Prior & Posterior Analyses
Should ACME Consult BBA?
Posterior analysis refers to using the revised probabilies (the posteriors obtained from Bayes' Theorem) to determine if acquiring additional information about the uncertain states of nature is warranted. Prior analysis refers to the original expected value model (EMV or EOL) that employed the decision maker's prior distribution to recommend (tentatively) a course of action.
Let's put things in perspective. ACME's decision problem is to decide what size their new manufacturing plant should be. The size chosen for the plant should match the expected demand for the plant's product. Market demand is uncertain, but ACME's managers have derived a probability distribution (the prior distribution) that depicts their beliefs about possible market demand. The managers realize, however, that conducting market research could lead them to improve their prior demand distribution, thus lowering their risk. But conducting the research costs money, and the research results are by no means infallible. So a second, related decision must be taken: should ACME conduct the market research?
Clearly, market research should be conducted only if the findings translate directly into a higher expected net gain, that is, a higher expected payoff after paying for the cost of the research. That is what posterior analysis attempts to determine. If such research is not justified (at the quoted cost given its degree of reliability) then management should look for other alternatives to acquire additional information (less costly or more reliable). If no other alternatives are feasible, then management should proceed with the decision recommended by the prior analysis.
We already performed ACME's prior analysis with both the EMV and EOL criteria using decision matrices. Posterior analyses, however, tend to be more elaborate and detailed, and decision trees become much more useful than static matrices. So let's redo the prior analysis with a decision tree. Not only will we refresh our understanding of the prior analysis, we will also have the opportunity to examine the decision-tree building process, a skill we will need on the next page.
Prior Analysis Revisited
Recall that decision trees are drawn from left to right depicting the chronological order of the problem being analyzed. Square nodes denote decision points and circle nodes indicate points in time where the uncertainty of nature is resolved. Branches emanating from uncertainty nodes require probabilities (which must always add up to one at every circle node) and payoffs must be assigned to every terminal branch. These conventions are followed in the tree below:

o analyze a decision tree we perform rollback. Rollback is conducted from right to left, taking expected values at every uncertainty node and selecting the best action alternative at every decision node. The payoffs can be either actual monetary returns (yielding an EMV model) or opportunity losses (for EOL). It is customary to write down the expected value of uncertainty nodes just above the circle (shown here in yellow). For decision nodes, the action alternatives that are not selected are crossed out with double-tic marks, and the value of the chosen action branch is jotted above the square node. The red border indicates that value is the optimal EMV*.

The posterior analysis is shown next. What we already know, however, is that the net expected monetary value of the posterior analysis must be greater than $2.7 million if the cost of the market research is to be justified. And that is very useful to know.

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