Probabilistic Logics and Probabilistic Networks
While probabilistic logics in principle might be applied to solve a range of problems, in practice they are rarely applied --- perhaps because they seem disparate, complicated, and computationally intractable. This programmatic book argues that several ap
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Bayesian Statistical Inference
Bayesian statistics is much more easily connected to the inferential problem of Schema (1.1) than classical statistics. The feature that distinguishes Bayesian statistical inference from classical statistics is that it also employs probability assignments over statistical hypotheses. It is therefore possible to present a Bayesian statistical procedure as an inference concerning probability assignments over hypotheses. Recall that we called the inference of probability assignments over data on the assumption of a statistical hypothesis direct. Because in Bayesian inference we derive a probability assignment over hypotheses on the basis of data, it is sometimes called indirect inference. The basic structure of Bayesian statistical inference extends to Bayesian inference outside the statistical domain, as it is for example used in philosophical and psychological modelling. Any such inference starts with a combination of probability assignments, from which further probability assignments are derived using Bayes’ theorem. And because this is a theorem of probability, Bayesian inference is also very close to the inferences dealt with in §2, which are based solely on the axioms as well. However, links to the standard semantics and to the role of Bayesian inference in philosophy and psychology are not discussed here.
6.1 Background While, as illustrated in Howson and Urbach (1993), Bayesian inference has made its mark on numerous domains of probabilistic inference in science, its earliest application, in the work of Bayes himself, is in statistical inference (Earman, 1992). Nevertheless, in the development of statistical science during the previous century, Bayesian inference played a minor role. It is only with the development of the proper computational tools and machinery that this has taken a turn for the better. Presently, Bayesian statistics enjoys a growing popularity, as witnessed by numerous good introductory textbooks Bernardo and Smith (2000); Press (2003); Gelman et al. (2003)
R. Haenni et al., Probabilistic Logics and Probabilistic Networks, Synthese Library 350, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0008-6 6,
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6 Bayesian Statistical Inference
It should be emphasized that not everyone is equally enthusiastic about the recent popularity of Bayesian techniques. Mayo (1996) argues forcefully against the Bayesian use of priors and the Bayesian reliance on likelihoods as the sole mediator between evidence and hypotheses, pointing to counterintuitive consequences of the latter view, e.g., on the issue of optional stopping. And even within the Bayesian camp itself, views diverge widely on what exactly characterizes Bayesian statistics: some authors side with strict subjectivists like De Finetti, interpreting all probabilities as degrees of belief, others propose to interpret at least some probabilities as objective, or pertaining to frequencies, and adhere to so-called mixed Bayesianism Jeffrey (1992). But none of these matters need concern us
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