Abstract
This paper draws together four perspectives that contribute to a new understanding of probability and solving problems involving probability. The first is the Subjective Bayesian perspective that probability is affected by one’s knowledge, and that it is updated as one’s knowledge changes. The main criticism of the Bayesian perspective is the problem of assigning prior probabilities; this problem disappears with our Information Theory perspective, in which we take the bold new step of equating probability with information. The main point of the paper is that the formal perspective (formalize, calculate, unformalize) is beneficial to solving probability problems. And finally, the programmer’s perspective provides us with a suitable formalism. To illustrate the benefits of these perspectives, we completely solve the hitherto open problem of the two envelopes.
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References
Dijkstra EW (1989) Fair gambling with a biased coin. http://www.cs.utexas.edu/users/EWD/ewd10xx/EWD1069.PDF
Dietrich F, List C (2005) The two envelope paradox: an axiomatic approach. Mind 114(454): 239–248
Gardner M (1982) Aha! Gotcha! Paradoxes to puzzle and delight. Freeman, New York
Hehner ECR (1977) Information content of programs and operation encoding. JACM 24(2):290–297. http://www.cs.utoronto.ca/~hehner/ICPOE.pdf
Hehner (1993) A practical theory of programming. Springer, Berlin. http://www.cs.utoronto.ca/~hehner/aPToP
Hehner ECR (2004) Probabilistic predicative programming. Mathematics of Program Construction, Stirling Scotland, Springer LNCS 3125:169–185. http://www.cs.utoronto.ca/~hehner/PPP.pdf
Hehner ECR (2007) Unified algebra. Int J Math Sci 1(1):20–37. http://www.cs.utoronto.ca/~hehner/UA.pdf
Kozen DC (1981) Semantics of probabilistic programs. J Comput Syst Sci 22: 328–350
Kraïtchik M (1930) La mathématique des jeux. Stevens, Bruxelles
Katz B, Olin D (2007) A tale of two envelopes. Mind 116(464): 903–926
The Monty Hall problem. en.wikipedia.org/wiki/Monty_Hall_problem
McIver AK, Morgan CC (2005) Abstraction, refinement and proof for probabilistic systems. Springer, Berlin
Morgan CC, McIver AK, Seidel K, Sanders JW (1996) Probabilistic predicate transformers. ACM Trans Program Lang Syst 18(3): 325–353
Pascal’s wager. plato.stanford.edu/entries/pascal-wager
Rosenthal JS (2006) A first look at rigorous probability theory, 2nd edn. World Scientific Publishing, Singapore
Rosenthal JS (2008) Monty Hall, Monty Fall, Monty Crawl. Math Horizons, pp 5–7. probability.ca/jeff/writing/montyfall.pdf
de Roever WP, Engelhardt K (1998) Data refinement: model-oriented proof methods and their comparisons. Tracts in Theoretical Computer Science, vol 47. Cambridge University Press, Cambridge
Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656
Shannon CE, Weaver W (1949) The mathematical theory of communication. University of Illinois Press, Illinois
Sanders JW, Zuliani P (2000) Quantum programming. Mathematics of program construction, Ponte de Lima Portugal, Springer LNCS, vol 1837
Tafliovich A, Hehner ECR (2006) Predicative quantum programming. Mathematics of program construction, Kuressaare Estonia, Springer LNCS 4014: 433–454
Zuliani P (2004) Non-deterministic quantum programming. In: Second international workshop on quantum programming languages, pp 179–195
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I. Hayes and J. Woodcock
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Hehner, E.C.R. A probability perspective. Form Asp Comp 23, 391–419 (2011). https://doi.org/10.1007/s00165-010-0157-0
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DOI: https://doi.org/10.1007/s00165-010-0157-0