15.4 Measuring Probability 363
Figure 15.3 Examples of information combination processes. The process on the left
side combines two random variables that have a mutual dependency. The process on
the right side combines random variables that are not directly observable.
15.4 Measuring Probability
So far we have focused on PDs as a means of expressing an observation.
The range of possibilities for what one can observe is very large, including
concentrations, temperatures, pressures, and simple Boolean observations.
However, some of the most important observations are measurements of
PDs, and a large array of statistical tests (such ast-, chi-square, andF-tests)
are concerned with such measurements. When observing a PD, one must
be careful to distinguish the PD that is being measured from the one that is
used for expressing the observation of the PD. It can get confusing because
the observation is the PD of a PD.
To understand what this means, consider the problem of determining the
body mass index (BMI) of individuals in a population. If one just focuses
on the BMI measurement, then one will not capture the variation of BMI in
the population. As data are accumulated, one will get more accurate mea-
surements of the average BMI, and that is all. In practice, one is interested
in the distribution of values in a population. In other words, one is mea-
suring a PD. Although population distributions cannot be exactly normally
distributed, since they are finite distributions, they can usually be approxi-
mated by a normal distribution.
