Abstract
The theory of probability is the mathematical framework for the study of the probability of occurrence of events. The first step is to establish a method to assign the probability of an event, for example, the probability that a coin lands heads up after a toss. The frequentist—or empirical—approach and the subjective—or Bayesian— approach are two methods that can be used to calculate probabilities. The fact that there is more than one method available for this purpose should not be viewed as a limitation of the theory, but rather as the fact that for certain parts of the theory of probability, and even more so for statistics, there is an element of subjectivity that enters the analysis and the interpretation of the results. It is therefore the task of the statistician to keep track of any assumptions made in the analysis, and to account for them in the interpretation of the results. Once a method for assigning probabilities is established, the Kolmogorov axioms are introduced as the “rules” required to manipulate probabilities. Fundamental results known as Bayes’ theorem and the theorem of total probability are used to define and interpret the concepts of statistical independence and of conditional probability, which play a central role in much of the material presented in this book.
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Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York (1970)
Akritas, M.G., Bershady, M.A.: Linear regression for astronomical data with measurement errors and intrinsic scatter. Astrophys. J. 470, 706 (1996). doi:10.1086/177901
Bayes, T., Price, R.: An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53 (1763)
Bonamente, M., Swartz, D.A., Weisskopf, M.C., Murray, S.S.: Swift XRT observations of the possible dark galaxy VIRGOHI 21. Astrophys. J. Lett. 686, L71–L74 (2008). doi:10.1086/592819
Bonamente, M., Hasler, N., Bulbul, E., Carlstrom, J.E., Culverhouse, T.L., Gralla, M., Greer, C., Hawkins, D., Hennessy, R., Joy, M., Kolodziejczak, J., Lamb, J.W., Landry, D., Leitch, E.M., Marrone, D.P., Miller, A., Mroczkowski, T., Muchovej, S., Plagge, T., Pryke, C., Sharp, M., Woody, D.: Comparison of pressure profiles of massive relaxed galaxy clusters using the Sunyaev–Zel’dovich and X-ray data. New. J. Phys. 14 (2), 025010 (2012). doi:10.1088/1367-2630/14/2/025010
Brooks, S.P., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455 (1998)
Bulmer, M.G.: Principles of Statistics. Dover, New York (1967)
Carlin, B., Gelfand, A., Smith, A.: Hierarchical Bayesian analysis for changepoint problems. Appl. Stat. 41, 389–405 (1992)
Cash, W.: Parameter estimation in astronomy through application of the likelihood ratio. Astrophys. J. 228, 939–947 (1979)
Cowan, G.: Statistical Data Analysis. Oxford University Press, Oxford (1998)
Cramer, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)
Emslie, A.G., Massone, A.M.: Bayesian confidence limits of electron spectra obtained through regularized inversion of solar hard X-ray spectra. Astrophys. J. 759, 122 (2012)
Fisher, R.A.: On a distribution yielding the error functions of several well known statistics. Proc. Int. Congr. Math. 2, 805–813 (1924)
Fisher, R.A.: The use of multiple measurements in taxonomic problems. Ann. Eugenics 7, 179–188 (1936)
Gamerman, D.: Markov Chain Monte Carlo. Chapman and Hall/CRC, London/New York (1997)
Gehrels, N.: Confidence limits for small numbers of events in astrophysical data. Astrophys. J. 303, 336–346 (1986). doi:10.1086/164079
Gelman, A., Rubin, D.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–511 (1992)
Gosset, W.S.: The probable error of a mean. Biometrika 6, 1–25 (1908)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 (1), 97–109 (1970). doi:10.1093/biomet/57.1.97. http://biomet.oxfordjournals.org/content/57/1/97.abstract
Helmert, F.R.: Die genauigkeit der formel von peters zur berechnung des wahrscheinlichen fehlers director beobachtungen gleicher genauigkeit. Astron. Nachr. 88, 192–218 (1876)
Hubble, E., Humason, M.: The velocity-distance relation among extra-galactic nebulae. Astrophys. J. 74, 43 (1931)
Isobe, T., Feigelson, Isobe, T., Feigelson, E.D., Akritas, M.G., Babu, G.J.: Linear regression in astronomy. Astrophys. J. 364, 104 (1990)
Jeffreys, H.: Theory of Probability. Oxford University Press, London (1939)
Kelly, B.C.: Some aspects of measurement error in linear regression of astronomical data. Astrophys. J. 665, 1489–1506 (2007). doi: 10.1086/519947
Kolmogorov, A.: Sulla determinazione empirica di una legge di distribuzione. Giornale dell’ Istituto Italiano degli Attuari 4, 1–11 (1933)
Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea, New York (1950)
Lampton, M., Margon, B., Bowyer, S.: Parameter estimation in X-ray astronomy. Astrophys. J. 208, 177–190 (1976). doi:10.1086/154592
Lewis, S.: gibbsit. http://lib.stat.cmu.edu/S/gibbsit
Madsen, R.W., Moeschberger, M.L.: Introductory Statistics for Business and Economics. Prentice-Hall, Englewood Cliffs (1983)
Marsaglia, G., Tsang, W., Wang, J.: Evaluating Kolmogorov’s distribution. J. Stat. Softw. 8, 1–4 (2003)
Mendel, G.: Versuche über plflanzenhybriden (experiments in plant hybridization). Verhandlungen des naturforschenden Vereines in Brünn, pp. 3–47 (1865)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953). doi: 10.1063/1.1699114
Pearson, K., Lee, A.: On the laws on inheritance in men. Biometrika 2, 357–462 (1903)
Plummer, M., Best, N., Cowles, K., Vines, K.: CODA: convergence diagnosis and output analysis for MCMC. R News 6 (1), 7–11 (2006). http://CRAN.R-project.org/doc/Rnews/
Press, W., Teukolski, S., Vetterling, W., Flannery, B.: Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press, Cambridge (2007)
Protassov, R., van Dyk, D.A., Connors, A., Kashyap, V.L., Siemiginowska, A.: Statistics, handle with care: detecting multiple model components with the likelihood ratio test. Astrophys. J. 571, 545–559 (2002). doi:1086/339856
Raftery, A., Lewis, S.: How many iterations in the gibbs sampler? Bayesian Stat. 4, 763–773 (1992)
Ross, S.M.: Introduction to Probability Models. Academic, San Diego (2003)
Thomson, J.J.: Cathode rays. Philos. Mag. 44, 293 (1897)
Tremaine, S., Gebhardt, K., Bender, R., Bower, G., Dressler, A., Faber, S.M., Filippenko, A.V., Green, R., Grillmair, C., Ho, L.C., Kormendy, J., Lauer, T.R., Magorrian, J., Pinkney, J., Richstone, D.: The slope of the black hole mass versus velocity dispersion correlation. Astrophys. J. 574, 740–753 (2002). doi:10.1086/341002
Von Eye, A., Schuster, C.: Regression Analysis for Social Sciences. Academic, New York (1998)
Wilks, S.S.: Mathematical Statistics. Princeton University Press, Princeton (1943)
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Bonamente, M. (2017). Theory of Probability. In: Statistics and Analysis of Scientific Data. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6572-4_1
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