# Decision theory and decision networks

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```					            Bayesian Networks (BAN)

Decision theory and
decision networks

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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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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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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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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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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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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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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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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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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
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 views: 28 posted: 3/10/2010 language: English pages: 14
Description: Decision theory and decision networks