Databases Reference
In-Depth Information
the estimate of the probability, of turning on a television set in the middle of a commercial is
0.8. Our experiment
E
here is turning on a television set, and the outcomes are
commercial
and
no commercial
. We could have been more careful with noting what was on when we turned
on the television set and noticed whether the program was a news program (2000 times), a
newslike program (20,000 times), a comedy program (40,000 times), an adventure program
(18,000 times), a variety show (20,000 times), a talk show (90,000 times), or a movie (10,000
times), and whether the commercial was for products or services. In this case the outcomes
would be
product commercial, service commercial, comedy, adventure, news, pseudonews,
variety, talk show
, and
movie
. We could then define an
event
as a set of outcomes. The event
commercial
would consist of the outcomes
product commercial
, and
service commercial
;the
event
no commercial
would consist of the outcomes
comedy, adventure, news, pseudonews,
variety, talk show
, and
movie
. We could also define other events such as
programs that may
contain news
. This set would contain the outcomes
news, pseudonews
, and
talk shows
, and
the frequency of occurrence of this set is 0.112.
Formally, when we define an experiment
E
, we also define a
sample space S
associated
with the experiment that consists of the
outcomes
. We can then combine these outcomes
into sets that are called
events
, and assign probabilities to these events. The largest subset
of
S
(event) is
S
itself, and the probability of the event
S
is simply the probability that the
experiment will have an outcome. The way we have defined things, this probability is one;
that is,
P
{
ω
i
}
(
S
)
=
1.
A.1.2 A Measure of Belief
Sometimes the idea that the probability of an event is obtained through the repetitions of an
experiment runs into trouble. What, for example, is the probability of your getting from Logan
Airport to a specific address in Boston in a specified period of time? The answer depends on a
lot of different factors, including your knowledge of the area, the time of day, the condition of
your transport, and so on. You cannot conduct an experiment and get your answer because the
moment you conduct the experiment, the conditions have changed, and the answer will now be
different. We deal with this situation by defining a priori and a posteriori probabilities. The a
priori probability is what you think or believe the probability to be before certain information
is received or certain events take place; the a posteriori probability is the probability after
you have received further information. Probability is no longer as rigidly defined as in the
frequency of occurrence approach but is a somewhat more fluid quantity, the value of which
changes with changing experience. For this approach to be useful we have to have a way
of describing how the probability evolves with changing information. This is done through
the use of
Bayes' rule
, named after the person who first described it. If
P
(
)
A
is the a priori
probability of the event
A
and
P
(
A
|
B
)
is the a posteriori probability of the event
A
given that
the event
B
has occurred, then
P
(
A
,
B
)
P
(
A
|
B
)
=
(A.1)
P
(
B
)
