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Decision theory and decision networks

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					            Bayesian Networks (BAN)




         Decision theory and
          decision networks



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                                      1
               Action, Utility and Probability



Actions You have a set of possible actions you can take


Utilities Each end result has a utility —a measure of how desir-
    able it is


Unknowns These will (hopefully) have probability distributions
   over possible values



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                                                          2
                   Doing the right thing




 • You have £1 to bet and have two choices (possible actions)

    1. Win £100 with a probability 0.0002

    2. Win £10 with a probability 0.001


 • Which one to go for?




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                   Expected values
We have two random variables, B1 and B2, with the following
distributions.



         P (B1 = 0) = 0.9998   P (B1 = 100) = 0.0002



          P (B2 = 0) = 0.999   P (B2 = 100) = 0.001


They have expected values:
          E(B1) = 0.9998 × 0 + 0.0002 × 100 = 0.02
          E(B2) = 0.999 × 0 + 0.001 × 10 = 0.01
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                        Expected utility

 • The utility of an action is the utility of the end result it leads
   to


 • But this usually depends on unknowns


 • So go for the action that maximises expected utility


 • Economists prefer “maximise expected utility”, statisticians
   prefer “minimise expected loss”

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                                                               5
                     Utility and money


Which would you prefer?


1. Getting £30000 for sure


2. Getting £40000 with probability 0.8


What about if you were faced with this choice, say, 20 times in
a row?


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                                                         6
                 The Students’ Dilemma
              (after the Prisoners’ dilemma)

 • You and your friend are accused of collusion.


 • If neither of you implicates the other then you are both
   docked 10 marks


 • If both of you implicate each other then you are both docked
   40 marks


 • If one of you implicates and the other does not, then the
   implicator gets away scot free and the implicated is docked
   100 marks
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                                                         7
                Pascal’s wager (pre-1662)




Actions To believe or not believe in God


Utilities Salvation, Damnation (−∞) or Neither


Unknowns God’s existence




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                                                 8
              The politics of decision theory




 • What price a human life?


 • A micromort is a 1 in a million chance of death


 • Apparently, a micromort is worth $20 in 1980 dollars (thanks
   to Daphne Koller)




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              Estimation and decision theory

 • We have been using BUGS to produce (samples of) posterior
   distributions over various unknown variables


 • We have then used the mean of this posterior distribution as
   our best guess for the unknown value, Why?


If the loss you suffer by estimating incorrectly is the squared
difference between you guess and the true value,

Then the mean minimises your expected loss.

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                                                         10
                     Decision networks



We can integrate decision making into Bayesian networks with
two extra sorts of nodes, in addition to chance nodes


Decision nodes Points where we can choose a course of action—
  represented by rectangles


Utility nodes These represent utilities—represented by diamonds




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                                                        11
                                        Wildcatter
  Oil
                      Drill




         Utility




 Oil               Drill      Utility
 dry                  y          -70
                                                     Oil       Prob
 dry                  n             0
                                                     dry        0.5
 wet                  y           50
                                                     wet        0.3
 wet                  n             0
                                                     soaking    0.2
 soaking              y         200
 soaking              n             0

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                        Causal links?



These two represents the same conditional independencies


    Flu             Fever          Sleepy




    Flu             Fever               Sleepy




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                   Intervening and causing
               Aspirin            Temp



    Flu                  Fever              Sleepy




    Flu                   Fever                Sleepy



                Aspirin              Temp



The impact of intervening actions can only follow the direction
of the causal links
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Description: Decision theory and decision networks