Mutual Influence between Different Views of Probability and Statistical Inference
DOI:
10.37618/PARADIGMA.1011-2251.2021.p221-256.id1024Palabras clave:
Intuitive views, Foundations of probability, Bayesian controversy, Justification of probability, Bayes theorem, Statistical testsResumen
In this paper, we analyse the various meanings of probability and its different applications, and we focus especially on the classical, the frequentist, and the subjectivist view. We describe the different problems of how probability can be measured in each of the approaches, and how each of them can be well justified by a mathematical theory. We analyse the foundations of probability, where the scientific analysis of the theory that allows for a frequentist interpretation leads to unsolvable problems. Kolmogorov’s axiomatic theory does not suffice to establish statistical inference without further definitions and principles. Finally, we show how statistical inference essentially determines the meaning of probability and a shift emerges from purely objectivist views to a complementary conception of probability with frequentist and subjectivist constituents. For didactical purpose, the result of the present analyses explains basic problems of teaching, originating from a biased focus on frequentist aspects of probability. It also indicates a high priority for the design of suitable learning paths to a complementary conception of probability. In the applications, modellers use information in a pragmatic way processing this information regardless of its connotation into formal mathematical models, which are always thought as essentially wrong but useful.Descargas
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Barnett, V. (1982). Comparative statistical inference. New York: Wiley.
Batanero, C. (2000). Controversies around the role of statistical tests in experimental research. Mathematical Thinking and Learning, 2(1-2), 75-97. https://doi.org/10.1207/S15327833MTL0202_4.
Batanero, C. & Borovcnik, M. (2016). Statistics and probability in high school. Rotterdam: Sense Publishers. https://doi.org/10.1007/978-94-6300-624-8.
Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on teaching and learning probability. ICME-13 Topical Surveys. Cham: Springer. https://doi.org/10.1007/978-3-319-31625-3_1.
Batanero, C., Henry, M., & Parzysz, B. (2005). The nature of chance and probability. In A. G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning. Mathematics Education Library, Vol. 40 (pp. 15-37). New York: Springer. https://doi.org/10.1007/0-387-24530-8_2.
Bayes, T. (1763). An essay towards solving a problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society, 53, 370-418.
Bellhouse, D. R. (2000). De Vetula: A medieval manuscript containing probability calculations. International Statistical Review, 68(2), 123-136. https://doi.org/10.1111/j.1751-5823.2000.tb00317.x.
Berger, J. O. (1985). Statistical decision theory and Bayesian analysis. New York: Springer.
Bernoulli, J. (1987). Ars conjectandi. Basel: Impensis Thurnisiorum. Originally published in 1713.
Birnbaum, A. (1962). On the foundations of statistical inference (with discussion). Journal of the American Statistical Association, 57(298), 269-326.
Borovcnik, M. (1984). Was bedeuten statistische Aussagen. Vienna: Hölder-Pichler-Tempsky.
Borovcnik, M. (2006). Probabilistic and statistical thinking. In M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 484-506). Barcelona: European Society for Research in Mathematics Education.
Borovcnik, (2015). Risk and decision making: The “logic” of probability. The Mathematics Enthusiast, 12(1,2&3), 113-139.
Borovcnik, (2019). Informal and “informal” inference. In J. M. Contreras, M. M. Gea, M. M. López-Martín, & E. Molina-Portillo (Eds.), Actas del Tercer Congreso International Virtual de Educación Estadística. Available from www.ugr.es/local/fqm126/civeest.html.
Borovcnik, M., Fejes-Tóth, P., Jánvári, Z., & Vancsó, Ö. (2020). Experimente zur Einführung von Ideen und Denkweisen statistischer Inferenz im Gymnasium. Stochastik in der Schule, 40(1), 18-27.
Borovcnik, M. & Kapadia, R. (2014). A historical and philosophical perspective on probability. In E. J. Chernoff, B. Sriraman (Eds. (2014). Probabilistic thinking: Presenting plural perspectives (p. 7-34). New York: Springer. https://doi.org/10.1007/978-94-007-7155-0_2.
Çinlar, E. (2011). Probability and stochastics. Berlin, New York: Springer.
Collins, D., Freels, J., Huzurbazar, A., Warr, R., & Weaver, B. (2013). Accelerated test methods for reliability prediction. Journal of Quality Technology, 45, 244-259. https://doi.org/10.1080/00224065.2013.11917936.
David, F. N. (1962). Games, gods and gambling. London: Griffin.
Edwards, A. W. F. (1978). Commentary on the arguments of Thomas Bayes. Scandinavian Journal of Statistics, 5, 116-118.
Fine, T.L. (1973). Theories of probability. New York: Academic Press.
Finetti, B. de (1937). La prévision: ses lois logiques, ses sources subjectives. Annales Institut Henri Poincaré, 7, 1-68.
Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh: Oliver and Boyd.
Fisher, R. A. (1935). The design of experiments. Edinburgh: Oliver and Boyd.
Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Hamburg: Perthes und Besser.
Good, I. J. (1965). The estimation of probabilities: An essay on modern Bayesian methods. Cambridge, MA: MIT Press.
Good, I. J. (1971). The probabilistic explication of information, evidence, surprise, causality, explanation, and utility (with discussion). In V. P. Godambe & D. A. Sprott (Eds.), Foundations of statistical inference (pp. 108-141). Toronto: Holt, Rinehart, and Winston.
Good, I. J. (1983). Good thinking. The foundations of probability and its applications. Mineola, NY: Dover Publications.
Gorard, S., & White, P. (2017). Still against inferential statistics: Rejoinder to Nicholson and Ridgway. Statistics Education Research Journal, 16(1), 70-75.
Graunt, J. (1662). Natural and political observations upon the Bills of Mortality, chiefly with reference to the government, religion, trade, growth, air, diseases etc. of the City of London. London: Royal Society of London.
Hacking, I. (1965). The logic of statistical inference. Cambridge: Cambridge University Press.
Hacking, I. (1975). The emergence of probability. Cambridge: Cambridge University Press.
Hald, A. (2007). A history of parametric statistical inference from Bernoulli to Fisher, 1713-1935. New York: Springer.
Hartley, D. (1749). Observations on man, his frame, his duty, and his expectations. London: Richardson.
Hilbert, D. (1900). Mathematische Probleme. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 253–297.
Jeffreys, H. (1948). Theory of probability. 2nd ed. Oxford: Clarendon.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291
Kolmogorov, A. N. (1956). Foundations of the theory of probability. New York: Chelsea.
Kolmogorov, A. N. (1977). Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebnisse der Mathematik, 2. Band, Heft 3. Berlin: Springer (original work published in 1933).
Laplace, P. S. de (1951). A philosophical essay on probabilities (extended version). New York: Dover (original work published in 1812).
Laplace, P. S. de (1995). Théorie analytique des probabilités, 2nd ed. Paris: Courcier (original work published in 1814).
Mises, R. v. (1919). Grundlagen der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 5, 52-99.
Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Transactions of the Royal Statistical Society, 97, 558-625.
Neyman, J., & Pearson, E. S. (1967). On the use and interpretation of certain test criteria for purposes of statistical inference. Part I and II. Biometrika 20A, 175-240; 263-294 (original work published in 1928)
Popper, K. R. (1959). The propensity interpretation of probability. British Journal of the Philosophy of Science, 10, 25-42.
Popper, K. R. (1962). Logic of scientific discovery. London: Routledge (original work published in 1935).
Porter, T. (1986). The rise of statistical thinking, 1820-1900. Princeton, NJ: Princeton University Press.
Savage, L. J. (Ed.) (1962). The foundation of statistical inference. London: Methuen.
Schnorr, C.P. (1971). Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie. Berlin-New York: Springer.
Seidenfeld, T. (1979). Philosophical problems of statistical inference – Learning from R. A. Fisher. Dordrecht: D. Reidel.
Stegmüller, W. (1973). Probleme und Resultate der Wissenschaftstheorie und Analytischen Philosophie, Vol. 4. Berlin: Springer.
Steinbring, H. (1980). Zur Entwicklung des Wahrscheinlichkeitsbegriffs. Bielefeld: IDM.
Steinbring, H. (1991). The theoretical nature of probability in the classroom. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters (pp. 135-168). Dordrecht: Kluwer.
Venn, J. (1866/1962). The logic of chance. Reprinted. New York: Chelsea.
Barnett, V. (1982). Comparative statistical inference. New York: Wiley.
Batanero, C. (2000). Controversies around the role of statistical tests in experimental research. Mathematical Thinking and Learning, 2(1-2), 75-97. https://doi.org/10.1207/S15327833MTL0202_4.
Batanero, C. & Borovcnik, M. (2016). Statistics and probability in high school. Rotterdam: Sense Publishers. https://doi.org/10.1007/978-94-6300-624-8.
Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on teaching and learning probability. ICME-13 Topical Surveys. Cham: Springer. https://doi.org/10.1007/978-3-319-31625-3_1.
Batanero, C., Henry, M., & Parzysz, B. (2005). The nature of chance and probability. In A. G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning. Mathematics Education Library, Vol. 40 (pp. 15-37). New York: Springer. https://doi.org/10.1007/0-387-24530-8_2.
Bayes, T. (1763). An essay towards solving a problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society, 53, 370-418.
Bellhouse, D. R. (2000). De Vetula: A medieval manuscript containing probability calculations. International Statistical Review, 68(2), 123-136. https://doi.org/10.1111/j.1751-5823.2000.tb00317.x.
Berger, J. O. (1985). Statistical decision theory and Bayesian analysis. New York: Springer.
Bernoulli, J. (1987). Ars conjectandi. Basel: Impensis Thurnisiorum. Originally published in 1713.
Birnbaum, A. (1962). On the foundations of statistical inference (with discussion). Journal of the American Statistical Association, 57(298), 269-326.
Borovcnik, M. (1984). Was bedeuten statistische Aussagen. Vienna: Hölder-Pichler-Tempsky.
Borovcnik, M. (2006). Probabilistic and statistical thinking. In M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 484-506). Barcelona: European Society for Research in Mathematics Education.
Borovcnik, (2015). Risk and decision making: The “logic” of probability. The Mathematics Enthusiast, 12(1,2&3), 113-139.
Borovcnik, (2019). Informal and “informal” inference. In J. M. Contreras, M. M. Gea, M. M. López-Martín, & E. Molina-Portillo (Eds.), Actas del Tercer Congreso International Virtual de Educación Estadística. Available from www.ugr.es/local/fqm126/civeest.html.
Borovcnik, M., Fejes-Tóth, P., Jánvári, Z., & Vancsó, Ö. (2020). Experimente zur Einführung von Ideen und Denkweisen statistischer Inferenz im Gymnasium. Stochastik in der Schule, 40(1), 18-27.
Borovcnik, M. & Kapadia, R. (2014). A historical and philosophical perspective on probability. In E. J. Chernoff, B. Sriraman (Eds. (2014). Probabilistic thinking: Presenting plural perspectives (p. 7-34). New York: Springer. https://doi.org/10.1007/978-94-007-7155-0_2.
Çinlar, E. (2011). Probability and stochastics. Berlin, New York: Springer.
Collins, D., Freels, J., Huzurbazar, A., Warr, R., & Weaver, B. (2013). Accelerated test methods for reliability prediction. Journal of Quality Technology, 45, 244-259. https://doi.org/10.1080/00224065.2013.11917936.
David, F. N. (1962). Games, gods and gambling. London: Griffin.
Edwards, A. W. F. (1978). Commentary on the arguments of Thomas Bayes. Scandinavian Journal of Statistics, 5, 116-118.
Fine, T.L. (1973). Theories of probability. New York: Academic Press.
Finetti, B. de (1937). La prévision: ses lois logiques, ses sources subjectives. Annales Institut Henri Poincaré, 7, 1-68.
Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh: Oliver and Boyd.
Fisher, R. A. (1935). The design of experiments. Edinburgh: Oliver and Boyd.
Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Hamburg: Perthes und Besser.
Good, I. J. (1965). The estimation of probabilities: An essay on modern Bayesian methods. Cambridge, MA: MIT Press.
Good, I. J. (1971). The probabilistic explication of information, evidence, surprise, causality, explanation, and utility (with discussion). In V. P. Godambe & D. A. Sprott (Eds.), Foundations of statistical inference (pp. 108-141). Toronto: Holt, Rinehart, and Winston.
Good, I. J. (1983). Good thinking. The foundations of probability and its applications. Mineola, NY: Dover Publications.
Gorard, S., & White, P. (2017). Still against inferential statistics: Rejoinder to Nicholson and Ridgway. Statistics Education Research Journal, 16(1), 70-75.
Graunt, J. (1662). Natural and political observations upon the Bills of Mortality, chiefly with reference to the government, religion, trade, growth, air, diseases etc. of the City of London. London: Royal Society of London.
Hacking, I. (1965). The logic of statistical inference. Cambridge: Cambridge University Press.
Hacking, I. (1975). The emergence of probability. Cambridge: Cambridge University Press.
Hald, A. (2007). A history of parametric statistical inference from Bernoulli to Fisher, 1713-1935. New York: Springer.
Hartley, D. (1749). Observations on man, his frame, his duty, and his expectations. London: Richardson.
Hilbert, D. (1900). Mathematische Probleme. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 253–297.
Jeffreys, H. (1948). Theory of probability. 2nd ed. Oxford: Clarendon.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291
Kolmogorov, A. N. (1956). Foundations of the theory of probability. New York: Chelsea.
Kolmogorov, A. N. (1977). Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebnisse der Mathematik, 2. Band, Heft 3. Berlin: Springer (original work published in 1933).
Laplace, P. S. de (1951). A philosophical essay on probabilities (extended version). New York: Dover (original work published in 1812).
Laplace, P. S. de (1995). Théorie analytique des probabilités, 2nd ed. Paris: Courcier (original work published in 1814).
Mises, R. v. (1919). Grundlagen der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 5, 52-99.
Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Transactions of the Royal Statistical Society, 97, 558-625.
Neyman, J., & Pearson, E. S. (1967). On the use and interpretation of certain test criteria for purposes of statistical inference. Part I and II. Biometrika 20A, 175-240; 263-294 (original work published in 1928)
Popper, K. R. (1959). The propensity interpretation of probability. British Journal of the Philosophy of Science, 10, 25-42.
Popper, K. R. (1962). Logic of scientific discovery. London: Routledge (original work published in 1935).
Porter, T. (1986). The rise of statistical thinking, 1820-1900. Princeton, NJ: Princeton University Press.
Savage, L. J. (Ed.) (1962). The foundation of statistical inference. London: Methuen.
Schnorr, C.P. (1971). Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie. Berlin-New York: Springer.
Seidenfeld, T. (1979). Philosophical problems of statistical inference – Learning from R. A. Fisher. Dordrecht: D. Reidel.
Stegmüller, W. (1973). Probleme und Resultate der Wissenschaftstheorie und Analytischen Philosophie, Vol. 4. Berlin: Springer.
Steinbring, H. (1980). Zur Entwicklung des Wahrscheinlichkeitsbegriffs. Bielefeld: IDM.
Steinbring, H. (1991). The theoretical nature of probability in the classroom. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters (pp. 135-168). Dordrecht: Kluwer.
Venn, J. (1866/1962). The logic of chance. Reprinted. New York: Chelsea.
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22-02-2021
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Borovcnik, M. . (2021). Mutual Influence between Different Views of Probability and Statistical Inference. PARADIGMA, 41(e1), 221–256. https://doi.org/10.37618/PARADIGMA.1011-2251.2021.p221-256.id1024
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