Abstract
A decisive stimulus to the development of formal decision theory was provided by the formulation of the principle of expected gain. We have seen how this principle was used to settle the question of how the stakes of an interrupted gambling game were to be divided. The term ‘moral expectation’ given to the principle at the time reflected a normative interpretation of expected gain: what the player could ‘justly’ expect.
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Notes
Parish records show that William Shakespear was baptized on April 26, 1564. April 23 is commonly surmised to be the date of his birth. The exact date is unknown. To continue with this bizarre but instructive example of a misconception about the meaning of probability, recall that in the Julian calendar April was the second month (hence September, October, etc. were seventh, eighth, etc.). If we write Shakespear’s birthdate 23.2 or 2.23 and interpret the period as a sign for multiplication, we again get 46! How much more ‘evidence’ is required to endow this number with some mystical property somehow connected with Shakespear?
The expression on the right side of (5.12) is derived from the normalizing condition. Let x/a = y. Then dy=(1/a)dx. f f 0, (1/a) f(x/a)dx= f °°, f (y)dy = 1, as required.
A function that maps a function on a scalar is called a functional. The integrand of a definite integral is a function; the value of the integral is a scalar. Thus, a definite integral is a functional. ° The entropy of a density is a measure of the uncertainty associated with the random variable it represents. A density function concentrated around a particular value has low entropy. A density representing a uniform distribution has maximum entropy. It represents maximum ignorance about the value that a random variable will assume.
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© 1989 Springer Science+Business Media Dordrecht
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Rapoport, A. (1989). Subjective Aspects of Risk. In: Decision Theory and Decision Behaviour. Theory and Decision Library, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7840-0_6
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DOI: https://doi.org/10.1007/978-94-015-7840-0_6
Publisher Name: Springer, Dordrecht
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