Part I Frameworks for Probability § 2 Variables

This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.

[1]  Jeff B. Paris,et al.  Common sense and stochastic independence , 2001 .

[2]  B. D. Finetti,et al.  Foresight: Its Logical Laws, Its Subjective Sources , 1992 .

[3]  Jon Williamson.  From Bayesianism to the Epistemic View of Mathematics: Remarks motivated by Richard Jeffrey’s ‘Subjective Probability: The Real thing' , 2006 .

[4]  Hans Reichenbach.  Book Reviews: The Theory of Probability: An Inquiry into the Logical and Mathematical Foundations of the Calculus of Probability , 1950 .

[5]  William Harper,et al.  Motivating Objective Bayesianism: From Empirical Constraints to Objective Probabilities , 2006 .

[6]  J. Marquis.  Abstract Mathematical Tools and Machines for Mathematics , 1997 .

[7]  Karl Raimund Sir Popper,et al.  Realism and the aim of science , 1983 .

[8]  Jon Williamson,et al.  From Bayesianism to the Epistemic View of Mathematics Richard Jeffrey. Subjective Probability: The Real Thing. Cambridge: Cambridge University Press, 2004. ISBN 0-521-82971-2 (hbk), 0-521-53668-5 (pbk). Pp. xvi + 124. , 2006 .

[9]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[10]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[11]  Martín Abadi,et al.  Decidability and Expressiveness for First-Order Logics of Probability , 1994, Inf. Comput..

[12]  David Lewis,et al.  Papers in metaphysics and epistemology: Humean Supervenience debugged , 1999 .

[13]  Johann Schumann,et al.  Automated Theorem Proving in Software Engineering , 2001, Springer Berlin Heidelberg.

[14]  Jeff B. Paris,et al.  A note on the inevitability of maximum entropy , 1990, Int. J. Approx. Reason..

[15]  Karl R. Popper,et al.  A World of Propensities , 1993, Popper's Views on Natural and Social Science.

[16]  Jon Williamson,et al.  Objective Bayesianism, Bayesian conditionalisation and voluntarism , 2009, Synthese.

[17]  John Venn,et al.  The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability, with Especial Reference to Its Logical Bearings and Its Application to Moral and Social Science , 1876 .

[18]  Edwin T. Jaynes,et al.  Prior Probabilities , 1968, Encyclopedia of Machine Learning.

[19]  E. B. Davies.  TOWARDS A PHILOSOPHY OF REAL MATHEMATICS , 2011 .

[20]  David Miller,et al.  Critical Rationalism: A Restatement and Defence , 1998 .

[21]  A. Church.  An Unsolvable Problem of Elementary Number Theory , 1936 .

[22]  G. Faè,et al.  The physical review , 1895 .

[23]  Jeff B. Paris,et al.  In defense of the maximum entropy inference process , 1997, Int. J. Approx. Reason..

[24]  John Earman,et al.  Bayes or bust , 1992 .

[25]  E. T. Jaynes,et al.  Where do we Stand on Maximum Entropy , 1979 .

[26]  Daniel Hunter,et al.  Causality and maximum entropy updating , 1989, Int. J. Approx. Reason..

[27]  Gai CarSO.  A Logic for Reasoning about Probabilities * , 2004 .

[28]  Jon Williamson,et al.  Objective Bayesianism with predicate languages , 2008, Synthese.

[29]  Jon Williamson,et al.  Objective Bayesian Nets for Systems Modelling and Prognosis in Breast Cancer , 2008, Innovations in Bayesian Networks.

[30]  James O. Berger,et al.  Objective Bayesian Methods for Model Selection: Introduction and Comparison , 2001 .

[31]  K. Popper.  The Propensity Interpretation of Probability , 1959 .

[32]  Colin Howson,et al.  Probability and logic , 2003, J. Appl. Log..

[33]  D. Gillies.  Philosophical Theories of Probability , 2000 .

[34]  Colin Howson,et al.  The Logic of Bayesian Probability , 2001 .

[35]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[36]  D. Lewis.  A Subjectivist’s Guide to Objective Chance , 1980 .

[37]  A. Paseau.  Naturalism in Mathematics and the Authority of Philosophy , 2005, The British Journal for the Philosophy of Science.

[38]  P. Laplace.  A Philosophical Essay On Probabilities , 1902 .

[39]  Colin Howson,et al.  Can logic be combined with probability? Probably , 2009, J. Appl. Log..

[40]  Jeff B. Paris,et al.  The emergence of reasons conjecture , 2003, J. Appl. Log..

[41]  Jon Williamson,et al.  Objective Bayesian probabilistic logic , 2008, J. Algorithms.

[42]  Jon Williamson,et al.  Objective Bayesian Nets , 2005, We Will Show Them!.

[43]  H. Gaifman.  Concerning measures in first order calculi , 1964 .

[44]  J. B. Paris,et al.  The Uncertain Reasoner's Companion: Bibliography , 1995 .

[45]  J. W. Addison,et al.  Automated development of fundamental mathematical theories , 1992 .

[46]  Stephen Senn.  Scientific reasoning, the bayesian approach. Colin Howson and Peter Urbach, Open Court, La Salle Illinois, 1989. No. of pages: xii + 312. Price: $34.95 , 1991 .

[47]  Donald C. Williams,et al.  The Theory of Probability: An Inquiry into the Logical and Mathematical Foundations of the Calculus of Probability. , 1950 .

[48]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[49]  Edwin T. Jaynes,et al.  Inference, Method, and Decision: Towards a Bayesian Philosophy of Science. , 1979 .

[50]  John D. Lafferty,et al.  Inducing Features of Random Fields , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[51]  J. Keynes.  A Treatise on Probability. , 1923 .

[52]  Ian Hacking,et al.  The Emergence of Probability: Duality , 2006 .

[53]  A. Wald,et al.  Probability, statistics and truth , 1939 .

[54]  Jon Williamson,et al.  Countable Additivity and Subjective Probability , 1999, The British Journal for the Philosophy of Science.

[55]  Richard Von Mises,et al.  Mathematical Theory of Probability and Statistics , 1966 .

[56]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[57]  Joseph Y. Halpern,et al.  Representation Dependence in Probabilistic Inference , 1995, IJCAI.

[58]  Steve Awodey.  An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’† , 2004 .

[59]  Marc Snir,et al.  Probabilities over rich languages, testing and randomness , 1982, Journal of Symbolic Logic.

[60]  E. Jaynes.  The well-posed problem , 1973 .

[61]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[62]  Gary James Jason,et al.  The Logic of Scientific Discovery , 1988 .

[63]  P. Spreij.  Probability and Measure , 1996 .

[64]  James H. Fetzer.  Probabilistic Explanations , 1982, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[65]  David Hilbert.  Philosophy of mathematics: On the infinite , 1984 .

[66]  R. Baierlein.  Probability Theory: The Logic of Science , 2004 .

[67]  Jon Williamson.  Inductive Influence , 2007, The British Journal for the Philosophy of Science.

[68]  Hilary Putnam,et al.  Philosophy of mathematics : selected readings , 1984 .

[69]  Rudolf Carnap,et al.  The continuum of inductive methods , 1952 .

[70]  Sean R. Eddy,et al.  Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids , 1998 .

[71]  R. V. Mises,et al.  Mathematical Theory of Probability and Statistics , 1966 .

[72]  Jon Williamson,et al.  Bayesian Nets and Causality: Philosophical and Computational Foundations , 2005 .

[73]  E. T. Jaynes,et al.  The Relation of Bayesian and Maximum Entropy Methods , 1988 .

[74]  Joseph Y. Halpern.  An Analysis of First-Order Logics of Probability , 1989, IJCAI.