ABSTRACT
The traditional frequentist approach to determining the probability of an event is to calculate a ratio of the number of occurrences of an event to the total number of trials. The Bayesian approach is a different way of thinking about probability; it is based instead on a subjective interpretation of probability. In the Bayesian approach, it is assumed that there is a prior probability or belief that a person already holds about the likelihood of occurrence of an event, even before gathering information about the event. The person then changes his or her own belief based on new information, and obtains a posterior probability. The objective here will be to explore the way in which updating prior probabilities takes place in light of new information. The foundation underlying this belief-updating inferencing is known as Bayes' rule. If the prior probability follows a certain distribution, Bayesian inferencing can be made by summarizing this new information in a probability distribution. Having the correct assumptions about a prior probability is crucial in the absence of any concrete evidence. We devote a section to the state-of-the-art in constructing priors, and we emphasize inferencing based on conjugate priors in particular.
Recall that the multiplication rule gives the following joint probabilities of two events X and Y :
