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Part 2: Philosophy of Science: Induction and Bayes' Theorem


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Philosophy of Science: Induction and Bayes' Theorem

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Part 2: Philosophy of Science: Induction and Bayes' Theorem

  2. 2. THE PROBLEM OF INDUCTION • So far… • In part 1, we’ve seen how formal logic works • We’ve seen some fallacies. • We’ve seen the difference between induction, deduction and abduction.
  3. 3. THE PROBLEM OF INDUCTION • Introduction • We need to use induction/abduction because we can’t deduce what will happen in a scientific experiment, unless we already have known laws. • We look at evidence and make generalisations. E.g. effects of heat on substances, say.
  4. 4. THE PROBLEM OF INDUCTION • In this part, we ask: Why is induction a problem? • Will the sun rise tomorrow? We don’t know. • Formal fallacy. Sun always rises = S. Sun rose/ will rise = R. 
 R ⊃ S, S, ∴ R… FAC. • What about Boyle’s Law? / Ideal Gas Equation? Is it just coincidence?
  5. 5. THE PROBLEM OF INDUCTION • Hume • Hume argues that we cannot go from what we already know, to future events. • Just a habit of mind, and is not given in nature. • Nothing “given” in our observations to show any causal link.
  6. 6. THE PROBLEM OF INDUCTION • Hume • Didn’t clearly distinguish between epistemological claim and ontological claim whether our inability to see causal links was a fact about the world or whether we just cannot see it. • We cannot take it, Hume says,“that instances of which we have had no experience, must resemble those of which we have had experience, and that the course of nature continues always uniformly the same.”
  7. 7. THE PROBLEM OF INDUCTION • Goodman • New Riddle of Induction. He gives an example of space aliens who have time concepts embedded in their colour concepts; that is, they consider colours to be inherently temporal as well as visual. So, he argues, while we think emeralds are persistently green, to a space alien, they’re grue at first, and bleen later on.Thus, he argues, we can’t even assume that any property (such as “greenness”) can be “projected” into the future; e.g.“this emerald will remain green”. Induction, he concludes, is unreliable, because nothing is “projectible”. • The fact that a given man in a room is a third son does not increase the probability that other men in this room are third sons.
  8. 8. THE PROBLEM OF INDUCTION • So how do we justify induction? • “Because it has worked in the past, it will carry on working”. Nope. • “Had we not survived, induction would have been shown to be unreliable.Yet, because we survive, it must be reliable.” Nope.Also, Evolution is an inductive argument.
  9. 9. THE PROBLEM OF INDUCTION • Popper on induction • Rejects induction. • There’s an infinite amount of evidence which could support a theory, and so, it can never be supported fully or conclusively. Instead, it’s easier to just reject a theory if there’s counter-evidence, and that can be conclusive. • Even if we see 1000 white swans, we cannot prove that all swans are white. However, one black swan will disprove the theory that swans are all white.
  10. 10. THE PROBLEM OF INDUCTION • Carnap on induction • Scientific theories are probabilistic. • Induction is probabilistic. • Induction is just the use of statistics applied to observations. • This leads us to Bayesianism.
  11. 11. P(h|e) = P(h) P(e|h) P(e) BAYES’THEOREM • Introduction • Tells us how to update our beliefs given evidence. • We’re trying to find evidence to support hypotheses • Good evidence more strongly supports a hypothesis. • Some hypotheses seem improbable a priori. E.g. Relativity. • Some evidence is surprising without an explanation. E.g. starlight bending. (Gravitational lensing).
  12. 12. BAYES’THEOREM • Formula • Where P is probability of, h is hypothesis and e is evidence: • P(h|e) = P(h) P(e|h) / P(e) • Where k is background knowledge: • P(h|e&k) = P(h|k) P(e|h&k) / P(e|k) • Let’s explain.What happens if P(e) is low, etc…?
  13. 13. BAYES’THEOREM P(h|e) = P(h) P(e|h) P(e) • Technical terms • P(h|e) = posterior probability • P(h) = prior probability of h • P(e) = prior probability of e • P(e|h) = likelihood of e, or probability of e given h
  14. 14. BAYES’THEOREM • Example 1: Theory of Relativity • Let’s suppose we thinkTR is improbable, because it’s hard to believe that time distorts, so P(h) is low.And let’s suppose that we would not expect to see light bend around a star. So P(e) is low.And let’s suppose that P(e|h), the case of light bending around a star, or e, is high, ifTR is true.This means that P(e|h) is high. If we put some values in, we can see that e, the evidence, supports or confirms the hypothesis h (TR); hence, although the prior probability, is low at first (becauseTR is hard to believe), the posterior (P(h|e)) is high, because of the evidential support. Let’s put some numbers in. Suppose P(e) is 0.3.And suppose P(h) is 0.2.And then suppose that P(e|h) is 0.8. • P(h|e) = 0.8 x 0.2 / 0.3 = 0.533. …The evidence supportsTR.
  15. 15. P(h|e) = P(h) P(e|h) P(e) BAYES’THEOREM • Example 2: Disease • Base rate problem: how common is a phenomenon, datum, event, trait, disease, etc., in a general population vs a sample population? • Reference class problem: which sample population? Are they representative? • Many Bayesian arguments fail to adequately set the prior because of neglecting these two issues.
  16. 16. BAYES’THEOREM • Example 2: Disease • Suppose that 3.4% of university students in South Africa are HIV-positive, that is, for every 1000 students, 34 have HIV. Suppose that an over-the-counter test returns a false HIV- positive result in 5% of cases. Suppose further that a student goes for a medical test and the test reports he is HIV-positive. Assume, further, that we know nothing more about this student.What are the odds that he is in fact HIV-positive?
  17. 17. P(h|e) = P(h) P(e|h) P(e) BAYES’THEOREM • Example 2: Disease • Most people will respond to this by saying, well, the test is wrong 5% of the time, so it’s probably right 95% of the time, meaning that the odds are 95% or 0.95 that he is in fact HIV-positive. However, the actual answer is 39%.
  18. 18. BAYES’THEOREM • Example 2: Disease • P(HIV|e) = P(e|HIV) P(HIV) / P(e) • Given P(HIV) = 34 / 1000 = 0.034 • P(¬HIV) = 0.966 • in other words, the prior probability that the person does not have HIV is 0.966, almost certain. Now, taking et, the evidence that the test says that the person does in fact have HIV, and the test is only 95% accurate, we get
  19. 19. BAYES’THEOREM • Example 2: Disease • P(e|HIV) = 0.95 • Therefore P(e|¬HIV) = 0.05 • To calculate P(e), the prior probability of et, we use theTotal Probability Rule, which says that P(e) = P(e|h)P(h) + P(e|¬h)P(¬h): • P(e) = P(e|HIV) P(HIV) + P(e|¬HIV) P(¬HIV)
 = (0.95)(0.034) + (0.05)(0.966) ≈ 0.081
  20. 20. BAYES’THEOREM • Example 2: Disease • Substituting P(e) back in: • P(HIV|e) = P(e|HIV) P(HIV) / P(e) • P(HIV|e) = (0.95)(0.034) / 0.081 = 0.3987 • So the student only has a 39% chance of HIV, not 95%.
  21. 21. P(h|e) = P(h) P(e|h) P(e) BAYES’THEOREM • Problem of the Priors • How do we assess P(e) and P(h) objectively? In a known population with known incidences, such as disease, it’s easy. It’s not so easy in other cases when we want to support a new theory.
  22. 22. BAYES’THEOREM • Frequentism • Frequentism is a possible way then to set the priors for Bayes’Theorem. Inductions are mathematically entailed from probability theory. So, if, for example, we know the relative frequency of a phenomenon in a large sample of a finite population, we can project the frequency into other samples of the same population by mathematical entailment.The frequency of sample 1 applies to sample 2.
  23. 23. BAYES’THEOREM • Frequentism • It’s the position that a probability is the actual measure of frequency of an event within a population of events of the same type; or, a measure of the frequency of phenomena or entities within a population of phenomena or entities of the same type. So, the chances of getting an Ace in a deck of cards is 4/52, because there are four aces in a population of 52 cards; i.e. 1:13.
  24. 24. BAYES’THEOREM • Frequentism: problems • Sample size: how big a sample of which run? • Sample representivity: which run in a sample? HHHHT? orTTTHH? Even if the run is ultimately “even” or “fair” • Finite vs Infinite/Hypothetical frequentism: which run of heads/tails • Single case problem: destroy the coin Image: S.Acharya, Stanford University
  25. 25. BAYES’THEOREM • Solution: Popper’s Propensity • A coin just has a propensity (tendency) to turn up heads or tails evenly. And a die just has a propensity to turn up a six one out of six times. And an experiment in quantum mechanics just has a propensity to turn up certain results. • Removes the scientist’s interference in quantum mechanics • Removes frequentist problems • Allows us to explain induction as probability theory.
  26. 26. BAYES’THEOREM • Evidential Relevance and the Raven Paradox (Hempel) (1) All ravens are black. (2) Everything that is not black is not a raven [entailed by (1)]. (3) “My pet raven is black” is evidence that supports the hypothesis (1) (4)This green (non-black) thing is an apple (not a raven).This evidence supports (2). (5) Since (4) supports (2), (4) supports (1), because (2) entails (1) (6)Therefore, green apples provide evidence that all ravens are black. Image: D. Hoffman wikimeda commons
  27. 27. BAYES’THEOREM • Evidential Relevance and the Raven Paradox (Hempel) • Two responses. Evidential relevance. If a piece of evidence is not relevant to the hypothesis being investigated, we can discard it. (Rowe, 1998, p550). • Sleight of hand. Notice how (1) talks about “ravens” but (2) talks about “all things”. So, if we want to reverse the inference, we can’t draw an inference from “all things” being “non-black” to a specific thing (a raven) being black. • What’s the point?The point is that background knowledge is relevant to assessing priors.
  28. 28. BAYES’THEOREM • The Problem of Background 
 Knowledge • k in Bayes’Theorem • The Green/Yellow taxi example. • Inferences can’t be made if our k tells us that there are confounders, e.g. contamination in bacteriological experiments • Background knowledge determines which hypotheses we exclude or pursue up-front, e.g. if a theory entails violation of a law.