Abstract
De Finetti suggested that scoring rules – namely, loss functions by which a forecaster is virtually charged depending on the degree of inaccuracy of his predictions – could be employed also to provide a compelling argument for probabilism. However, De Finetti’s choice of a specific scoring rule for this purpose (Brier’s quadratic rule) appears somewhat arbitrary, and the general pragmatic flavour of the argument – which makes it a variant of the well-known “Dutch Book Theorem” – has been deemed unsuitable for an epistemic justification of probabilism. In this paper we suggest how Brier’s rule may be justified on epistemic grounds by means of a strategy that is different from the one usually adopted for this purpose (e.g., in Joyce 1998), taking advantage of a recent characterization result concerning distance functions between real-valued vectors (D’Agostino and Dardanoni 2008).
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- 1.
Joyce (1998), p. 584.
- 2.
As Murphy and Winkler put it: “Categorical or deterministic forecasts suffer from two serious deficiencies. First, a forecaster or forecasting system is seldom, if ever, certain of which event will occur. Second, categorical forecasts do not provide users of the forecasts with the information that they need to make rational decisions in uncertain situations.”(Murphy and Winkler 1984, p. 489).
- 3.
Our use of the expression “distance function”, in this paper, is rather loose and refers to any continuous function of two real-valued vectors (of the same finite size) which can be intuitively regarded as measuring their distance, including functions which do not fully satisfy the standard textbook definition. Such non-standard “distance” functions have often been used to solve measuring problems in several applications areas.
- 4.
Some authors prefer to define scoring rules so as the forecaster’s goal is that of maximizing his score. Trivial algebraic manipulation is sufficient to turn one scenery into the other.
- 5.
For this purpose, it is sufficient to add \((1 - (x + y))/2\) to both x and y.
- 6.
- 7.
On this point, see the already cited (Joyce 1998).
- 8.
- 9.
Observe that any scoring rule based on the generalized α-power (with α > 1) of the absolute differences between subjective estimates and observed outcomes is proper. On this point see (Selten 1998).
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Acknowledgements
We wish to thank Wolfgang Spohn for very useful suggestions.
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D’Agostino, M., Sinigaglia, C. (2009). Epistemic Accuracy and Subjective Probability. In: Suárez, M., Dorato, M., Rédei, M. (eds) EPSA Epistemology and Methodology of Science. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3263-8_8
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