Bayesian updating is particularly important in the dynamic analysis of a sequence of data.
Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law.
For example, if one does not know whether the newborn baby next door is a boy or a girl, the color of decorations on the crib in front of the door may support the hypothesis of one gender or the other; but if in front of that door, instead of the crib, a dog kennel is found, the posterior probability that the family next door gave birth to a dog remains small in spite of the "evidence", since one's prior belief in such a hypothesis was already extremely small.
The critical point about Bayesian inference, then, is that it provides a principled way of combining new evidence with prior beliefs, through the application of Bayes' rule.
Bayesian theory calls for the use of the posterior predictive distribution to do predictive inference, i.e., to predict the distribution of a new, unobserved data point.