Chapter 19 Statistical Decision Theory ©

Framework for a Decision Problem action i.Decision maker has available K possible courses of action : a 1, a 2,..., a K. Actions are sometime called alternatives. states of nature ii.There are H possible uncertain states of nature : s 1, s 2,..., s H. States of nature are the possible outcomes over which the decision maker has no control. Sometimes states of nature are called events. payoff payoff table iii.For each possible action-state of nature combination, there is an associated outcome representing either profit or loss, called the monetary payoff, M ij, that corresponds to action a i and state of nature s j. The table of all such outcomes for a decision problem is called a payoff table.

Payoff Table for a Decision Problem with K Possible Actions and H Possible States of Nature (Table 19.1) a i s i s1s1 s2s2...sHsH a1a2...aKa1a2...aK M 11 M 21. M K1 M 12 M 22. M K M 1H M 2H. M KH ACTIONSSTATES OF NATURE

Decision Rule Based on Maximin Criterion Suppose that a decision maker has to choose from K admissible actions a 1, a 2,..., a K, given H possible states of nature s 1, s 2,..., s H. Let M ij denote the payoff corresponding to the i th action and j th state of nature. For each action, seek the smallest possible payoff. For action a 1, for example, this is the smallest of M 11, M 12,...M 1H. Let us denote the minimum M 1 * where More generally, the smallest possible payoff for action a i is given by The maximin criterion then selects the action a i for which the corresponding M i * is largest (that is, the action for which the minimum payoff is highest).

Regret or Opportunity Loss Table regret tableopportunity loss table Suppose that a payoff table is arranged as a rectangular array, with rows corresponding to actions and columns to states of nature. If each payoff in the table is subtracted from the largest payoff in its column, the resulting array is called a regret table, or opportunity loss table.

Decision Rule Based on the Minimax Criterion minimax regret criterion Given the regret table, the action dictated by the minimax regret criterion is found as follows: (i)For each row (action), find the maximum regret. (ii)Choose the action corresponding to the minimum of these maximum regrets. The minimax criterion selects the action for which the maximum regret is smallest; that is, the minimax regret criterion produces the smallest possible opportunity loss that can be guaranteed.

Payoff s with State-of-Nature Probabilities (Table 19.6) a i s i s1(1)s1(1) s2(2)s2(2)...sH(H)sH(H) a1a2...aKa1a2...aK M 11 M 21. M K1 M 12 M 22. M K M 1H M 2H. M KH ACTIONSSTATES OF NATURE

Expected Monetary Value (EMV) Criterion Suppose that a decision maker has K possible actions, a 1, a 2,..., a K and is faced with H states of nature. Let M ij denote the payoff corresponding to the i th action and j th state and  j the probability of occurrence of the j th state of nature with. expected monetary value of action The expected monetary value of action a i, EMV(a i ), is Expected Monetary Value Criterion The Expected Monetary Value Criterion adopts the action with the largest expected monetary value; that is, given a choice among alternation actions, the EMV criterion dictates the choice of the action for which the EMV is highest.

Decision Trees tree diagram The tree diagram is a graphical device that forces the decision-maker to examine all possible outcomes, including unfavorable ones.

Decision Trees All decision trees contain: Decision (or action) nodes Event (or state-of-nature) nodes Terminal nodes

Decision Trees (Figure 19.3) Process A Process B Process C States of natureActions

Bayes’ Theorem Bayes’ Theorem Let s 1, s 2,..., s H be H mutually exclusive and collectively exhaustive events, corresponding to the H states of nature of a decision problem. Let A be some other event. Denote the conditional probability that s i will occur, given that A occurs, by P(s i |A), and the probability of A, given s i, by P(A|s i ). Bayes’ Theorem states that the conditional probability of s i, given A, can be expressed as In the terminology of this section, P(s i ) is the prior probability of s i and is modified to the posterior probability, P(s i |A), given the sample information that event A has occurred.

Expected Value of Perfect Information, EVPI Perfect information Suppose that a decision maker has to choose from among K possible actions, in the face of H states of nature, s 1, s 2,..., s H. Perfect information corresponds to knowledge of which state of nature will arise. The expected value of perfect information is obtained as follows: (i)Determine which action will be chosen if only the prior probabilities P(s 1 ), P(s 2 ),..., P(s H ) are used. (ii)For each possible state of nature, s i, find the difference, W i, between the payoff for the best choice of action, if it were known that state would arise, and the payoff for the action chosen if only prior probabilities are used. This is the value of perfect information, when it is known that s i will occur. expected value of perfect information, EVPI (iii)The expected value of perfect information, EVPI, is then

Expected Value of Sample Information, EVSI Suppose that a decision maker has to choose from among K possible actions, in the face of H states of nature, s 1, s 2,..., s H. The decision-maker may obtain sample information. Let there be M possible sample results, A 1, A 2,..., A M. The expected value of sample information is obtained as follows: (i)Determine which action will be chosen if only the prior probabilities were used. (ii)Determine the probabilities of obtaining each sample result:

Expected Value of Sample Information, EVSI (continued) (iii) For each possible sample result, A i, find the difference, V i, between the expected monetary value for the optimal action and that for the action chosen if only the prior probabilities are used. This is the value of the sample information, given that A i was observed. expected value of sample information, EVSI (iv)The expected value of sample information, EVSI, is then:

Obtaining a Utility Function Suppose that a decision maker may receive several alternative payoffs. The transformation from payoffs to utilities is obtained as follows: (i)The units in which utility is measured are arbitrary. Accordingly, a scale can be fixed in any convenient fashion. Let L be the lowest and H the highest of all the payoffs. Assign utility 0 to payoff L and utility 100 to payoff H. (ii)Let I be any payoff between L and H. Determine the probability  such that the decision-maker is indifferent between the following alternatives: (a) Receive payoff I with certainty (b) Receive payoff H with probability  and payoff L with probability (1 -  ) utility function (iii)The utility to the decision-maker of payoff I is then 100 . The curve relating utility to payoff is called a utility function.

Utility Functions; (a) Risk Aversion; (b) Preference for Risk; (c) Indifference to Risk (Figure 19.11) Payoff Utility Payoff Utility Payoff Utility (a) Risk aversion (b) Preference for risk (c) Indifference to risk

The Expected Utility Criterion expected utility, EU(a i ), Suppose that a decision maker has K possible actions, a 1, a 2,..., a K and is faced with H states of nature. Let U ij denote the utility corresponding to the i th action and j th state and  j the probability of occurrence of the j th state of nature. Then the expected utility, EU(a i ), of the action a i is expected utility criterion Given a choice between alternative actions, the expected utility criterion dictates the choice of the action for which expected utility is highest. Under generally reasonable assumptions, it can be shown that the rational decision-maker should adopt this criterion. If the decision-maker is indifferent to risk, the expected utility criterion and expected monetary value criterion are equivalent.

Key Words 4Action 4Admissible action 4Aversion to Risk 4Bayes’ Theorem 4Decision nodes 4Decision Trees 4EMV 4Event nodes 4EVPI 4EVSI 4Expected Monetary Value Criterion 4 Expected net Value of Sample Information 4 Expected Utility Criterion 4 Inadmissible Action 4 Indifference to Risk 4 Maximin Criterion 4 Minimax Regret Criterion 4 Opportunity Loss Table 4 Payoff Table 4 Perfect Information

Key Words (continued) 4Preference for Risk 4Regret Table 4Sensitivity Analysis 4States of Nature 4Terminal Nodes 4Tree Plan 4Utility Function 4Value of Perfect Information 4Value of Sample Information