Notebook per
***An Intuitive Explanation of Bayes' Theorem
Citation (APA): Yudkowsky, E. (2017). ***An Intuitive Explanation of Bayes' Theorem [Kindle Android version]. Retrieved from Amazon.com
Parte introduttiva
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An Intuitive Explanation of Bayes' Theorem By Elizer Yudkowsky
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"Bayes' Theorem"
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Bayesian reasoning.
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Just one equation.
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why your friends would be interested in it.
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Maybe you understand it in theory,
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in practice you get mixed up
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you can't understand why your friends and/ or research colleagues seem to think it's the secret of the universe.
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Why does a mathematical concept generate this strange enthusiasm in its students?
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What is the so-called Bayesian Revolution
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subsume even the experimental method itself as a special case?
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Bayesian reasoning is very counterintuitive.
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People do not employ Bayesian reasoning intuitively,
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rapidly forget Bayesian methods
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like quantum mechanics
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...
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is inherently difficult
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c
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Here's a story
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t
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1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
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x PROBLEMA DEL CANCRO
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come up with your own answer before continuing.
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most doctors get the same wrong answer
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around 15% of doctors get it right.
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See Casscells, Schoenberger, and Grayboys 1978; Eddy 1982; Gigerenzer and Hoffrage 1995;
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x TEST SUI DOTTORI
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result which is easy to replicate,
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most doctors estimate the probability to be between 70% and 80%,
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alternate version of the problem
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10 out of 1000 women at age forty who participate in routine screening have breast cancer. 800 out of 1000 women with breast cancer will get positive mammographies. 96 out of 1000 women without breast cancer will also get positive mammographies. If 1000 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?
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x ALTRO ES
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here's the problem on which doctors fare best of all,
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46%
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100 out of 10,000 women at age forty who participate in routine screening have breast cancer. 80 of every 100 women with breast cancer will get a positive mammography. 950 out of 9,900 women without breast cancer will also get a positive mammography. If 10,000 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?
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x TERZO FRAMING
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The correct answer is 7.8%,
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Out of 10,000 women, 100 have breast cancer; 80 of those 100 have positive mammographies. From the same 10,000 women, 9,900 will not have breast cancer and of those 9,900 women, 950 will also get positive mammographies. This makes the total number of women with positive mammographies 950 + 80 or 1,030. Of those 1,030 women with positive mammographies, 80 will have cancer. Expressed as a proportion, this is 80/ 1,030 or 0.07767 or 7.8%.
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x IL RAGIONAMENTO X LA RISPOSTA CORRETTA
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x DUE GRUPPI
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x 4 GRUPPI
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If you administer a mammography to 10,000 patients, then out of the 1030 with positive mammographies, 80 of those positive-mammography patients will have cancer.
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1 out of those 13 will have cancer.
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The most common mistake is to ignore the original fraction of women with breast cancer, and the fraction of women without breast cancer who receive false positives, and focus only on the fraction of women with breast cancer who get positive results. For example, the vast majority of doctors in these studies seem to have thought that if around 80% of women with breast cancer have positive mammographies, then the probability of a women with a positive mammography having breast cancer must be around 80%.
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x L ERRORE PIÙ COMUNE
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three pieces of information
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the final answer always depends on the original fraction of women with breast cancer,
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consider an alternate universe in which only one woman out of a million has breast cancer.
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the probability goes from 1: 1,000,000 to 1: 100,000.
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AUMENTO TRASCURABILE
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the mammography result doesn't replace your old information
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the mammography slides the estimated probability in the direction of the result.
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a positive result on the mammography slides the 1% chance upward to 7.8%.
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L AGGIORNAMENTO
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replacing the original 1% probability with the 80% probability
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L ERRORE COMUNE
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It may seem like a good idea, but it just doesn't work.
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"The probability that a woman with a positive mammography has breast cancer" is not at all the same thing as "the probability that a woman with breast cancer has a positive mammography"; they are as unlike as apples and cheese.
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x DUE PROBLEMI DIFFERENTI
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uses all three pieces of problem information
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"the prior probability that a woman has breast cancer",
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"the probability that a woman with breast cancer gets a positive mammography",
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"the probability that a woman without breast cancer gets a positive mammography".
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consider an alternate test, mammography +. Like the original test, mammography + returns positive for 80% of women with breast cancer. However, mammography + returns a positive result for only one out of a million women without breast cancer - mammography + has the same rate of false negatives, but a vastly lower rate of false positives. Suppose a patient receives a positive mammography +. What is the chance that this patient has breast cancer?
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x ALTRA VERSIONE: FALSI NEGATIVI BASSISSIMI
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99.988%,
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Mammography + is thus a better test in terms of its total emotional impact
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if you have a positive mammography +, your chance of having cancer is a virtual certainty.
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mammography + does not generate as many false positives
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case where the chance of a true positive and the chance of a false positive are the same,
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NO INFO
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Take, for example, the "test" of flipping a coin;
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The original proportion of patients with breast cancer is known as the prior probability.
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The chance that a patient with breast cancer gets a positive mammography, and the chance that a patient without breast cancer gets a positive mammography, are known as the two conditional probabilities.
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CONDITIONAL
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Collectively, this initial information is known as the priors.
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revised probability or the posterior probability.
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if the two conditional probabilities are equal, the posterior probability equals the prior probability.
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How can I find the priors for a problem? A. Many commonly used priors are listed in the Handbook of Chemistry and Physics.
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Where do priors originally come from? A. Never ask that question.
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SOGGETTO
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Priors for scientific problems are established by annual vote of the AAAS.
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I suggest visualizing Bayesian evidence as sliding the probability
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Gigerenzer and Hoffrage
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some ways of phrasing story problems are much more evocative of correct Bayesian reasoning.
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The least evocative phrasing used probabilities.
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A slightly more evocative phrasing used frequencies
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instead of saying that 1%
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one would say that 1 out of 100
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saying "1 out of 100 women" encourages you to concretely visualize X women with cancer,
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most effective presentation found so far is what's known as natural frequencies
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A natural frequencies presentation is one in which the information about the prior probability is included in presenting the conditional probabilities.
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x NATURAL FREQUENCY
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When problems are presented in natural frequences, the proportion of people using Bayesian reasoning rises to around half.
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visualization,
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p( A& B) is the same as p( B& A), but p( A | B) is not the same thing as p( B | A), and p( A& B) is completely different from p( A | B).
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x CONFUSIONE
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The probability that a test gives a true positive divided by the probability that a test gives a false positive is known as the likelihood ratio of that test.
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x ATTENDIBILITÀ DEL TEST
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The likelihood ratio sums up everything there is to know about the meaning of a positive result
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but the meaning of a negative result on the test is not specified,
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If the prior probability is 1%, then knowing only the likelihood ratio is enough to determine the posterior probability after a positive result.
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X È SUFFICIENTE
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The Reverend Thomas Bayes, by far the most enigmatic figure in mathematical history. Almost nothing is known of Bayes's life, and very few of his manuscripts survived. Thomas Bayes was born in 1701 or 1702 to Joshua Bayes and Ann Carpenter, and his date of death is listed as 1761. The exact date of Thomas Bayes's birth is not known for certain because Joshua Bayes, though a surprisingly wealthy man, was a member of an unusual, esoteric, and even heretical religious sect, the "Nonconformists". The Nonconformists kept their birth registers secret, supposedly from fear of religious discrimination; whatever the reason, no true record exists of Thomas Bayes's birth. Thomas Bayes was raised a Nonconformist and was soon promoted into the higher ranks of the Nonconformist theosophers, whence comes the "Reverend" in his name. In 1742 Bayes was elected a Fellow of the Royal Society of London, the most prestigious scientific body of its day, despite Bayes having published no scientific or mathematical works at that time. Bayes's nomination certificate was signed by sponsors including the President and the Secretary of the Society, making his election almost certain. Even today, however, it remains a mystery why such weighty names sponsored an unknown into the Royal Society. Bayes's sole publication during his known lifetime was allegedly a mystical book entitled Divine Benevolence, laying forth the original causation and ultimate purpose of the universe. The book is commonly attributed to Bayes, though it is said that no author appeared on the title page, and the entire work is sometimes considered to be of dubious provenance. Most mysterious of all, Bayes' Theorem itself appears in a Bayes manuscript presented to the Royal Society of London in 1764, three years after Bayes's supposed death in 1761! Despite the shocking circumstances of its presentation, Bayes' Theorem was soon forgotten, and was popularized within the scientific community only by the later efforts of the great mathematician Pierre-Simon Laplace. Laplace himself is almost as enigmatic as Bayes; we don't even know whether it was "Pierre" or "Simon" that was his actual first name. Laplace's papers are said to have contained a design for an AI capable of predicting all future events, the so-called "Laplacian superintelligence". While it is generally believed that Laplace never tried to implement his design, there remains the fact that Laplace presciently fled the guillotine that claimed many of his colleagues during the Reign of Terror. Even today, physicists sometimes attribute unusual effects to a "Laplacian Operator" intervening in their experiments. In summary, we do not know the real circumstances of Bayes's birth, the ultimate origins of Bayes' Theorem, Bayes's actual year of death, or even whether Bayes ever really died. Nonetheless "Reverend Thomas Bayes", whatever his true identity, has the greatest fondness and gratitude of Earth's scientific community.
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why is it that some people are so excited about Bayes' Theorem?
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"Do you believe that a nuclear war will occur in the next 20 years? If no, why not?"
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UNA DOMANDA
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I went ahead and asked the above question in an IRC channel, #philosophy on EFNet.
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One EFNetter who answered replied "No" to the above question, but added that he believed biological warfare would wipe out "99.4%" of humanity within the next ten years. I then asked whether he believed 100% was a possibility. "No," he said. "Why not?", I asked. "Because I'm an optimist,"
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x UNA RISPOSTA
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Another person who answered the above question said that he didn't expect a nuclear war for 100 years, because "All of the players involved in decisions regarding nuclear war are not interested right now." "But why extend that out for 100 years?", I asked. "Pure hope," was his reply.
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x UN ALTRA
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What is it exactly that makes these thoughts "irrational"
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DOMANDA
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"It is not rational to believe things only because they are comforting."
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is equally irrational to believe things only because they are discomforting;
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to be an optimist has nothing to do with
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There is also a mathematical reply
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This mathematical reply is known as Bayes' Theorem.
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as we have earlier seen, when the two conditional probabilities are equal, the revised probability equals the prior probability.
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But suppose you are arguing with someone who is verbally clever
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"Ah, but since I'm an optimist, I'll have renewed hope for tomorrow, work a little harder at my dead-end job, pump up the global economy a little, eventually, through the trickle-down effect, sending a few dollars into the pocket of the researcher who ultimately finds a way to stop biological warfare - so you see, the two events are related after all, and I can use one as valid evidence about the other." In one sense, this is correct - any correlation, no matter how weak, is fair prey for Bayes' Theorem;
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X TUTTO CONTA
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Bayes' Theorem not only tells us when to revise our probabilities, but how much to revise
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In cognitive science, Bayesian reasoner is the technically precise codeword that we use to mean rational mind.
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you may hear cognitive psychologists saying that people do not take prior frequencies sufficiently into account,
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A related error is to pay too much attention to p( X | A) and not enough to p( X | ~ A)
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x ALTRO ERRORE
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For example, if it is raining, this very strongly implies the grass is wet - p( wetgrass | rain) ~ 1 - but seeing that the grass is wet doesn't necessarily mean that it has just rained; perhaps the sprinkler was turned on,
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x ERBA BAGNATA
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science itself is a special case of Bayes' Theorem;
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experimental evidence is Bayesian evidence.
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when you perform an experiment and get evidence that "confirms" or "disconfirms" your theory, this confirmation and disconfirmation is governed by the Bayesian rules. For example, you have to take into account, not only whether your theory predicts the phenomenon, but whether other possible explanations also predict the phenomenon.
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x LA SCIENA È BAYESIANA
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Karl Popper's falsificationism
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the old philosophy that the Bayesian revolution is currently dethroning.
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Karl Popper's idea that theories can be definitely falsified, but never definitely confirmed, is yet another special case of the Bayesian rules; if p( X | A) ~ 1 - if the theory makes a definite prediction - then observing ~ X very strongly falsifies A. On the other hand, if p( X | A) ~ 1, and we observe X, this doesn't definitely confirm the theory; there might be some other condition B such that p( X | B) ~ 1, in which case observing X doesn't favor A over B. For observing X to definitely confirm A, we would have to know, not that p( X | A) ~ 1, but that p( X | ~ A) ~ 0, which is something that we can't know because we can't range over all possible alternative explanations.
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x POPPER UN CASO SPECIALE DI BAYES
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For example, when Einstein's theory of General Relativity toppled Newton's incredibly well-confirmed theory of gravity, it turned out that all of Newton's predictions were just a special case of Einstein's predictions.
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x BAYES POPPER... NEWTON E EINSTEIN
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Falsification is much stronger than confirmation.
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Popper's idea that there is only falsification and no such thing as confirmation turns out to be incorrect.
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Bayes' Theorem shows that falsification is very strong evidence compared to confirmation, but falsification is still probabilistic in nature;
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x TUTTO È PROBAB
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not governed by fundamentally different rules from confirmation, as Popper argued.
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Hence the Bayesian revolution.
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A is the thing we want to know about. X is how we're observing it;
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X is the evidence
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In a sense, p( Q | P) really means p( Q& P | P), but
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p( Q), the frequency of "things that have property Q within the entire sample",
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Bayes' Theorem binds reasoning into the physical universe.
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MENTE&MONDO. RAZIONALITÀ APPLICATA
