Titel_SS06

The decision/event tree is illustrated in Figure 3.3 together with the expected costs (in boxes).
It is seen that the action alternative yields the smallest expense (largest expected utility) so

this action alternative is the optimal decision.

a 1

(^11)
Figure 3.3: Simple decision problem with assigned prior probabilities and utility.
3.7 Decision Analysis with Additional Information - Posterior
Analysis
When additional information becomes available, the probability structure in the decision
problem may be updated. Having updated the probability structure the decision analysis is
unchanged in comparison to the situation with given - prior information.
Given the result of an experiment the updated probability structure (or just the posterior
probability) is denoted and may be evaluated by the use of Bayes’ rule:
zk
P''[ ]:
' 
''[ ]
'
k i
i
kjj
j
Pz P
P
Pz P

::

: i

::







 

(3.5)

which may be explained as:

Posterior probability of Normalising Sample likelihood prior probability

with given sample outcome constant given of

i

ii

:

::

!!!

"#"#""# 

$%$%$$%

!

%

(3.6)

The normalizing factor is to ensure that ''[ ]P :i forms a proper probability. The mixing of new

and old information appears through the sample likelihood Pzki: and the prior

probability '[ ]P :i. The likelihood is the probability of obtaining the observation given the

true state of nature

zk

:i.