RE: About Probability

[David A. Heiser]

I would like to enter the arena.   I see the original question as two questions, one about probability in a general sense, and the second about probability as used within Bayes Theorem. This is in line with the historical arguments.   Most statisticians (from Fisher down to the present) recognize an objective probabilility and a subjective probability. In technical terms it is the ontological and epistemic viewpoints. The following probability classes have aspects of both ontological and epistemic viewpoints, which muddies the whole thing. There really is nothing very clear in any "definition" of probability. A frequentist has belief in that the replications "are random, identical draws, events, rolls, etc.". This is axiomatic to a frequentist definition of probability. It is an epistemic viewpoint. Von Mises clearly saw that any frequentist definition of probability is circular. So we are back to the starting line.   Lets look at a set of definitions:   1. The outcome of an independent replicate from a defined population. This is the basis of the frequentist definition, where probability comes from a large number of replicates and probability is the ratio of the number of desired events to the total number of replicates.   The following are mathematical, and are highly ontological. 2. The greatest upper bound on a set of significance levels. 3. The degree that an observation (or experiment) supports a hypothesis. 4. A confidence, tolerance or prediction interval. 5. The weights (to and from) the hidden layers in a neural network. 6. The outcome from a mathematical expression/equation.   The following are subjective and highly epistemic: 7. A degree of belief based on past experience. 8. The degree of membership in a fuzzy set. 9. A measure of truth (in the religious/judicial/legal sense). 10. A measure of what will happen based on a religious belief in cause and effect.   The only criteria is that the outcome of any of the above from definitions 1 to 10, is that it can be represented by a real number from zero to one, where both zero and one represent states that (outside of mathematics) can never be reached.   Definitions 2-4 have epistemic aspects in our belief in what is an appropriate distribution or test.   Some of the experts further take the position that probability only can be interpreted and used where "chance" is involved.   Bayes entry in 1763 profoundly affected our viewpoints on probability, which even in 2000 has not settled down. In a mathematical sense, any mathematical expression that represents a probability (density function) can be used as a prior in the Bayesian approach. This varies from the non-informative to the informative. Most of the historical theoretical work on understating the Bayesian concept was based on the binomial distribution.   To a Bayesian (Martz and many others), the cornerstone in Bayesian inference is subjective probability, to which a "degree of belief" can be attached. Therefore there is a lack of consistency among different investigators in the outcome or posterior conclusions.   In addition decision theory can be applied to a Bayesian analysis, which gives a different viewpoint on making conclusions.   In Bayesian statistics, anything goes as long as the fundamentals are followed:       Posterior Distribution = (Prior Distribution) * (Likelihood function) / (Marginal distribution)   The subjective aspect allows one to have great freedom in defining what probability is. If you can construct a prior on your belief in the actions of God, you can use it.   DAH