Definition
The two most important concepts used in Bayesian modeling are probability and utility. Probabilities are used to model our belief about the state of the world and utilities are used to model the value to us of different outcomes, thus to model costs and benefits. Probabilities are represented in the form of p(x | C), where C is the current known context and x is some event(s) of interest from a space χ. The left and right arguments of the probability function are in general propositions (in the logical sense). Probabilities are updated based on new evidence or outcomes y using Bayes rule, which takes the form
where χ is the discrete domain of x. More generally, any measurable set can be used for the domain χ. An integral or mixed sum and integral can replace the sum. For a utility function u(x) of some event x, for instance the benefit of a particular outcome, the expected value of u() is
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Recommended Reading
A good introduction to the problems of uncertainty and philosophical issues behind the Bayesian treatment of probability is in Lindley (2006). From the statistical machine learning perspective, a good introductory text is by MacKay (2003) who carefully covers information theory, probability, and inference but not so much statistical machine learning. Another alternative introduction to probabilities is the posthumously completed and published work of Jaynes (2003).
Discussions from the frequentist versus Bayesian battlefront can be found in works such as (Rosenkrantz and Jaynes, 1983), and in the approximate artificial intelligence versus probabilistic battlefront in discussion articles such as Cheeseman’s (1988) and the many responses and rebuttals. It should be noted that it is the continued success in applications that have really led these methods into the mainstream, not the entertaining polemics.
Good mathematical statistics text books, such as Casella (2001) cover the breadth of statistical methods and therefore handle basic Bayesian theory. A more comprehensive treatment is given in Bayesian texts such as Gelman et al. (2003).
Most advanced statistical machine learning text books cover Bayesian methods, but to fully understand the subtleties of prior beliefs and Bayesian methodology one needs to view more advanced Bayesian literature. A detailed theoretical reference for Bayesian methods is Bernardo and Smith (1994).
Bernardo, J., & Smith, A. (1994). Bayesian theory. Chichester: Wiley.
Casella, G., & Berger, R. (2001). Statistical inference (2nd ed.). Pacific Grove: Duxbury.
Cheeseman, P. (1988). An inquiry into computer understanding. Computational Intelligence, 4(1), 58–66.
Gelman, A., Carlin, J., Stern, H., & Rubin, D. (2003). Bayesian data analysis (2nd ed.). Boca Raton: Chapman & Hall/CRC Press.
Horvitz, E., Heckerman, D., & Langlotz, C. (1986). A framework for comparing alternative formalisms for plausible reasoning. Fifth National Conference on Artificial Intelligence, Philadelphia, pp. 210–214.
Jaynes, E. (2003). Probability theory: the logic of science. New York: Cambridge University Press.
Lindley, D. (2006). Understanding uncertainty. Hoboken: Wiley.
MacKay, D. (2003). Information theory, inference, and learning algorithms. Cambridge: Cambridge University Press.
Rosenkrantz, R. (Ed.). (1983). E.T. Jaynes: papers on probability, statistics and statistical physics. Dordrecht: D. Reidel.
Wainwright, M. J., & Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Hanover: NowPublishers.
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Buntine, W. (2011). Bayesian Methods. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30164-8_63
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