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Empiricism and the Interpretation of Probability Federica Russo <Russo@lofs.ucl.ac.be> UCL
Overview Stating the Problem : Why is the Interpretation of Probability relevant for Empiricism ? Empirical Data and Inferential Procedures Inferential Procedures and the Interpretation of Probability Working Hypothesis : De dicto-De re Probability 1
2 Stating the problem : why is the Interpretation of Probability relevant for Empiricism? Subjectivist interpretations Objectivist interpretations What if the subjectivists were right ? Would our ambition to know through experience drop down ? Do probability statements yield knowledge ?
3 The duality of probability In connection with the degree of belief warranted by evidence I suggest : don’t contrapose subjective and objective probability let’s think of probability as a polysemic concept In connection with the tendencies displayed by some chance device to produce stable relative frequencies (~ subjective probability) (~ objective probability)
4 Whithin the empiricist perspective you treat empirical data by inferential procedures to achieve knowledge. What’s empirical data ? What’s inferential procedures ? Observations, i.e. evidence such as the number of successes or failures in a binomial experiment; e.g.: coin tossing, after N tosses, the frequency of heads is n/m Statistical tools to estimate a parameter or to make a prediction; e.g.: coin tossing, to infer P(next H) |e Empirical Data and Inferential Procedures Do statistical inferences yield knowledge ?
5 Some inferential procedures P = population S i = sample a, b, c = individuals A, B = properties Universal Inference (Sample Population) n/m in S are A n/m in P are A Direct Inference (Population Sample) n/m in P are A a is A with probability n/m n/m in P are A n/m in S are A Predictive Inference (Next Case) n/m in S 1 are A n/m in S 2 are A n/m in S are A a is A with probability n/m
6 These inferential (statistical) procedures are inductive inferences In fact, step premisses conclusion known not Known The conclusion is not certain, just probable Do statistical inferences yield knowledge ? Do probability statements yield knowledge ? Inferential Procedures and the Interpretation of Probability
7 Let’s analyse the carnapian c-function c (h,e) = p What kind of probability is involved within ? Ex. : “the probability that the next toss of this coin will yield heads is 1/2” What is logical c (h,e) = p What is empirical “the coin will hand heads” “ my total body of evidence is e” Our scientific knowledge about the world; i.e. long-run frequencies, probability distributions Quantitative representation of the degree of confirmation or degree of belief
8 Working Hypothesis : De dicto - De re Probability Probability Language : Kolgomorov’s axiomatization Interpretation of Probability (in analogy with modality) Syntactic Level Semantic Level Probability pertains to what is said -de dicto- probability represents quantitatively personal opinion probability represents our knowledge of observed frequencies or distributions Probability pertains to the object -de re-
9 Conclusion <ul><li>the problem of the interpretation of probability is relevant for empiricism </li></ul><ul><li>whithin the empiricist perspective we treat (empirical) data by inferential procedures essentially inductive </li></ul><ul><li>probability statements involved in these inferences do yield knowledge </li></ul><ul><li>I proposed a working hypothesis on the basis of the duality of the concept of probability </li></ul>
References Carnap R. (1950), The Logical Foundations of Probability , University of Chicago Press. Edwards A (1976), Likelihood , Cambridge University Press. Hacking I. (1965), Logic of Statistical Inference , Cambridge University Press. - (1975), The Emergence of Probability , Cambridge University Press. Jeffrey R. (1971), Studies in Inductive Logic and Probability , University of California Press. Kyburg H. (1974), The Logical Foundations of Statistical Inference , Reidel Publishing Company. - (1983), Epistemology and Inference , University of Minnesota Press. Salmon W.C. (1967), The Foundations of Scientific Inference , University of Pittsburgh Press. Savage L.J. (1972), The Foundations of Statistics, New York. Comments? Mailto : Russo@lofs.ucl.ac.be