Abstract
This chapter provides definitions of the statistical properties of Bayes Theorem and Bayesian Belief Network (BBN). These areas include: (1) Axioms of Probability, (2) Base-Rate Fallacy, (3) Bayes Theorem probabilities, (4) Joint and Disjoint BBN Structures, (5) Bayesian Updating, (6) Certain Event, (7) Categorical Variable, (8) Chain (Product) Rule, (9) Collectively Exhaustive, (10) Combinations and Permutations, (11) Complement and Complement Rule, (12) Conditional and Unconditional Probability, (13) Counting and Countable Set and Uncountable Set, (14) Complement and Complement Rule, (15) Discrete Variable, (16) Disjoint or Mutually Exclusive Events (Sets), (17) Event, (18) Factorial, (19) Intersection and Union (of Sets), (20) Independence and Pairwise Independence, (21) Joint and Marginal Probability Distribution and Marginalization, (22) Law of Total Probability, (23) Mean (Arithmetic Mean), (24) Ordinal Variable, (25) Outcome (Outcome Space), (26) Parameter, (27) Partition, (28) Population, (29) Probability and Probability Sample, (30) Sample, Sample Size, Sample Space, Random Sample, Simple Random Sample, Random Experiment (Event), and Random Variable, (31) Real Number, (32) Set, Subset, Member of a Set, and Empty Set, (33) Theories of Probability, (34) Unit, (35) Venn Diagram, (36) The Algebra of Sets, and (37) Zero Sum Marginal Difference.
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Notes
- 1.
Bolstad (2007).
- 2.
Meaning when these events occur in nature.
- 3.
Except in the case of a two-node BBN where there is only one possible path.
- 4.
Disjoint BBN are subset of parent joint BBN.
- 5.
Taken from:https://en.wikipedia.org/wiki/Disjoint_sets.
- 6.
We also use the comma to represent the joint symbol ∩ for simplicity. For example: P(A 1 ∩ A 2) = P(A 1, A 2).
- 7.
This is the null set.
- 8.
We used the online combinations calculator @ https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html (14 September 2016).
- 9.
Note that the special cases A, B = (A and B are mutually exclusive) and A, B = B (B is a subset of A) agree with our intuition, as described at the top of this paragraph.
- 10.
- 11.
We derived this definition from Hebert et al. (2007).
- 12.
In this manual, strikethroughs represent disjoint events, that are marginalized away, because they are independent of the other events.
- 13.
Hebert et al. (2007) also suggests that it is important to note that these probabilities are conditional. They specify the degree of belief in some proposition or propositions, based on an initial assumption that some other propositions are true.
- 14.
- 15.
In this manual strikethroughs, represent disjoint events that are marginalized away because they are independent of the other events.
- 16.
- 17.
See Stoll (1979) for the proofs to these identities.
- 18.
Here, \(\bar{A} = U - A\).
- 19.
Here, \(\bar{A} = U - A\).
- 20.
This verifies that the elements of the joints in the True and False events sum to zero.
References
Bolstad, W.M. (2007). Introduction to Bayesian statistics (2nd ed.). Hoboken, NJ: Wiley.
Disjoint sets. (n.d.). Retrieved November 06, 2016, from https://en.wikipedia.org/wiki/Disjoint_sets
Hebert, S., Lee, V., Morabito, M., & Polan, J. (2007). Bayesian network theory. https://controls.engin.umich.edu/wiki/index.php/Bayesian_network_theory. Accessed 23 July 2012.
Law of total probability (n.d). Retrieved November 06, 2016, from https://en.wikipedia.org/wiki/Law_of_total_probability
Stark, P. B. (2012). Glossary of statistical terms. http://statistics.berkeley.edu/~stark/SticiGui/Text/gloss.htm. Accessed 14 September 2016.
Statistics Glossary – probability. (n.d.). Retrieved November 06, 2016, from http://www.stats.gla.ac.uk/steps/glossary/probability.html/#event
Stoll, R. R. (1979). Set theory and logic. Mineola, NY: Dover Publications.
Acknowledgements
I would like to thank Dr. Philip B. Stark, Department of Statistics, University of California, Berkeley for allowing me to use his statistical terminology and definitions liberally from his website, SticiGui, found @ http://www.stat.berkeley.edu/~stark/SticiGui/Text/gloss.htm (See Stark 2012).
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Grover, J. (2016). Statistical Properties of Bayes’ Theorem. In: The Manual of Strategic Economic Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-319-48414-3_3
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