This article is a summary of the International Society for Bayesian Analysis's (ISBA) definition of the basics of the Bayesian theorem and analysis. The according to the ISBA website, the organization "was founded in 1992 to promote the development and application of Bayesian analysis useful in the solution of theoretical and applied problems in science, industry and government. By sponsoring and organizing meetings, publishing the electronic journal of Bayesian statistics Bayesian Analysis, and other activities ISBA provides a focal point for those interested in Bayesian analysis and its applications".
Bayesian analysis, a statistical tool for handling probability distributions, got its start in the mid 18th century. It was not, however, until the 1980s when modern computers were able to handle the complex computations that made Bayesian implementation difficult, that Bayesian analysis gained more widespread acceptance. Since then, its use has increased in popularity, being used in many different applications--from healthcare, to weather, to criminal justice. Despite the many nuanced manifestations of Bayesian analysis, it serves a common application: to analyze the probability of unknown and uncertain occurences.
The left side of the equation expresses the known quantities--"parameters"--as a probability of the current data--"prior distribution". 'y' represents the new data that enters into the calculation. Thus, the "'likelihood,' [is] proportional to the distribution of the observed data given the model parameters.
On the right, the equation's new probablity distribution (posterior distribution) is read: "posterior is proportional to the prior times the likelihood".
- Many diverse applications
- "Philosophical consistency"
- Lacks problems that are associated with other 'frequentist' methods
- Produce clear answers, products
- Reformulates for each variable
- Subjective nature of prior probabilities-"your prior information is different from mine".
- More complex problems require more powerful computational tools