Tuesday, 12 March 2013

Probability preferences

In order to support my claim that Pascal (and to some extent, Fermat) are too highly praised in the history of probability theory, I'd like to make a claim about what I see as important in the constellation of ideas around the birth of probability theory.  This is my opinion, and is based on what I know that has happened in the subject of probability theory since the time of Cardano, Pascal, Fermat and Huygens.

Concepts of primary importance in probability theory (in the pre-Kolmogorov world of Cardano, Fermat, Pascal)
  1. Event Space
  2. Independence. 
  3. Conjunction and disjunction.     
  4. Equivalence class.  
  5. Parallel/sequential irrelevance of future outcomes.  
  6. A relation between historical observed regularities and multiple future possible worlds.  
  7. A clear separation between the implementation of the random process(es) and the implementation of the activity, game, contract, etc. which utilises the source of randomness.

Concepts of secondary importance.
  1. Equi-probable event space. 
  2. Expectation.  
  3. Single versus multiple random sources.  
  4. Law of large numbers (though it is of primary importance to the dependent subject of statistics).
  5. i.i.d. (two or more random sources which are independent and identically distributed)
  6. A Bernoulli scheme
  7. The binomial distribution
  8. Stirling's approximation for n factorial
  9. The normal distribution
  10. Information content of a random device
  11. Identification of the activity, game, contract, etc, as purely random, or additionally strategic. 

I'd like to say something about each of these in turn.

Before I do, I'd like to say this - the Greeks didn't develop probability theory, as Bernstein and also David suggest, due to a preference for theory over experimentation, but perhaps because probabilities are ratios, and the Indians didn't invent base ten positional number notation until the eighth century A.D., making subsequent manipulations of these ratios more notationally bearable.  No doubt the early renaissance love of experimentation (Bacon and Galileo) may have assisted in drawing the parallel between the outcome of a scientific experiment and the outcome of a randomisation machine.