1. Zachary Blomme University of Nebraska, Lincoln Pattern Recognition
Influence Diagrams
Zachary Blomme
University of Nebraska, Lincoln
Pattern Recognition

2. Outline Decision Trees Influence Diagrams Dynamic Networks
Mathematically equivalent to influence diagrams, and simpler when the problem instance is small
Influence Diagrams
A Bayesian network that has been slightly modified to make a decision
Dynamic Networks
Models dynamic aspects of a Bayesian network (usually time)

3. Decision Trees Decision Node Chance Node Utility Expected Utility
Represents decisions to be made
Chance Node
Represents random variables
Utility
Value of outcome to decision make
Expected Utility
The expected utility of a chance node is the expected value of the node’s outcomes
Decision
Node
Chance
Node

4. Example 1
You are considering purchasing 100 shares of NASDIP(n) currently valued at $10. This is your model for the value of the stock at the end of the month:
p(n=$5) = .25
p(n=$10) = .25
p(n=$20) = .5

5. Example 1 $5 $500 .25 $10 NASDIP $1000 .25 Buy NASDIP $20 $2000 .5 D
Money in
the bank
$1005
.5% interest

6. Example 1 Solved $5 $500 $1375 .25 $10 NASDIP $1000 .25 Buy NASDIP $20
$2000 D .5
$1375
Money in
the bank
$1005
.5% interest

7. Example 2
As example 1, except that we now also have the chance to buy an option on NASDIP for $ This option would allow you to buy 500 shares of NASDIP for $11 in one month.

8. Example 2 $5 $500 .25 $10 NASDIP1 $1000 .25 Buy NASDIP $20 $2000 .5 D$0
.25
Buy option
NASDIP2
$10 $0
.25
$20
$4500 .5

9. Example 2 solved $5 $500 .25 $1375 NASDIP1 $10 $1000 .25 Buy NASDIP
$20
$2000 D .5
$2250 $5 $0
.25
Buy option
$2250
NASDIP2
$10 $0
.25
$20
$4500

10. Utility Function
We don’t have to use expected value to calculate utility
One alternate method is the exponential utility function
The parameter R is the risk tolerance. As it becomes smaller the function will become more risk adverse
UR(x) = 1 – e-x/R

11. Example 3 $5 $500 .25 $10 NASDIP $1000 .25 Buy NASDIP $20 $2000 .5 D
Money in
the bank
$1005
.5% interest

12. Solving Decision Trees
Starting at the right
Proceed to the left
Passing expected utilities to chance nodes
Passing maxima to decision nodes
Until the root is reached

13. Example 6 Buying and selling a Car Know someone who’ll sell for $10000
Know someone who’ll buy for $11000
Problem: 20% of these cars are known to have a bad transmission
There exists a test that can identify bad transmissions with a 30% false negative rate and a 10% false positive rate
The test costs $200

14. Example 6 Good .571 $10800 Tran1 Bad .428 D1 $7800 Bad .42 $9800
Keep Money
Good
.965
$10800
Tran2
Test D2
Bad
.034
$7800
Good
.58
Run Test
$9800
Keep Money
Good .8
$11000
Buy Car
Tran3 D3
Bad .2
$8000
$10000
Keep Money

15. Example 7 Should Leonardo take an umbrella when he goes out?
Leonardo finds carrying an umbrella to be inconvenient
There is a 40% chance of rain
The possible outcomes for this event are, in increasing preference:
His new suit will be ruined
Suit will not be ruined, inconvenience
Suit will not be ruined

16. Example 7 rain Suit ruined .4 Rain Don’t take Umbrella No rain
Suit not ruined .6 D
Take
Umbrella
Suit not ruined,
inconvenience

17. Assign numeric values to outcomes
Assign a utility of 0 to worst outcome
Assign a utility of 1 to best outcome
To determine the utility of an unknown intermediate outcomes run a lottery with two known, flanking outcomes
The utility of the unknown intermediate outcome is defined as the expected utility of the lottery where you would be indifferent to the lottery and the assured unknown intermediate outcome

18. Example 7 U(suit not ruined) = 1 U(suit ruined) = 0
U(inconvenience) = EU(Lotteryp’)
= p’U(suit not ruined) + (1-p’)U(suit ruined)
=p’(1) + (1-p’)(0)
=p’
For this example we’ll choose p’ = .8
U(inconvenience) = .8

19. Example 7 rain Suit ruined .4 Rain Don’t take Umbrella No rain.4
Rain
Don’t take
Umbrella
No rain
Suit not ruined 1 .6 D
Take
Umbrella
Suit not ruined,
Inconvenience .8

20. Example 8
Amit is 15 years old and has a streptococcal infection (sore throat)
There is a treatment available that reduces the number of days with a sore throat from 4 to 3
The treatment has a chance of death from anaphylaxis.
Amit’s remaining life expectancy is 60 years

21. Example 8 No death 3 sore throat days .999997 A Treat death dead
Don’t
treat
4 sore throat
days

22. How do we quantify Amit’s life?
We can do this using quality adjusted life expectancies(QALE)
Amit feels that 1 year of life with a sore throat is worth .9 years of life without
So Amit’s QALE will be:
L is Amit’s remaining life expectancy and t is his time with the disease
QALE(L,t) = (L-t) + .9t

23. Example 8 3 days = .008219 years 4 days = .010959 years
QALE(60, 3 days) = years
QALE(60, 4 days) = years

24. Example 8 No death 3 sore throat days .999997 59.998998 yrs
Treat
death
Dead
0 yrs D
Don’t
treat
4 sore throat
Days

25. Influence Diagrams Decision Node Chance Node Utility Node
Represents decisions to be made
Chance Node
Represents random variables
Utility Node
Random variable whose values are the utilities of the outcomes
Chance
Node
Utility Node

26. Edges
Decision
Node
Value of parent is known at time of decision, so edge represents a sequence
Value of the node is probabilistically dependent upon parent
Value of node is deterministically dependent upon parent
Chance
Node
Utility Node

27. Influence Diagrams
Chance nodes in an influence diagram satisfy the Markov condition
Because of this, an influence diagram is a Bayesian network with decision nodes and a utility node

28. Example 10/17
You are considering purchasing 100 shares of NASDIP(n) currently valued at $10. This is your model for the value of the stock at the end of the month:
p(n=$5) = .25
p(n=$10) = .25
p(n=$20) = .5

29. Example 1 Solved $5 $500 $1375 .25 $10 NASDIP $1000 .25 Buy NASDIP $20
$2000 D .5
$1375
Money in
the bank
$1005
.5% interest

30. Example 10/17 P(NASDIP = $5) = .25 P(NASDIP = $10) = .25
Utility D
d1 = Buy
d2 = Bank
U(d1, $5) = $500
U(d1, $10) = $1000
U(d1, $20) = $2000
U(d1, n) = $1005

31. Example 12
This problem illustrates how chance nodes can depend on decision nodes
In this example buying a great deal of stock can influence market price
Dow
ICK
Utility D

32. Example 14 Buying and selling a Car
Know someone who’ll sell for $10000
Know someone who’ll buy for $11000
Problem: 20% of these cars are known to have a bad transmission
There exists a test that can identify bad transmissions with a 30% false negative rate and a 10% false positive rate
The test costs $200

33. Example 6 Good .571 $10800 Tran1 Bad .428 D1 $7800 Bad .42 $9800
Keep Money
Good
.965
$10800
Tran2
Test D2
Bad
.034
$7800
Good
.58
Run Test
$9800
Keep Money
Good .8
$11000
Buy Car
Tran3 D3
Bad .2
$8000
$10000
Keep Money

34. Example 14/20 P(Test=positive|tran = good) = .3
P(Test=positive|tran = bad) = .9
P(tran = good) = .8
P(tran = bad) = .2
Test
Tran
Utility D
Run
Test
U(r1, d1, good) = $10800
U(r1, d1, bad) = $7800
U(r1, d2, t) = $9800
U(r2, d, good) = $11000
U(r2, d, bad) = $8000
U(r3, d, t) = $10000
r1 = run test
r2 = buy car
r3 = don’t buy
d1 = Buy
d2 = Don’t Buy

35. Dynamic Bayesian Networks
Uses a Bayesian network to modify dynamic aspects of a problem, usually time.
To do this, we must define a few terms and make a few assumptions

36. Definitions Random Vector Random Matrix Independence
A column vector whose variables are random variables
Random Matrix
Like random vector, only a matrix
Independence
A random vector is independent if the set of it’s variables are independent

37. Definitions
X is a random vector and the set of variables that constitute X
x is a vector value of X and the set of variables that constitute x
P(x) is the joint probability distribution P(x1, …, xn)

38. Dynamic Bayesian Networks
Assume changes occur between discrete time points.
Assume there is a finite number T of these time points.
A simple dynamic Bayesian network contains the variables that constitute the T random vectors X[t], and has the following specifications

39. Initial Bayesian Network
X1[0]
A DAG containing variables in X[0]
Also contains a probability distribution for these variables
X2[0]
X3[0]

40. Transition Bayesian Network
X1[t]
X1[t+1]
X2[t]
X2[t+1]
X3[t]
X3[t+1]

41. Dynamic Bayesian Network
X1[0]
X1[1]
X1[2]
X2[0]
X2[1]
X2[2]
X3[0]
X3[1]
X3[2]

42. Further Assumptions
All you need to predict a world state at time t is the world state at time t-1
Because of this we satisfy the Markov property
This process is stationary. P(x[t+1] | x[t]) is the same for all t

43. D[t]
D[t+1]
D[t+2]
LR[t]
LR[t+1]
LR[t+2]
LR[t+3]
ER[t]
ER[t+1]
ER[t+2]
ER[t+3]
EA[t]
EA[t+1]
EA[t+2]
EA[t+3]
LA[t]
LA[t+1]
LA[t+2]
LA[t+3]