15 Oct '12 13:16

Originally posted by kaminsky
Popper (pro-evolutionist)called evolution a metaphysical research program , since it could not be shown to be false. This does not mean evolution is nonsense, but strictly speaking not scientific . I am pro-evolution.

Karl Popper's philosophy of science is actually just a subset of Bayesian theory which really does determine whether something is or is not science.

Evolution however is very much falsifiable, and is very much science.

http://yudkowsky.net/rational/bayes

Bayes' Theorem describes what makes something "evidence" and how much evidence it is. Statistical models are judged by comparison to the Bayesian method because, in statistics, the Bayesian method is as good as it gets - the Bayesian method defines the maximum amount of mileage you can get out of a given piece of evidence, in the same way that thermodynamics defines the maximum amount of work you can get out of a temperature differential. This is why you hear cognitive scientists talking about Bayesian reasoners. In cognitive science, Bayesian reasoner is the technically precise codeword that we use to mean rational mind.

There are also a number of general heuristics about human reasoning that you can learn from looking at Bayes' Theorem.

For example, in many discussions of Bayes' Theorem, you may hear cognitive psychologists saying that people do not take prior frequencies sufficiently into account, meaning that when people approach a problem where there's some evidence X indicating that condition A might hold true, they tend to judge A's likelihood solely by how well the evidence X seems to match A, without taking into account the prior frequency of A. If you think, for example, that under the mammography example, the woman's chance of having breast cancer is in the range of 70%-80%, then this kind of reasoning is insensitive to the prior frequency given in the problem; it doesn't notice whether 1% of women or 10% of women start out having breast cancer. "Pay more attention to the prior frequency!" is one of the many things that humans need to bear in mind to partially compensate for our built-in inadequacies.

A related error is to pay too much attention to p(X|A) and not enough to p(X|~A) when determining how much evidence X is for A. The degree to which a result X is evidence for A depends, not only on the strength of the statement we'd expect to see result X if A were true, but also on the strength of the statement we wouldn't expect to see result X if A weren't true. 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, or you're looking at the early morning dew. Since p(wetgrass|~rain) is substantially greater than zero, p(rain|wetgrass) is substantially less than one. On the other hand, if the grass was never wet when it wasn't raining, then knowing that the grass was wet would always show that it was raining, p(rain|wetgrass) ~ 1, even if p(wetgrass|rain) = 50%; that is, even if the grass only got wet 50% of the times it rained. Evidence is always the result of the differential between the two conditional probabilities. Strong evidence is not the product of a very high probability that A leads to X, but the product of a very low probability that not-A could have led to X.

The Bayesian revolution in the sciences is fuelled, not only by more and more cognitive scientists suddenly noticing that mental phenomena have Bayesian structure in them; not only by scientists in every field learning to judge their statistical methods by comparison with the Bayesian method; but also by the idea that science itself is a special case of Bayes' Theorem; experimental evidence is Bayesian evidence. The Bayesian revolutionaries hold that 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. Previously, the most popular philosophy of science was probably Karl Popper's falsificationism - this is the old philosophy that the Bayesian revolution is currently dethroning. 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 favour 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. 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.

You can even formalize Popper's philosophy mathematically. The likelihood ratio for X, p(X|A)/p(X|~A), determines how much observing X slides the probability for A; the likelihood ratio is what says how strong X is as evidence. Well, in your theory A, you can predict X with probability 1, if you like; but you can't control the denominator of the likelihood ratio, p(X|~A) - there will always be some alternative theories that also predict X, and while we go with the simplest theory that fits the current evidence, you may someday encounter some evidence that an alternative theory predicts but your theory does not. That's the hidden gotcha that toppled Newton's theory of gravity. So there's a limit on how much mileage you can get from successful predictions; there's a limit on how high the likelihood ratio goes for confirmatory evidence.

On the other hand, if you encounter some piece of evidence Y that is definitely not predicted by your theory, this is enormously strong evidence against your theory. If p(Y|A) is infinitesimal, then the likelihood ratio will also be infinitesimal. For example, if p(Y|A) is 0.0001%, and p(Y|~A) is 1%, then the likelihood ratio p(Y|A)/p(Y|~A) will be 1:10000. -40 decibels of evidence! Or flipping the likelihood ratio, if p(Y|A) is very small, then p(Y|~A)/p(Y|A) will be very large, meaning that observing Y greatly favours ~A over A. Falsification is much stronger than confirmation. This is a consequence of the earlier point that very strong evidence is not the product of a very high probability that A leads to X, but the product of a very low probability that not-A could have led to X. This is the precise Bayesian rule that underlies the heuristic value of Popper's falsificationism.

Similarly, Popper's dictum that an idea must be falsifiable can be interpreted as a manifestation of the Bayesian conservation-of-probability rule; if a result X is positive evidence for the theory, then the result ~X would have disconfirmed the theory to some extent. If you try to interpret both X and ~X as "confirming" the theory, the Bayesian rules say this is impossible! To increase the probability of a theory you must expose it to tests that can potentially decrease its probability; this is not just a rule for detecting would-be cheaters in the social process of science, but a consequence of Bayesian probability theory. On the other hand, Popper's idea that there is only falsification and no such thing as confirmation turns out to be incorrect. Bayes' Theorem shows that falsification is very strong evidence compared to confirmation, but falsification is still probabilistic in nature; it is not governed by fundamentally different rules from confirmation, as Popper argued.

So we find that many phenomena in the cognitive sciences, plus the statistical methods used by scientists, plus the scientific method itself, are all turning out to be special cases of Bayes' Theorem. Hence the Bayesian revolution.

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