MBA 7020 Business Analysis Foundations Decision Tree

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MBA 7020Business Analysis Foundations
Decision Tree Bayes Theorem July 25, 2005
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Agenda
Bayes Theorem
Problems
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Decision Trees

• A method of visually structuring the problem
• Effective for sequential decision problems
• Two types of branches
• Decision nodes
• Choice nodes
• Terminal points
• Solving the tree involves pruning all but the
  best decisions
• Completed tree forms a decision rule

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Decision Nodes

• Decision nodes are represented by Squares
• Each branch refers to an Alternative Action
• The expected return (ER) for the branch is
• The payoff if it is a terminal node, or
• The ER of the following node
• The ER of a decision node is the alternative with
  the maximum ER

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Chance Nodes

• Chance nodes are represented by Circles
• Each branch refers to a State of Nature
• The expected return (ER) for the branch is
• The payoff if it is a terminal node, or
• The ER of the following node
• The ER of a chance node is the sum of the
  probability weighted ERs of the branches
• ER ? P(Si) Vi

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Terminal Nodes

• Terminal nodes are optionally represented by
  Triangles
• The node refers to a payoff
• The value for the node is the payoff

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Problem 1

• Jenny Lind is a writer of romance novels. A movie
  company and a TV network both want exclusive
  rights to one of her more popular works. If she
  signs with the network, she will receive a single
  lump sum, but if she signs with the movie company
  the amount she will receive depends on the market
  response to her movie.
• Jenny Lind Potential Payouts
• Movie company
• Small box office - 200,000
• Medium box office - 1,000,000
• Large box office - 3,000,000
• TV Network
• Flat rate - 900,000
• Questions
• How can we represent this problem?
• What decision criterion should we use?

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Jenny Lind Payoff Table
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Jenny Lind Decision Tree
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Problem 2 Solving the Tree

• Start at terminal node at the end and work
  backward
• Using the ER calculation for decision nodes,
  prune branches (alternative actions) that are not
  the maximum ER
• When completed, the remaining branches will form
  the sequential decision rules for the problem

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Jenny Lind Decision Tree (Solved)
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Decision Tree Activation Test Source Delta
Airlines SkyMiles Program
SkyMiles Enrollment Message A
Returned within xx days Message B
Did not return within xx days Message C
Returned within xx days
Did not return within xx days
Did not return within xx days
Graduate to SOW
If Vc xx, send Message D
If Vc lt xx, no more messages
Graduate to SOW
If Vc lt xx, no more messages
If Vc xx, send Message D
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Probability

• The Three Requirements of Probabilities
• All Probabilities must lie with the range of 0 to
  1.
• The sum of the individual probabilities equal to
  the probability of their union
• The total probability of a complete set of
  outcomes must be equal to 1.

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Direct Marketing Campaign Platform
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Communication Variables
Vehicles E-mail Kits 2 Statement (
Telephone Direct Mail (USPS)

• Message / Offer (incentive)
• Hurdle (SOW)
• trip x get y
• Next trip (Re-Activation)
• Rate of trip triggers
• Points (double/flat?)
• Miles (front back-end)
• Other

• Creative Execution
• Can test several executions tailored to
  clusters/segments

• Timing/Frequency
• Monthly (statements)
• Repeat/Follow-up Mailings

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Measuring Effectiveness Lift/Gains Chart
Targeting
100
90
Percent of potential responders captured
Random mailing
45
0
45
100
Percent of population targeted
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Example Direct Mail OptimizationSource
InterContinental Hotels Group Priority Club
Rewards Program

• Using multivariate model we are able to maximize
  profit while minimizing costs
• In comparison to methodology used last year model
  savings XXX
• Savings attributable to reduced mailing to
  achieve last years result (variable cost
  savings).
• Other benefits - Customer Behavior, Planning Tool

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Agenda
Bayes Theorem
Decision Tree
Problems
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Bayes' Theorem

• Bayes' Theorem is used to revise the probability
  of a particular event happening based on the fact
  that some other event had already happened.
• Probabilities Involved
• P(Event)
• Prior probability of this particular situation
• P(Prediction Event)
• Predictive power (Likelihood) of the information
  source
• P(Prediction ? Event)
• Joint probabilities where both Prediction and
  Event occur
• P(Prediction)
• Marginal probability that this prediction is made
• P(Event Prediction)
• Posterior probability of Event given Prediction

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Bayes Theorem

• Bayes's Theorem begins with a statement of
  knowledge prior to performing the experiment.
  Usually this prior is in the form of a
  probability density. It can be based on physics,
  on the results of other experiments, on expert
  opinion, or any other source of relevant
  information. Now, it is desirable to improve this
  state of knowledge, and an experiment is designed
  and executed to do this. Bayes's Theorem is the
  mechanism used to update the state of knowledge
  to provide a posterior distribution. The
  mechanics of Bayes's Theorem can sometimes be
  overwhelming, but the underlying idea is very
  straightforward Both the prior (often a
  prediction) and the experimental results have a
  joint distribution, since they are both different
  views of reality.

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Bayes Theorem

• Let the experiment be A and the prediction be B.
  Both have occurred, AB. The probability of both A
  and B together is P(AB). The law of conditional
  probability says that this probability can be
  found as the product of the conditional
  probability of one, given the other, times the
  probability of the other. That is
• P(AB) P(B) P(AB) P(BA) P(A)
• if both P(A) and P(B) are non zero.
• Simple algebra shows that
• P(BA) P(AB) P(B) / P(A)      equation 1
• This is Bayes's Theorem. In words this says that
  the posterior probability of B (the updated
  prediction) is the product of the conditional
  probability of the experiment, given the
  influence of the parameters being investigated,
  times the prior probability of those parameters.
  (Division by the total probability of A assures
  that the resulting quotient falls on the 0, 1
  interval, as all probabilities must.)

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Bayes Theorem
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Conditional Probability
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Bayes' Theorem
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Probability Information

• Prior Probabilities
• Initial beliefs or knowledge about an event
  (frequently subjective probabilities)
• Likelihoods
• Conditional probabilities that summarize the
  known performance characteristics of events
  (frequently objective, based on relative
  frequencies)

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Circumstances for using Bayes Theorem

• You have the opportunity, usually at a price, to
  get additional information before you commit to a
  choice
• You have likelihood information that describes
  how well you should expect that source of
  information to perform
• You wish to revise your prior probabilities

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Problem

• A company is planning to market a new product.
  The companys marketing vice-president is
  particularly concerned about the products
  superiority over the closest competitive product,
  which is sold by another company. The marketing
  vice-president assessed the probability of the
  new products superiority to be 0.7. This
  executive then ordered a market survey to
  determine the products superiority over the
  competition.
• The results of the survey indicated that the
  product was superior to its competitor.
• Assume the market survey has the following
  reliability
• If the product is really superior, the
  probability that the survey will indicate
  superior is 0.8.
• If the product is really worse than the
  competitor, the probability that the survey will
  indicate superior is 0.3.
• After completion of the market survey, what
  should the vice-presidents revised probability
  assignment to the event new product is superior
  to its competitors?

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Joint Probability Table
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Agenda
Bayes Theorem
Decision Tree
Problems
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What kinds of problems?

• Alternatives known
• States of Nature and their probabilities are
  known.
• Payoffs computable under different possible
  scenarios

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Basic Terms

• Decision Alternatives
• States of Nature (eg. Condition of economy)
• Payoffs ( outcome of a choice assuming a state
  of nature)
• Criteria (eg. Expected Value)

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Example Problem 1- Expected Value Decision
Tree
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Expected Value
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Decision Tree
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Example Problem 2- Sequential Decisions

• Would you hire a consultant (or a psychic) to get
  more info about states of nature?
• How would additional info cause you to revise
  your probabilities of states of nature occurring?
• Draw a new tree depicting the complete problem.
• Consultants Track Record

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Example Problem 2- Sequential Decisions (Ans)
Open MBA7020Joint_Probabilities_Table.xls

• First thing you want to do is get the information
  (Track Record) from the Consultant in order to
  make a decision.
• This track record can be converted to look like
  this
• P(F/S1) 0.2 P(U/S1) 0.8
• P(F/S2) 0.6 P(U/S2) 0.4
• P(F/S3) 0.7 P(U/S3) 0.3
• F Favorable UUnfavorable
• Next, you take this information and apply the
  prior probabilities to get the Joint Probability
  Table/Bayles Theorum

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Example Problem 2- Sequential Decisions (Ans)
Open MBA7020Joint_Probabilities_Table.xls

• Next step is to create the Posterior
  Probabilities (You will need this information to
  compute your Expected Values)
• P(S1/F) 0.06/0.49 0.122
• P(S2/F) 0.36/0.49 0.735
• P(S3/F) 0.07/0.49 0.143
• P(S1/U) 0.24/0.51 0.47
• P(S2/U) 0.24/0.51 0.47
• P(S3/U) 0.03/0.51 0.06
• Solve the decision tree using the posterior
  probabilities just computed.

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