IFSM 300 – Class 6

February 13, 2002 

Preliminaries

·        Class Web site reminder: http://www.gl.umbc.edu/~rrobinso/

·        Homework 2 (in assignment 5): due Monday, February 18.

·        Assignment 6: read chapter 7. 

Decision Trees With Revising of Probabilities (Part 2)

Introduced in Lecture 5, Completing in This Lecture

·        Refer to Marketing Research, situation 5.2, page 173.

·        Complete notes at the end. Summary here.

·        Continue toothpaste example to consider whether or not to purchase information (summarized in situation 5.2, page 173, and in Figure 5.10, page 187).

·        Calculate the revised probabilities in Figure 5.10.

·        Review the probability tree of Figure 5.3 (page 175).

·        Develop a probability table that shows the joint and marginal probabilities.

·        From the probability table, calculate the revised probabilities.

·        Review how this is the same as using Bayes’ theorem.

·        Find the revised EV*.

·        Calculate and interpret EVS, EVSI, E, and ENGS. 

Software for Decision Analysis

·        The rise of spreadsheet add-ins notwithstanding, serious DA requires industrial strength software.

·        Examples are given on the Web site of the Decision Analysis Society of INFORMS (software list at http://faculty.fuqua.duke.edu/daweb/dasw.htm).

·        Our class Web site has a link to the Decision Analysis Society’s homepage. 

Millionaire Q1 (overhead)

In a Bayesian probability revision, the data we need to get started are:

A.     P(s|I) and P(I).

B.     P(I|s) and P(s).(!!)

C.     P(sÇI) and P(I).

D.    P(IÇs) and P(s). 

Millionaire Q2 (overhead)

In a full decision tree with Bayesian revision, the order (starting at the left side) of the tree section that involves purchasing information is:

A.     Make management decision, learn management outcome s, decide to purchase information, learn information outcome I|s.

B.     Decide to purchase information, learn management outcome s, make management decision, learn information outcome I|s.

C.     Decide to purchase information, learn information outcome I, make management decision, learn management outcome s|I. (!!) 

Millionaire Q3 (overhead)

Step 1 of the 4 steps to prepare a decision tree for Bayesian analysis consists of obtaining joint probabilities from:

A.     A 2-stage probability tree with s in the first stage and I|s in the second stage. (!!)

B.     A 2-stage probability tree with I in the first stage and s|I in the second stage.

C.     A table of joint and marginal probabilities.

D.    Bayes’ Theorem. 

Millionaire Q4 (overhead)

In the table of joint and marginal probabilities (prepared in step 2 of the 4 steps), besides displaying data we already were given or we already calculated, we determine one type of quantity not previously known:

A.     P(sÇI).

B.     P(I). (!!)

C.     P(s).

D.    P(s|I). 

Millionaire Q5 (overhead)

In step 3, we calculate the one type of data needed in the decision tree but not yet determined:

A.     P(I|s).

B.     P(I Çs).

C.     P(s|I). (!!)

D.    P(sÇI). 

Millionaire Q6 (overhead)

Among the various summary measures, one that INCLUDES the cost of information is:

A.     EVS.

B.     EVSI.

C.     E.

D.    ENGS. (!!) 

Complete Notes

·        On the pages that follow. 

Notes: Decision Trees with Revising of Probabilities (Part 2)

Marketing Research

Conditional and Joint Probabilities. Before continuing to discuss the question of whether or not to purchase market research in the Brite-and-Kist example, we must review more about probability. 

Suppose you have two different wheels of fortune. One wheel contains 2 red spokes and 3 blue spokes, for a total of 5 equally likely spokes. The second wheel contains 4 red spokes and 1 blue spoke, for a total of 5 equally likely spokes.  

Imagine an experiment in which you first toss a fair coin and then select a wheel of fortune to spin: 

A complete probability tree for this experiment is:         

Note that the probability 2/5 for red after heads is not the overall probability of red, because red also may occur after tails. Rather, 2/5 is the probability of red given that heads happened. The notation for this conditional probability is P(R½H), where the vertical line means given or assuming.

The symbol Ç (cap or intersection) represents together. So HÇR means we got heads and then red, a so-called joint outcome. The probability P(HÇR) is called a joint probability.

How do you calculate a joint probability? Answer: Multiply together the probabilities on the associated tree branches. For instance,  

P(HÇR) = P(H) P(R½H) = (1/2) (2/5) = 2/10 

P(HÇB) = P(H) P(B½H) = (1/2) (3/5) = 3/10. 

Why does this make sense? What you are doing is starting with the probability of heads and then spreading it out over the subsequent possible red-blue outcomes in proportion to the red-blue probabilities: 

P(HÇR) = 2/5 of 1/2  = (2/5) (1/2) = 2/10 

P(HÇB) = 3/5 of 1/2  = (3/5) (1/2) = 3/10. 

Next we learn one more important fact: 

P(HÇR) = P(RÇH) – if and only if H is exactly the same event in both 

                                    expressions, and R is exactly the same event in both  

                                    expressions. 

In other words, suppose we reverse the process and spin a wheel first, then toss the coin. If we still are working with the first of the two wheels, multiplying the probabilities on the branches of the reversed tree would give us: 

            P(R) P(H½R) = (2/5) (1/2) = 2/10. 

This is P(RÇH). We see that it gives the same result as the P(HÇR) we calculated before.   

Equipped with this last-discovered fact, we now derive a cornerstone of probability. Suppose A and B are different uncertain events where A happens and then B. We learned earlier that  

P(AÇB) = P(A) P(B½A).  

We also learned that if we keep A the same and B the same,  

P(AÇB) = P(BÇA). 

Substituting from the second expression into the first, we can write 

P(BÇA) = P(A) P(B½A). 

Solve this for P(B½A): 

P(B½A) º P(BÇA) / P(A). 

This is the famous definition of conditional probability. It is used in probability theory as a definition (thus the symbol “º”), or fundamental building block, rather than something derived from other facts. 

Marketing Research Example. We now continue the Pucter and Simple example with an extension called marketing research, situation 5.2, page 173:

The main question now raised is whether or not to purchase the market research offered. If we do purchase it, we would plan to use the information it provides to revise our probability estimates (assignments), thereby, we would hope, improving the quality of those estimates. 

To purchase or not to purchase is a decision. You  know how to analyze decisions – in a decision tree. We can readily extend the tree already prepared so as to cover this new decision.  

Data Concerning Market Research. What will the market research tell us? In situation 5.2, it will give us one of two possible reports: the market is favorable for Brite (symbol I₁) or the market is unfavorable for Brite (symbol I₂).  

If we buy that research, the sequence of events will be: (1) we get the arket research report (I₁ or I₂); we decide whether to market Brite or Kist (a₁ or a₂); and (3) sales turn out to be low or high (s₁ or s₂).  

In other words, we’ll have an uncertain outcome I₁ or I₂ in the first chance stage and an uncertain outcome s₁ or s₂ in the second chance stage. To complete our decision tree, therefore, we require P(I₁) and P(I₂) for the first chance stage. And we require for the second chance stage conditional probabilities such as P(s₁½I₁) and P(s₂½I₂) because s₁ or s₂ happens after I₁ or I₂ happens.  

The pertinent data in the problem is information about the anticipated quality of the research. It comes in the form P(research report will be right) and P(research report will be wrong).  

Note that a report which says I₁ (favorable to Brite) is saying s₂ (high sales) will occur if we market Brite (choose a₁). Thus, 

P(favorable report will be right) = P(report says I₁ when in fact s₂ will occur) 

    = P(I₁ given s₂) = P(I₁½s₂).  

P(favorable report will be wrong) = P(report says I₁ when s₁ will occur) 

                                                       = P(I₁ given s₁) = P(I₁½s₁). 

Wording in the problem description of 5.2 may be confusing. But now you realize that it must be giving us probabilities in the form P(I½s). What isn’t stated in the problem, we calculate:

In addition to this information about the anticipated quality of the market research, we still have, of course, our original probability estimates (also called probability assignments):  P(s₁) = 0.45 and P(s₂) = 0.55. 

Four-Step Procedure.  The job ahead is to convert the data we have (in the form P(s) and P(I½s)) into the decision-tree inputs we require (in the form P(I) and P(s½I)). I recommend the following four-step procedure. Note well: This procedure differs a bit from the procedure in the book. 

Step 1. Draw a probability tree to obtain the joint probabilities P(sÇI). Since the data we have are P(s) and P(I½s), in the tree we’ll place s first and then I½s.

Step 2.  Prepare this table of joint and marginal probabilities: 

In this table, we calculate a marginal probability P(s) or P(I) by adding the joint probabilities across the same row or down the same column. We knew the P(s)’s already, so these just give us a check. The P(I)’s are new. And we’ll be using other numbers from the table in step 3. 

Step 3.   The table enables us to easily determine the inputs we require for the decision tree: 

P(I₁) = 0.580, P(I₂) = 0.420 from the bottom row of the table.

P(s₁½I₁) = P(s₁ÇI₁)/P(I₁) = 0.063/0.580 = 0.109 = 0.11

  From the definition of        from the

  conditional probability       table

P(s₂½I₁) = P(s₂ÇI₁)/P(I₁) = 0.517/0.580 = 0.891 = 0.89

  [Check: 1 – P(s₁½I₁) = 1 – 0.11 = 0.89] 

P(s₁½I₂) = P(s₁ÇI₂)/P(I₂) = 0.387/0.420 = 0.921 = 0.92 

P(s₂½I₂) = P(s₂ÇI₂)/P(I₂) = 0.033/0.420 = 0.079 = 0.08

  [Check: 1 - P(s₁½I₂) = 1 – 0.92 = 0.08]. 

Step 4. Enter the inputs found in step 3 into the decision tree. To optimize in the tree, we proceed in the usual way. 
Observe that the first decision node now has two branches --  a₃ (purchase the market  

research) and a₄ (don’t purchase).  

If we select a₄, we are back with the same analysis we did originally before considering market research. That begins with a decision node having two branches, a₁ (market Brite) and a₂ (market Kist), and an EV of $325,000. That’s the bottom section of the decision tree. 

The top section of the decision tree traces the sequence if we select a₃ (purchase the research).  That section is where we input the numbers we derived in step 3, above. 

Remember that the cost of the research must be subtracted from the expected profit, if we select a₃. It could be done anywhere along the line. In the tree shown, it is subtracted from the EV over the chance node labeled 2. 

In summary, the decision-tree analysis shows that the best policy a* and corresponding best expected profit EV* are:

a* = a₃,  a₁½I₁,  a₂½I₂  

EV* = $516,380. 

Bayes’ Formula. This is an alternative to steps 1, 2, and 3 above. It permits us to plug the data that we have straight into a formula to obtain a conditional probability we want. 

To illustrate, let’s derive a Bayes’ formula for P(s₁½I₁): 

P(s₁½I₁) = P(s₁ÇI₁) / P(I₁)                                From the definition of conditional probability.     

= P(s₁ÇI₁) / [P(s₁ÇI₁) + P(s₂ÇI₁)]      Substituting for P(I₁) using column 1 of the table in step 2 of the four-step procedure.

   = P(I₁Çs₁) / [P(I₁Çs₁) + P(I₁Çs₂)]     Using the P(AÇB) = P(BÇA) rule,

                                                            which really is using a “commutative

                                                            law” AÇB = BÇA of set algebra. 

P(s₁½I₁) = P(s₁) P(I₁½s₁) / [ P(s₁) P(I₁½s₁) + P(s₂) P(I₁½s₂)]  Applying the

                                                                        definition of conditional probability

                                                                        to each term.     

This result is a Bayes’ formula, or Bayes’ theorem, written for the P(s₁½I₁) in our          

example.  With this formula, we can confirm our previous calculation: 

P(s₁½I₁) =  (0.45)(0.14) / [(0.45)(0.14) + (0.55)(0.94)] = 0.063 / [0.063 + 0.517]

              = 0.063 / 0.580 = 0.109 = 0.11.  

To write a Bayes’ formula for any other conditional probability, in this or any other problem, we easily generalize the foregoing. The denominator contains terms of the form P(s) P(I½s), summed over all possible s’s with I always the same. The numerator contains the particular term from the denominator that has the same s as in the conditional probability we are evaluating. 

For example, suppose there are three s’s (s₁, s₂, s₃). And suppose we want the Bayes’ formula for P(s₃½I₃). Then 

            P(s₃½I₃) = P(s₃) P(I₃½s₃) / [ P(s₁) P(I₃½s₁) + P(s₂) P(I₃½s₂) + P(s₃) P(I₃½s₃)]. 

Related Performance Measures. The optimization analysis in the full decision tree showed that deciding to purchase market research increases EV* from $325,000 to $516,380. How might we further summarize the benefit of information from market research? 

First, we may establish summary measures similar to those introduced before to evaluate perfect information:

Second, we can compare actual to perfect, and we can adjust EVSI to recognize cost: