].

15-15.M’ = 22’= 2n = 14M = 24= 3

M” = = 23.72

”2 = = .5538” = .7442

The posterior distribution of the population mean is normal with mean M” = 23.72 and standard deviation ” = .7442.

15-16.n = 20M = 25= 2

M” = = 24.66

”2 = = .1469” = 0.3833

The posterior distribution of the population mean is normal with mean M” = 24.66 and standard deviation ” = 0.3833. A 99% HPD credible set for the population mean is: 24.66 2.576(.3833) = [23.67, 25.65].

Bayesian Revision
Normal
Prior
M / 22 / (M) / 2
New Evidence
n / 20 / x-bar / 25 / s / 2
Posterior / Credible sets
M / 24.8571 / (M) / 0.43644 / 1
99% / 24.85714 / + or - / 1.124186

15-17.Governor: D (largest s.d.)ARCO expert: C (smallest s.d.) Most embarrassed: C (prediction far off, with smallest s.d. implying greatest confidence).

15-18.Decision analysis is a quantitative method of evaluation decision problems and may be used as an aid in reaching an optimal decision in a given situation.

15-19.Human decision problems may not always be quantified, and when they are quantifiable, there is no assurance that values are correctly assessed.

15-20.Actions--taken by the decisionmaker; chance occurrencesthe “actions” of chance: probabilitiesassessments of the likelihood of chance occurrences; additional information used in further assessment of probabilities; final outcomethe aim of the decision problem. This is when the final payoff (or loss) is obtained.

15-21.Probabilities are needed as quantitative measures of likelihood of chance occurrences. They are obtained using the most appropriate method for any given situation; often these are subjective probabilities elicited from experts. The probabilities are used in evaluating expected values of outcomes.

15-22.A decision tree is a graphical display of the interconnection of actions and chance occurrences that captures the sequential nature of a decision problem.

15-23.Cost = 180($20,000) = $3.6 million.

15-24.

15.25.Buy Federated Stores: E = $65.5 million

Start own chain: E = $58 million

Make new computer: E = $86 million

The optimal decision is therefore to make a new computer.

15-26.

15-27.

E(L) = 4.8(.35) + (.2)(4) + (.45)10 = 6.98

15-28.The optimal decision is to invest, then pull out if necessary. The expected monetary value of the decision is $3,040. When you pull out, the expected value is $4,100.

15-29.The expected value of the limited partnership is $2,500. Hence the optimal decision is as in the previous problem: invest in wheat futures. The expected value is $3,040.

15-30.In a decision problem, it is often necessary to incorporate probabilities describing the reality of a test or survey. In order to transform such probabilities to the probabilities of states of nature given the test or survey results, Bayes’ theorem is needed.

15-31A predictive probability is the unconditional probability of a given test or survey result. It is obtained using the law of total probability.

15-32.The optimal decision now is to hire the consultants; the expected monetary value is $7.44 million. Then follow subsequent arrows in Figure 15-18 in each possible consultant’s finding.

15-33.Don’t test; go for it. E = 1.8. If you must test, still make the decision to advertise on television, regardless of the test results.

15-34.

15-35.

15-36.Test, then do as test recommends. E = $752,000.

15-37.Optimal decision: test and follow the test’s recommendation. E = $587,000

15-38.Go with discount broker. E = $225.

15-39.A utility function is a value-of-money function of an individual.

15-40.Since people’s attitudes toward risk and toward money should be accounted for, an analysis using utility rather than money may be more meaningful.

15-41.A risk-averse individual has a concave utility function. A risk-seeking individual has a convex utility function.

15-42.This investor is risk-averse.

15-44.Additional information in a decision problem helps us make better decisions. The value of such information may be assessed a priori.

15-45.The expected value of perfect information is computed as the expected monetary value of the decision situation when perfect information is available minus the expected value of the decision situation without any additional information. We use the expectation because the actual information is not known.

15-46.See Problem 15-47 below.

15-47.EVPI = E(payoff with perfect information)  E(payoff without info)

= 8(.2) + 9(.35) + 10(.45)  6.35 = 2.9 [in $100,000’s].

Do not buy information; cost exceeds expected value.

15-48.EVPI = 0 because the optimal decision is to go ahead in any case.

15-49. / x / P(x) / P(y | x) / P(x) P(y | x) / P(x | y)
.1 / .1 / .0137 / .00137 / .01142
.2 / .3 / .1201 / .03603 / .30043
.3 / .2 / .2099 / .04198 / .35004
.4 / .2 / .1623 / .03246 / .27066
.5 / .1 / .0667 / .00667 / .05561
.6 / .1 / .0142 / .00142 / .01184
1.0 / .11993 / 1.00000
Bayesian Revision
Binomial
p / 0.1 / 0.2 / 0.3 / 0.4 / 0.5 / 0.6 / Total
Prior / 0.1 / 0.3 / 0.2 / 0.2 / 0.1 / 0.1 / 1
New Evidence
n / 16 / x / 5
Total
Joint Prob. / 0.0014 / 0.0360 / 0.0420 / 0.0325 / 0.0067 / 0.0014 / 0.1199
Posterior / 0.0114 / 0.3004 / 0.3501 / 0.2707 / 0.0556 / 0.0119

15-50.The set [0.2, 0.4] has posterior probability 0.9211.

15-51. / x / P(x) / P(y | x) / P(x) P(y | x) / P(x | y)
.1 / .01142 / .0319 / .00036 / .0026
.2 / .30043 / .1746 / .05245 / .3844
.3 / .35004 / .1789 / .06262 / .4589
.4 / .27066 / .0746 / .02019 / .1480
.5 / .05561 / .0148 / .00082 / .0060
.6 / .01184 / .0013 / .00002 / .0001
1.00000 / .13646 / 1.0000

15-52.The Bayesian approach allows for the use of prior information; the classical approach uses only the data. In the Bayesian approach, parameters are viewed as random variables with which we may associate probability distributions.

15-53.The family of normal probability distributions is closed under the operation of Bayesian updating of information: if the prior is normal and the likelihood is normal, so is the posterior.

15-54.M’ = 45’= 5n = 100M = 102= 10

M” = = 99.81

”2 = = 0.9615” = .9806

Bayesian Revision
Normal
Prior
M / 45 / (M) / 5
New Evidence
n / 100 / x-bar / 102 / s / 10
Posterior / Credible sets
M / 99.8077 / (M) / 0.98058 / 1
99% / 99.80769 / + or - / 2.525814

15-55.95% HPD region = 99.81 1.96(.9806) = [97.888, 101.732]

15-56.n = 60M = 101.5

M” = = 100.43

”2 = = 0.6097” = 0.781

95% HPD region = 100.43 1.96(.781) = [98.90, 101.96]

15-57.A payoff table is a matrix of possible outcomes of a decision problem. A decision tree is a graphical way of showing a decision problem. A payoff table can certainly be used without a decision tree.

15-58.A subjective probability is one obtained by anything other than an objective frequency-based approach. Such probabilities may vary depending on the person assessing them. Lack of objectivity is the main limitation of these probabilities.

15-59.The main principle behind the de Finetti game is the gauging of a subjective probability against an objective hypothetical lottery with known probabilities. The game is somewhat simplistic as it does not allow for checking the coherence of the probability assessments.

15-60.The main complaint against Bayesian methods relates to their use of prior information, which may be of unknown reliability.

15-61.E(gamble) = (.2)5,000 + (.8)500 = 1,400 < 3,000, therefore, I am a risk taker.

15-62.The investment has expected monetary value $2,650. The alternative investment has expected monetary value $4,000. Hence, the alternative investment is optimal. Limitation: this analysis does not incorporate attitude toward risk.

15-63.Optimal decision: Merge. The expected monetary outcome is $2.45 million.

15-64.

15-65.EVPI = (.55)(6.5) + (.45)(0)  2.45 = $1.125 million.

15-66.

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