In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit pull in opposite directions. To solve this problem, two proposals, the first one based on Bayes's theorem criterion (BTC) and the second one advocated by Forster and Sober based on Akaike's Information Criterion (AIC) are discussed. We show that AIC, which is frequentist in spirit, is logically equivalent to BTC, provided that a suitable choice of priors is made. We evaluate the charges against Bayesianism and contend that AIC approach (...) has shortcomings. We also discuss the relationship between Schwarz's Bayesian Information Criterion and BTC. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit, pull in opposite directions. To this problem, we propose a solution that strikes a balance between simplicity and goodness-of-fit. Using Bayes' theorem we argue that the notion of prior probability represents a measurement of simplicity of a theory, whereas the notion of likelihood represents the theory's goodness-of-fit. We justify the use of prior probability and show how to calculate the likelihood of a family of curves. We diagnose the relationship (...) between simplicity of a theory and its predictive accuracy. (shrink)

The underdetermination thesis poses a threat to rational choice of scientific theories. We discuss two arguments for the thesis. One draws its strength from deductivism together with the existence thesis, and the other is defended on the basis of the failure of a reliable inductive method. We adopt a partially subjective/objective pragmatic Bayesian epistemology of science framework, and reject both arguments for the thesis. Thus, in science we are able to reinstate rational choice called into question by the underdetermination thesis.

This chapter introduces the definitions of the notion of a random sequence using the three main ideas described that have dominated algorithmic randomness. There are three key approaches for defining randomness in sequences: unpredictability, typicality, and incompressibility. A notion is set up notation for strings and sequences, the Cantor Space, which forms the basic framework for carrying out further discussions, and contains a brief and informal introduction to Lebesgue measure on the unit interval and the Cantor space, which one treats (...) as equivalent. Discussion of mathematical randomness defines a series of classical stochastic or frequency properties in an effort to distill out randomness. This chapter further forms the core material on algorithmic randomness: Von Mises randomness, Martin-Lof randomness, Kolmogorov complexity and randomness of finite strings, application to Godel incompleteness, and Schnorr's theorem. It forms an overview of other forms of related randomness notions that have been studied and indicates the current state of affairs. (shrink)