Lecture 5:
Decision and inference
Part 2
Influence diagrams
Decision-theoretic components can be added to a Bayesian network. The
complete network is then related to as a Bayesian Decision Network or more
common Influence diagram (ID)
Return to the example with banknotes. Let
H0: Dye is present
H1: Dye is not present
States of nature
E1: Method gives positive detection
E2: Method gives negative detection
Data
and let
A simple Bayesian network can be constructed for the relevance between the state
of nature and data:
H
H
Probabilities
H0
0.001
H1
0.999
Probabilities
E
H:
E
H0
H1
E1
0.99
0.02
E2
0.01
0.98
Now, we will add two nodes to the network, one for the actions that can be taken
and one for the loss function
A
A
a1
Destroy banknote
a2
Use banknote
A:
L
H:
L
a1
a2
H0
H1
H0
H1
0
100
500
0
Neither of the nodes are random nodes.
Node L must be a child node with nodes H and A as parents.
A
H
L
E
With this network, an influence diagram, we would like to be able to propagate
data from node E to a choice of decision in node A.
Hence, in the loss node L the expected posterior loss should be calculated.
Using GeNIe:
In GeNIe (and other software), this node is per
definition a utility node, but it can be used as a loss
node by representing losses as negative utilities
Run the network
Instantiate node E to E1
The expected posterior utility
(negative loss) can be read off in
node A (and also in node L )
About prior distributions
Conjugate prior distributions
Example: Assume the parameter of interest is , the proportion of some property of
interest in the population (i.e. the probability for this property to occur)
A reasonable prior density for  is the Beta density:
  1 1   
p  ;  ,   
B  ,  
 1
1
B  ,     x 1 1  x 
0
 1
, 0    1;   0,   0 hyperparam eters 
dx 
    
Beta function 
    
Beta(1,1)
Beta(5,5)
Beta(1,5)
Beta(5,1)
Beta(2,5)
Beta(5,2)
Beta(0.5,0.5)
Beta(0.3,0.7)
Beta(0.7,0.3)
0
0.5
1
Now, assume a sample of size n from the population in which y of the values
possess the property of interest.
The likelihood is
n y
L ; y      1   n  y
 y
 1
 1
n y

1  
n y 
   1    
y
B  ,  
L  | y  p  
 q  | y   1
  

 1
 1
1
0 Lx | y  px dx 0  n  x y 1  x n y  x 1  x  dx
B  ,  
 y
 y 1   
n y

 x 1  x 
1
0
y
n y
   1 1   
 y  1 1   
 1
x
 1
1  x 
 1
n  y   1

dx
1
x
0
y  1
1  x 
n  y   1

dx
 y  1 1   

B y   , n  y   
n  y   1
Thus, the posterior density is also a Beta density with hyperparameters y +  and
n – y + .
Prior distributions that combined with the likelihood gives a posterior in the same
distributional family are named conjugate priors.
(Note that by a distributional family we mean distributions that go under a
common name: Normal distribution, Binomial distribution, Poisson distribution
etc. )
A conjugate prior always go together with a particular likelihood to produce the
posterior.
We sometimes refer to a conjugate pair of distributions meaning
(prior distribution, sample distribution = likelihood)
In particular, if the sample distribution, i.e. f (x; ) belongs to the k-parameter
exponential family (class) of distributions:
k
 Aj θ B j  x C  x  D θ 
f  x; θ   e j1
we may put
 Aj θ  j  k 1  D θ  K 1 ,, k , k 1 
k
p θ   e j1
k
 Aj θ  j  k 1  D θ 
 e j1
where 1 , … , k + 1 are parameters of this prior distribution and K( ) is a function of
1 , … , k + 1 only .
Then
q θ | x   L θ | x   p θ  
k
n
n
k
 A j θ  B j  xi   C  xi  nD θ   A j θ  j  k 1 D θ  K 1 ,, k , k 1 
i 1
i 1
 e j 1
e j 1

n
e

e

A j θ 

j 1

k
e
C  xi 
i 1


A j θ 

j 1

k
K 1 ,, k , k 1 

e

B j  xi  j    n  k 1  D θ 

i 1

n



B j  xi  j    n  k 1  D θ 

i 1

n

i.e. the posterior distribution is of the same form as the prior distribution but with
parameters
n
n
i 1
i 1
1   B1  xi , ,  k   Bk xi ,  k 1  n
instead of
1,,  k ,  k 1
Some common cases (within or outside the exponential family):
Conjugate prior
Sample distribution
Posterior
Beta
Binomial
 ~ Beta  ,  
Beta
X ~ Bin n,  
Normal, known  2
Normal

 ~ N  , 2

Gamma

X i ~ N ,
2

Normal
2
2 2 
 2
n




 | x ~ N 2


x
,
2
2
2
2
   n 2
  n
  n 

Poisson
 ~ Gamma ,  
Pareto
Gamma
X i ~ Po 
Uniform
p      ;   
 | x ~ Beta   x,   n  x 
 |  xi ~ Gamma    xi ,   n 
Pareto
X i ~ U 0, 

q ; x      n  ;   max  , xn 

Non-informative priors (uninformative)
A prior distribution that gives no more information about  than possibly the
parameter space is called a non-informative or uninformative prior.
Example: Beta(1,1) for an unknown proportion  simply says that the parameter
can be any value between 0 and 1 (which coincides with its definition)
A non-informative prior is characterized by the property that all values in the
parameter space are equally likely.
Proper non-informative priors:
0.25
0.2
0.15
The prior is a true density or mass function
0.1
0.05
1.2
0
1
2
3
4
5
1
0.8
0.6
Improper non-informative priors:
0.4
0.2
0
0
The prior is a constant value over Rk
Example: N ( , ) for the mean of a normal population
0.2
0.4
0.6
0.8
1