Abstract
In this chapter some basic features of Bayesian statistics (BS) are described and some methods used in BS for making multinomial inferences are illustrated.
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Various rules of inductive acceptance have been proposed in statistics and epistemology. Furthermore, the meaning of the inductive acceptance of a hypothesis has been interpreted in different ways. While the `weakest’ type of acceptance presumably involves selection of a hypothesis for further scrutiny (see, for instance, Popper’s methodological rules and Fisher’s significance tests), the `strongest’ type presumably involves the incorporation of the accepted hypothesis within the available background knowledge which will be taken as guaranteed in subsequent inquiries (see, for instance, Levi’s acceptance rules (1976)). Other rules of inductive acceptance seem to lie somewhere between these two `extremes’. For instance, point estimation could be seen as a type of `as if-acceptance’ where the estimated value of a given parameter is tentatively used, especially in calculations. A stronger type of inductive acceptance appears to be represented by interval estimation, and an even stronger type by the Neyman-Pearson hypothesis testing.
The probabilistic approach has been supported by Carnap (1968a), Jeffrey (1956, 1968, 1970), Bar-Hillel (1968), Box and Tiao (1973, Appendix A5.6) and others. The supporters of the acceptational approach include, among others, Hempel (1962), Hintikka and Pietarinen (1966), Hilpinen (1968), Kyburg (1974), Levi (1967, 1976, 1980), Kaplan (1981), Harsanyi (1985), La Valle (1970) and Zellner (1971).
The origins of the predictivistic approach to statistical inference date back at least to 1774 with the famous Laplace’s rule of succession (cf. Zabel], 1989). The predictivistic approach received strong backing from Roberts (1965). A systematic exposition of the predictive methods used in statistics is given by Aitchison and Dunsmore (1975). In conflict with the monistic approaches mentioned in the text - which emphasize the methodological relevance of one of the different kinds of inferences - it could be argued that different kinds of inductive inferences are appropriate in different situations. For instance, it seems clear that predictive inferences play a crucial role in risk analysis while in theoretical physics global inferences are much more important. This suggests that a pluralistic approach would provide a more adequate image of the inductive inferences demanded in scientific practice.
The predictive distribution p(y/z) specifies the epistemic probabilities of the possible results of a future experiment y relative to the result z of a previously performed experiment z.
Winkler (1972, p. 143) points out, continuous distributions “are studied because (1) they are often mathematically easier to work than discrete models, and (2) they provide an excellent approximation to numerous discrete models.”
ere p(9/y) and p(9) are either both probability functions or both density functions, according to whether A is discrete or continuous, and p(y/B) and p(y) are likewise either both probability functions or density functions according to whether y is discrete or continuous.
Aitchison and Dunsmore (1975, pp. 1 and 17–23).
mula (42) follows from the sequence of equalities: VAR(q) = Evar(q,) (from (37)) = E(E(gi) - [E(q,)]2)(from (20)(b)) = EE(gi) - E[E(q,)]2 = E(Ice) - C[E(q)] = E[C(q)] - C[E(q)]. Taking into account that E[G(q)] = 1 - E[C(q)] and G[E(q)] = 1 - C[E(q)] it follows that VAR(q) = G[(E(q)] - E[G(q)]).
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© 1993 Springer Science+Business Media Dordrecht
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Festa, R. (1993). Bayesian Statistics and Multinomial Inferences: Basic Features. In: Optimum Inductive Methods. Synthese Library, vol 232. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8131-8_3
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DOI: https://doi.org/10.1007/978-94-015-8131-8_3
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