In the field of statistics, there are two main approaches to probability. The usual way probability is taught, for example in most typical introductory statistics courses, represents the *frequentist* or *classical* approach. In the frequentist approach to probability, the population parameters are fixed constants whose values are unknown. These probabilities are defined to be the relative frequencies of the various categories, where the experiment is repeated an indefinitely large number of times. For example, if we toss a fair coin 10 times, it may not be very unusual to observe 80% heads; but if we toss the fair coin 10 trillion times, we can be fairly certain that the proportion of heads will be near 50%. It is this “long-run” behavior that defines probability for the frequentist approach.

However, there are situations for which the classical definition of probability is unclear. For example, what is the probability that terrorists will strike New York City with a dirty bomb? As such an occurrence has never occurred, it is difficult to conceive what the long-run behavior of this gruesome experiment might be. In the frequentist approach to probability, the parameters are fixed, and the randomness lies in the data, which are viewed as a random sample from a given distribution with unknown but fixed parameters.

The *Bayesian* approach to probability turns these assumptions around. In Bayesian statistics, the parameters ...

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