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Bayesian probability

From Wikipedia, the free encyclopedia

Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation[1] representing a state of knowledge[2] or as quantification of a personal belief.[3]

The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses. That is to say, propositions whose truth or falsity is uncertain. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.

Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, relevant data (evidence).[4] The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.

The term Bayesian derives from the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference.[5]:131 Mathematician Pierre-Simon Laplace pioneered and popularised what is now called Bayesian probability.[5]:97–98

YouTube Encyclopedic

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  • ✪ The Bayesian Trap
  • ✪ Conditional probability explained visually (Bayes' Theorem)
  • ✪ What is Subjective (Bayesian) Probability?
  • ✪ A visual guide to Bayesian thinking
  • ✪ CRITICAL THINKING - Fundamentals: Bayes' Theorem [HD]


Picture this: You wake up one morning and you feel a little bit sick. No particular symptoms, just not 100%. So you go to the doctor and she also doesn't know what's going on with you, so she suggests they run a battery of tests and after a week goes by, the results come back, turns out you tested positive for a very rare disease that affects about 0.1% of the population and it's a nasty disease, horrible consequences, you don't want it. So you ask the doctor "You know, how certain is it that I have this disease?" and she says "Well, the test will correctly identify 99% of people that have the disease and only incorrectly identify 1% of people who don't have the disease". So that sounds pretty bad. I mean, what are the chances that you actually have this disease? I think most people would say 99%, because that's the accuracy of the test. But that is not actually correct! You need Bayes' Theorem to get some perspective. Bayes' Theorem can give you the probability that some hypothesis, say that you actually have the disease, is true given an event; that you tested positive for the disease. To calculate this, you need to take the prior probability of the hypothesis was true - that is, how likely you thought it was that you have this disease before you got the test results - and multiply it by the probability of the event given the hypothesis is true - that is, the probability that you would test positive if you had the disease - and then divide that by the total probability of the event occurring - that is testing positive. This term is a combination of your probability of having the disease and correctly testing positive plus your probability of not having the disease and being falsely identified. The prior probability that a hypothesis is true is often the hardest part of this equation to figure out and, sometimes, it's no better than a guess. But in this case, a reasonable starting point is the frequency of the disease in the population, so 0.1%. And if you plug in the rest of the numbers, you find that you have a 9% chance of actually having the disease after testing positive. Which is incredibly low if you think about it. Now, this isn't some sort of crazy magic. It's actually common sense applied to mathematics. Just think about a sample size of 1000 people. Now, one person out of that thousand, is likely to actually have the disease. And the test would likely identify them correctly as having the disease. But out of the 999 other people, 1% or 10 people would falsely be identified as having the disease. So, if you're one of those people who has a positive test result and everyone's just selected at random - well, you're actually part of a group of 11 where only one person has the disease. So your chances of actually having it are 1 in 11. 9%. It just makes sense. When Bayes first came up with this theorem he didn't actually think it was revolutionary. He didn't even think it was worthy of publication, he didn't submit it to the Royal Society of which he was a member, and in fact it was discovered in his papers after he died and he had abandoned it for more than a decade. His relatives asked his friend, Richard Price, to dig through his papers and see if there is anything worth publishing in there. And that's where Price discovered what we now know as the origins of Bayes' Theorem. Bayes originally considered a thought experiment where he was sitting with his back to a perfectly flat, perfectly square table and then he would ask an assistant to throw a ball onto the table. Now this ball could obviously land and end up anywhere on the table and he wanted to figure out where it was. So what he'd asked his assistant to do was to throw on another ball and then tell him if it landed to the left, or to the right, or in front, behind of the first ball, and he would note that down and then ask for more and more balls to be thrown on the table. What he realized, was that through this method he could keep updating his idea of where the first ball was. Now of course, he would never be completely certain, but with each new piece of evidence, he would get more and more accurate, and that's how Bayes saw the world. It wasn't that he thought the world was not determined, that reality didn't quite exist, but it was that we couldn't know it perfectly, and all we could hope to do was update our understanding as more and more evidence became available. When Richard Price introduced Bayes' Theorem, he made an analogy to a man coming out of a cave, maybe he'd lived his whole life in there and he saw the Sun rise for the first time, and kind of thought to himself: "Is, Is this a one-off, is this a quirk, or does the Sun always do this?" And then, every day after that, as the Sun rose again, he could get a little bit more confident, that, well, that was the way the world works. So Bayes' Theorem wasn't really a formula intended to be used just once, it was intended to be used multiple times, each time gaining new evidence and updating your probability that something is true. So if we go back to the first example when you tested positive for a disease, what would happen if you went to another doctor, get a second opinion and get that test run again, but let's say by a different lab, just to be sure that those tests are independent, and let's say that test also comes back as positive. Now what is the probability that you actually have the disease? Well, you can use Bayes formula again, except this time for your prior probability that you have the disease, you have to put in the posterior probability, the probability that we worked out before which is 9%, because you've already had one positive test. If you crunch those numbers, the new probability based on two positive tests is 91%. There's a 91% chance that you actually have the disease, which kind of makes sense. 2 positive results by different labs are unlikely to just be chance, but you'll notice that probability is still not as high as the accuracy, the reported accuracy of the test. Bayes' Theorem has found a number of practical applications, including notably filtering your spam. You know, traditional spam filters actually do a kind of bad job, there's too many false positives, too much of your email ends up in spam, but using a Bayesian filter, you can look at the various words that appear in e-mails, and use Bayes' Theorem to give a probability that the email is spam, given that those words appear. Now Bayes' Theorem tells us how to update our beliefs in light of new evidence, but it can't tell us how to set our prior beliefs, and so it's possible for some people to hold that certain things are true with a 100% certainty, and other people to hold those same things are true with 0% certainty. What Bayes' Theorem shows us is that in those cases, there is absolutely no evidence, nothing anyone could do to change their minds, and so as Nate Silver points out in his book, The Signal and The Noise, we should probably not have debates between people with a 100% prior certainty, and 0% prior certainty, because, well really, they'll never convince each other of anything. Most of the time when people talk about Bayes' Theorem, they discussed how counterintuitive it is and how we don't really have an inbuilt sense of it, but recently my concern has been the opposite: that maybe we're too good at internalizing the thinking behind Bayes' Theorem, and the reason I'm worried about that is because, I think in life we can get used to particular circumstances, we can get used to results, maybe getting rejected or failing at something or getting paid a low wage and we can internalize that as though we are that man emerging from the cave and we see the Sun rise every day and every day, and we keep updating our beliefs to a point of near certainty that we think that that is basically the way that nature is, it's the way the world is and there's nothing that we can do to change it. You know, there's Nelson Mandela's quote that: 'Everything is impossible until it's done', and I think that is kind of a very Bayesian viewpoint on the world, if you have no instances of something happening, then what is your prior for that event? It will seem completely impossible your prior may be 0 until it actually happens. You know, the thing we forget in Bayes' Theorem is that: our actions play a role in determining outcomes, and determining how true things actually are. But if we internalize that something is true and maybe we're a 100% sure that it's true, and there's nothing we can do to change it, well, then we're going to keep on doing the same thing, and we're going to keep on getting the same result, it's a self-fulfilling prophecy, so I think a really good understanding of Bayes' Theorem implies that experimentation is essential. If you've been doing the same thing for a long time and getting the same result that you're not necessarily happy with, maybe it's time to change. So is there something like that that you've been thinking about? If so, let me know in the comments. Hey, this episode of Veritasium was supported in part by viewers like you on Patreon and by Audible. Audible is a leading provider of spoken audio information including an unmatched selection of audiobooks: original, programming, news, comedy and more. So if you're thinking about trying something new and you haven't tried Audible yet, you should give them a shot, and for viewers of this channel, they offer a free 30-day trial just by going to: You know, the book I've been listening to on Audible recently is called: 'The Theory That Would Not Die' by Sharon Bertsch McGrayne, and it is an incredible in-depth look at Bayes' Theorem, and I've learned a lot just listening to this book, including the crazy fact that Bayes never came up with the mathematical formulation of his rule that was done independently by the mathematician Pierre-Simon Laplace so, really I think he deserves a lot of a credit for this theory, but Bayes gets naming rights because he was first, and if you want, you can download this book and listen to it, as I have, when I've just been driving in the car or going to the gym, which I'm doing again, and so if there's a part of your day that you feel is kind of boring then I can highly recommend trying out audiobooks from Audible. Just go to: So as always I want to thank: Audible for supporting me, and I want to thank you for watching.


Bayesian methodology

Bayesian methods are characterized by concepts and procedures as follows:

  • The use of random variables, or more generally unknown quantities,[6] to model all sources of uncertainty in statistical models including uncertainty resulting from lack of information (see also aleatoric and epistemic uncertainty).
  • The need to determine the prior probability distribution taking into account the available (prior) information.
  • The sequential use of Bayes' formula: when more data become available, calculate the posterior distribution using Bayes' formula; subsequently, the posterior distribution becomes the next prior.
  • While for the frequentist, a hypothesis is a proposition (which must be either true or false) so that the frequentist probability of a hypothesis is either 0 or 1, in Bayesian statistics, the probability that can be assigned to a hypothesis can also be in a range from 0 to 1 if the truth value is uncertain.

Objective and subjective Bayesian probabilities

Broadly speaking, there are two interpretations on Bayesian probability. For objectivists, interpreting probability as extension of logic, probability quantifies the reasonable expectation everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by Cox's theorem.[2][7] For subjectivists, probability corresponds to a personal belief.[3] Rationality and coherence allow for substantial variation within the constraints they pose; the constraints are justified by the Dutch book argument or by the decision theory and de Finetti's theorem.[3] The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability.


The term Bayesian refers to Thomas Bayes (1702–1761), who proved a special case of what is now called Bayes' theorem in a paper titled "An Essay towards solving a Problem in the Doctrine of Chances".[8] In that special case, the prior and posterior distributions were Beta distributions and the data came from Bernoulli trials. It was Pierre-Simon Laplace (1749–1827) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence.[9] Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called "inverse probability" (because it infers backwards from observations to parameters, or from effects to causes).[10] After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called frequentist statistics.[10]

In the 20th century, the ideas of Laplace developed in two directions, giving rise to objective and subjective currents in Bayesian practice. Harold Jeffreys' Theory of Probability (first published in 1939) played an important role in the revival of the Bayesian view of probability, followed by works by Abraham Wald (1950) and Leonard J. Savage (1954). The adjective Bayesian itself dates to the 1950s; the derived Bayesianism, neo-Bayesianism is of 1960s coinage.[11][12][13] In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed.[14] No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods and the consequent removal of many of the computational problems, and to an increasing interest in nonstandard, complex applications.[15] While frequentist statistics remains strong (as seen by the fact that most undergraduate teaching is still based on it [16][citation needed]), Bayesian methods are widely accepted and used, e.g., in the field of machine learning.[17]

Justification of Bayesian probabilities

The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

Axiomatic approach

Richard T. Cox showed that[7] Bayesian updating follows from several axioms, including two functional equations and a hypothesis of differentiability. The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite.[18] Other axiomatizations have been suggested by various authors with the purpose of making the theory more rigorous.[6]

Dutch book approach

The Dutch book argument was proposed by de Finetti; it is based on betting. A Dutch book is made when a clever gambler places a set of bets that guarantee a profit, no matter what the outcome of the bets. If a bookmaker follows the rules of the Bayesian calculus in the construction of his odds, a Dutch book cannot be made.

However, Ian Hacking noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example, Hacking writes[19][20] "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."

In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "probability kinematics"[21] following the publication of Richard C. Jeffreys' rule, which is itself regarded as Bayesian[22]). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial[23] and not universally seen as satisfactory.[24]

Decision theory approach

A decision-theoretic justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by Abraham Wald, who proved that every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.[25] Conversely, every Bayesian procedure is admissible.[26]

Personal probabilities and objective methods for constructing priors

Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent. Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience.[27] Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated ... [the question whether probabilities] might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. p. 19 of The Theory of Games and Economic Behavior). We did not carry this out; it was demonstrated by Pfanzagl ... with all the necessary rigor".[28]

Ramsey and Savage noted that the individual agent's probability distribution could be objectively studied in experiments. Procedures for testing hypotheses about probabilities (using finite samples) are due to Ramsey (1931) and de Finetti (1931, 1937, 1964, 1970). Both Bruno de Finetti[29][30] and Frank P. Ramsey[30][31] acknowledge their debts to pragmatic philosophy, particularly (for Ramsey) to Charles S. Peirce.[30][31]

The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.[32] This work demonstrates that Bayesian-probability propositions can be falsified, and so meet an empirical criterion of Charles S. Peirce, whose work inspired Ramsey. (This falsifiability-criterion was popularized by Karl Popper.[33][34])

Modern work on the experimental evaluation of personal probabilities uses the randomization, blinding, and Boolean-decision procedures of the Peirce-Jastrow experiment.[35] Since individuals act according to different probability judgments, these agents' probabilities are "personal" (but amenable to objective study).

Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act). To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities.

Indeed, some Bayesians have argued the prior state of knowledge defines the (unique) prior probability-distribution for "regular" statistical problems; cf. well-posed problems. Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to John Maynard Keynes, Harold Jeffreys, and Edwin Thompson Jaynes. These theorists and their successors have suggested several methods for constructing "objective" priors (Unfortunately, it is not clear how to assess the relative "objectivity" of the priors proposed under these methods):

Each of these methods contributes useful priors for "regular" one-parameter problems, and each prior can handle some challenging statistical models (with "irregularity" or several parameters). Each of these methods has been useful in Bayesian practice. Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like James Berger (Duke University) and José-Miguel Bernardo (Universitat de València), simply because such priors are needed for Bayesian practice, particularly in science.[36] The quest for "the universal method for constructing priors" continues to attract statistical theorists.[36]

Thus, the Bayesian statistician needs either to use informed priors (using relevant expertise or previous data) or to choose among the competing methods for constructing "objective" priors.

See also


  1. ^ Cox, R.T. (1946). "Probability, Frequency, and Reasonable Expectation". American Journal of Physics. 14 (1): 1–10. Bibcode:1946AmJPh..14....1C. doi:10.1119/1.1990764.
  2. ^ a b Jaynes, E.T. (1986). "Bayesian Methods: General Background". In Justice, J. H. (ed.). Maximum-Entropy and Bayesian Methods in Applied Statistics. Cambridge: Cambridge University Press.
  3. ^ a b c de Finetti, Bruno (2017). Theory of Probability: A critical introductory treatment. Chichester: John Wiley & Sons Ltd. ISBN 9781119286370.
  4. ^ Paulos, John Allen (5 August 2011). "The Mathematics of Changing Your Mind [by Sharon Bertsch McGrayne]". Book Review. New York Times. Retrieved 2011-08-06.
  5. ^ a b Stigler, Stephen M. (March 1990). The history of statistics. Harvard University Press. ISBN 9780674403413.
  6. ^ a b Dupré, Maurice J.; Tipler, Frank J. (2009). "New axioms for rigorous Bayesian probability". Bayesian Analysis (3): 599–606.
  7. ^ a b Cox, Richard T. (1961). The algebra of probable inference (Reprint ed.). Baltimore, MD; London, UK: Johns Hopkins Press; Oxford University Press [distributor]. ISBN 9780801869822.
  8. ^ McGrayne, Sharon Bertsch (2011). The Theory that Would not Die.  , p. 10, at Google Books.
  9. ^ Stigler, Stephen M. (1986). "Chapter 3". The History of Statistics. Harvard University Press.
  10. ^ a b Fienberg, Stephen. E. (2006). "When did Bayesian Inference become "Bayesian"?" (PDF). Bayesian Analysis. 1 (1): 5, 1–40. Archived from the original (PDF) on 10 September 2014.
  11. ^ Harris, Marshall Dees (1959). "Recent developments of the so-called Bayesian approach to statistics". Agricultural Law Center. Legal-Economic Research. University of Iowa: 125 (fn. #52), 126. The works of Wald, Statistical Decision Functions (1950) and Savage, The Foundation of Statistics (1954) are commonly regarded starting points for current Bayesian approaches
  12. ^ Annals of the Computation Laboratory of Harvard University. 31. 1962. p. 180. This revolution, which may or may not succeed, is neo-Bayesianism. Jeffreys tried to introduce this approach, but did not succeed at the time in giving it general appeal.
  13. ^ Kempthorne, Oscar (1967). The Classical Problem of Inference—Goodness of Fit. Fifth Berkeley Symposium on Mathematical Statistics and Probability. p. 235. It is curious that even in its activities unrelated to ethics, humanity searches for a religion. At the present time, the religion being 'pushed' the hardest is Bayesianism.
  14. ^ Bernardo, J.M. (2005). Reference analysis. Handbook of Statistics. 25. pp. 17–90. doi:10.1016/S0169-7161(05)25002-2. ISBN 9780444515391.
  15. ^ Wolpert, R.L. (2004). "A conversation with James O. Berger". Statistical Science. 9: 205–218.
  16. ^ Bernardo, José M. (2006). A Bayesian mathematical statistics primer (PDF). ICOTS-7. Bern.
  17. ^ Bishop, C.M. (2007). Pattern Recognition and Machine Learning. Springer.
  18. ^ Halpern, J. "A counterexample to theorems of Cox and Fine". Journal of Artificial Intelligence Research. 10: 67–85.
  19. ^ Hacking (1967), Section 3, page 316
  20. ^ Hacking (1988, page 124)
  21. ^ Skyrms, Brian (1 January 1987). "Dynamic Coherence and Probability Kinematics". Philosophy of Science. 54 (1): 1–20. CiteSeerX doi:10.1086/289350. JSTOR 187470.
  22. ^ Joyce, James (30 September 2003). "Bayes' Theorem". The Stanford Encyclopedia of Philosophy.
  23. ^ Fuchs, Christopher A.; Schack, Rüdiger (1 January 2012). Ben-Menahem, Yemima; Hemmo, Meir (eds.). Probability in Physics. The Frontiers Collection. Springer Berlin Heidelberg. pp. 233–247. arXiv:1103.5950. doi:10.1007/978-3-642-21329-8_15. ISBN 9783642213281.
  24. ^ van Frassen, Bas (1989). Laws and Symmetry. Oxford University Press. ISBN 0-19-824860-1.
  25. ^ Wald, Abraham (1950). Statistical Decision Functions. Wiley.
  26. ^ Bernardo, José M.; Smith, Adrian F.M. (1994). Bayesian Theory. John Wiley. ISBN 0-471-92416-4.
  27. ^ Pfanzagl (1967, 1968)
  28. ^ Morgenstern (1976, page 65)
  29. ^ Galavotti, Maria Carla (1 January 1989). "Anti-Realism in the Philosophy of Probability: Bruno de Finetti's Subjectivism". Erkenntnis. 31 (2/3): 239–261. doi:10.1007/bf01236565. JSTOR 20012239.
  30. ^ a b c Galavotti, Maria Carla (1 December 1991). "The notion of subjective probability in the work of Ramsey and de Finetti". Theoria. 57 (3): 239–259. doi:10.1111/j.1755-2567.1991.tb00839.x. ISSN 1755-2567.
  31. ^ a b Dokic, Jérôme; Engel, Pascal (2003). Frank Ramsey: Truth and Success. Routledge. ISBN 9781134445936.
  32. ^ Davidson et al. (1957)
  33. ^ Thornton, Stephen (7 August 2018). "Karl Popper". Stanford Encyclopedia of Philosophy.
  34. ^ Popper, Karl (2002) [1959]. The Logic of Scientific Discovery (2nd ed.). Routledge. p. 57. ISBN 0-415-27843-0 – via Google Books. (translation of 1935 original, in German).
  35. ^ Peirce & Jastrow (1885)
  36. ^ a b Bernardo, J. M. (2005). "Reference Analysis". In Dey, D.K.; Rao, C. R. (eds.). Handbook of Statistics (PDF). 25. Amsterdam: Elsevier. pp. 17–90.


  • Bessière, Pierre; Mazer, E.; Ahuacatzin, J.-M.; Mekhnacha, K. (2013). Bayesian Programming. CRC Press. ISBN 9781439880326.
  • de Finetti, Bruno (September 1989) [1931]. "Probabilism: A critical essay on the theory of probability and on the value of science". Erkenntnis. 31. (translation of de Finetti, 1931)
  • de Finetti, Bruno (1964) [1937]. "Foresight: Its logical laws, its subjective sources". In Kyburg, H.E.; Smokler, H.E. (eds.). Studies in Subjective Probability. New York, NY: Wiley. (translation of de Finetti, 1937, above)
  • McGrayne, S.B. (2011). The Theory that would not Die: How Bayes' rule cracked the Enigma code, hunted down Russian submarines, and emerged triumphant from two centuries of controversy. New Haven, CT: Yale University Press. ISBN 9780300169690. OCLC 670481486.
  • Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern-von Neumann Utility Theory". In Martin Shubik (ed.). Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press. pp. 237–251.
  • Pfanzagl, J.; Baumann, V. & Huber, H. (1968). "Events, Utility and Subjective Probability". Theory of Measurement. Wiley. pp. 195–220.
  • Stigler, S.M. (1999). Statistics on the Table: The history of statistical concepts and methods. Harvard University Press. ISBN 0-674-83601-4.
  • Stone, J.V. (2013). Bayes’ Rule: A tutorial introduction to Bayesian analysis. England: Sebtel Press. "Chapter 1 of Bayes' Rule".
  • Winkler, R.L. (2003). Introduction to Bayesian Inference and Decision (2nd ed.). Probabilistic. ISBN 978-0-9647938-4-2. Updated classic textbook. Bayesian theory clearly presented
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