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Hi Less Wrong. I found a link to this site a year or so ago and have been lurking off and on since. However, I've self identified as a rationalist since around junior high school. My parents weren't religious and I was good at math and science, so it was natural to me to look to science and logic to solve everything. Many years later I realize that this is harder than I hoped.
Anyway, I've read many of the sequences and posts, generally agreeing and finding many interesting thoughts. It's fun reading about zombies and Newcomb's problem and the like.
I guess this sounds heretical, but I don't understand why Bayes theorem is placed on such a pedestal here. I understand Bayesian statistics, intuitively and also technically. Bayesian statistics is great for a lot of problems, but I don't see it as always superior to thinking inspired by the traditional scientific method. More specifically, I would say that coming up with a prior distribution and updating can easily be harder than the problem at hand.
I assume the point is that there is more to what is considered Bayesian thinking than Bayes theorem and Bayesian statistics, and I've reread some of the articles with the idea of trying to pin that down, but I've found that difficult. The closest I've come is that examining what your priors are helps you to keep an open mind.
Bayesian theorem is just one of many mathematical equations, like for example Pythagorean theorem. There is inherently nothing magical about it.
It just happens to explain one problem with the current scientific publishing process: neglecting base rates. Which sometimes seems like this: "I designed an experiment that would prove a false hypothesis only with probability p = 0.05. My experiment has succeeded. Please publish my paper in your journal!"
(I guess I am exaggerating a bit here, but many people 'doing science' would not understand immediately what is wrong with this. And that would be those who even bother to calculate the p-value. Not everyone who is employed as a scientist is necessarily good at math. Many people get paid for doing bad science.)
This kind of thinking has the following problem: Even if you invent hundred completely stupid hypotheses; if you design experiments that would prove a false hypothesis only with p = 0.05, that means five of them would be proved by the experiment. If you show someone else all hundred experiments together, they may understand what is wrong. But you are more likely to send only the successful five ones to the journal, aren't you? -- But how exactly is the journal supposed to react to this? Should they ask: "Did you do many other experiments, even ones completely irrelevant to this specific hypothesis? Because, you know, that somehow undermines the credibility of this one."
The current scientific publishing process has a bias. Bayesian theorem explains it. We care about science, and we care about science being done correctly.
That's not neglecting base rates, that's called selection bias combined with incentives to publish. Bayes theorem isn't going to help you with this.
http://xkcd.com/882/
Uhm, it's similar, but not the same.
If I understand it correctly, selection bias is when 20 researchers make an experiment with green jelly beans, 19 of them don't find significant correlation, 1 of them finds it... and only the 1 publishes, and the 19 don't. The essence is that we had 19 pieces of evidence against the green jelly beans, only 1 piece of evidence for the green jelly beans, but we don't see those 19 pieces, because they are not published. Selection = "there is X and Y, but we don't see Y, because it was filtered out by the process that gives us information".
But imagine that you are the first researcher ever who has researched the jelly beans. And you only did one experiment. And it happened to succeed. Where is the selection here? (Perhaps selection across Everett branches or Tegmark universes. But we can't blame the scientific publishing process for not giving us information from the parallel universes, can we?)
In this case, base rate neglect means ignoring the fact that "if you take a random thing, the probability that this specific thing causes acne is very low". Therefore, even if the experiment shows a connection with p = 0.05, it's still more likely that the result just happened randomly.
The proper reasoning could be something like this (all number pulled out of the hat) -- we already have pretty strong evidence that acne is caused by food; let's say there is a 50% probability for this. With enough specificity (giving each fruit a different category, etc.), there are maybe 2000 categories of food. It is possible that more then one of them cause acne, and our probability distribution for that is... something. Considering all this information, we estimate a prior probability let's say 0.0004 that a random food causes acne. -- Which means that if the correlation is significant on level p = 0.05, that per se means almost nothing. (Here one could use the Bayes theorem to calculate that the p = 0.05 successful experiment shows the true cause of acne with probablity cca 1%.) We need to increase it to p = 0.0004 just to get a 50% chance of being right. How can we do that? We should use a much larger sample, or we should repeat the experiment many times, record all the successed and failures, and do a meta-analysis.
That's a different case -- you have no selection bias here, but your conclusions are still uncertain -- if you pick p=0.05 as your threshold, you're clearly accepting that there is a 5% chance of a Type I error: the green jelly beans did nothing, but the noise happened to be such that you interpreted it as conclusive evidence in favor of your hypothesis.
But that all is fine -- the readers of scientific papers are expected to understand that results significant to p=0.05 will be wrong around 5% of the times, more or less (not exactly because the usual test measures P(D|H), the probability of the observed data given the (null) hypothesis while you really want P(H|D), the probability of the hypothesis given the data).
People rarely take entirely random things and test them for causal connection to acne. Notice how you had to do a great deal of handwaving in establishing your prior (aka the base rate).
As an exercise, try to be specific. For example, let's say I want to check if the tincture made from the bark of a certain tree helps with acne. How would I go about calculating my base rate / prior? Can you walk me through an estimation which will end with a specific number?
And this is the base rate neglect. It's not "results significant to p=0.05 will be wrong about 5% of time". It's "wrong results will be significant to p=0.05 about 5% of time". And most people will confuse these two things.
It's like when people confuse "A => B" with "B => A", only this time it is "A => B (p=0.05)" with "B => A (p=0.05)". It is "if wrong, then in 5% significant". It is not "if significant, then in 5% wrong".
Yes, you are right. Establishing the prior is pretty difficult, perhaps impossible. (But that does not make "A => B" equal to "B => A".) Probably the reasonable thing to do would be simply to impose strict limits in areas where many results were proved wrong.
Um, what "strict limits" are you talking about, what will they look like, and who will be doing the imposing?
To get back to my example, let's say I'm running experiments to check if the tincture made from the bark of a certain tree helps with acne -- what strict limits would you like?
p = 0.001, and if at the end of the year too many researches fail to replicate, keep decreasing. (let's say that "fail to replicate" in this context means that the replication attempt cannot prove it even with p = 0.05 -- we don't want to make replications too expensive, just a simple sanity check)
a long answer would involve a lot of handwaving again (it depends on why do you believe the bark is helpful; in other words, what other evidence do you already have)
a short answer: for example, p = 0.001
Well, and what's magical about this particular number? Why not p=0.01? why not p=0.0001? Confidence thresholds are arbitrary, do you have a compelling argument why any particular one is better than the rest?
Besides, you're forgetting the costs. Assume that the reported p-values are true (and not the result of selection bias, etc.). Take a hundred papers which claim results at p=0.05. At the asymptote about 95 of them will turn out to be correct and about 5 will turn out to be false. By your strict criteria you're rejecting all of them -- you're rejecting 95 correct papers. There is a cost to that, is there not?
Lumifer, please update that at this moment you don't grok the difference between "A => B (p=0.05)" and "B => A (p = 0.05)", which is why you don't understand what p-value really means, which is why you don't understand the difference between selection bias and base rate neglect, which is probably why the emphasis on using Bayes theorem in scientific process does not make sense to you. You made a mistake, that happens to all of us. Just stop it already, please.
And don't feel bad about it. Until recently I didn't understand it too, and I had a gold medal from international mathematical olympiad. Somehow it is not explained correctly at most schools, perhaps because the teachers don't get it themselves, or maybe they just underestimate the difficulty of proper understanding and the high chance of getting it wrong. So please don't contibute to the confusion.
Imagine that there are 1000 possible hypotheses, among which 999 are wrong, and 1 is correct. (That's just a random example to illustrate the concept. The numbers in real life can be different.) You have an experiment that says "yes" to 5% of the wrong hypotheses (this is what p=0.05 means), and also to the correct hypothesis. So at the end, you have 50 wrong hypotheses and 1 correct hypothesis confirmed by the experiment. So in the journal, 98% of the published articles would be wrong, not 5%. It is "wrong => confirmed (p=0.05)", not "confirmed => wrong (p=0.05)".
That's not how p-values work. p=0.05 doesn't mean that the hypothesis is 95% likely to be correct, even in principle; it means that there's a 5% chance of seeing the same correlation if the null hypothesis is true. Pull a hundred independent data sets and we'd normally expect to find a p=0.05 correlation or better in at least five or so of them, no matter whether we're testing, say, an association of cancer risk with smoking or with overuse of the word "muskellunge".
This distinction's especially important to keep in mind in an environment where running replications is relatively low-status or where negative results tend to be quietly shelved -- both of which, as it happens, hold true in large chunks of academia. But even if this weren't the case, we'd normally expect replication rates to be less than one minus the claimed p-value, simply because there are many more promising ideas than true ones and some of those will turn up false positives.
No, they won't. You're committing base rate neglect. It's entirely possible for people to publish 2000 papers in a field where there's no hope of finding a true result, and get 100 false results with p<0.05 just by chance (along with 1900 other false results with p > 0.05).
I know a few answers to this question, and I'm sure there are others. (As an aside, these foundational questions are, in my opinion, really important to ask and answer.)
I think that the qualitative side of Bayes is super important but don't think we've found a good way to communicate it yet. That's an active area of research, though, and in particular I'd love to hear your thoughts on those four answers.
What is the qualitative side of Bayes?
Unfortunately, the end of that sentence is still true:
I think that What Bayesianism Taught Me is a good discussion on the subject, and my comment there explains some of the components I think are part of qualitative Bayes.
I think that a lot of qualitative Bayes is incorporating the insights of the Bayesian approach into your System 1 thinking (i.e. habits on the 5 second level).
Well, yes, but most of the things there are just useful ways to think about probabilities and uncertainty, proper habits, things to check, etc. Why Bayes? He's not a saint whose name is needed to bless a collection of good statistical practices.
It's more or less the same reason people call a variety of essentialist positions 'platonism' or 'aristotelianism'. Those aren't the only thinkers to have had views in this neighborhood, but they predated or helped inspire most of the others, and the concepts have become pretty firmly glued together. Similarly, the phrases 'Bayes' theorem' and 'Bayesian interpretation of probability' (whence, jointly, the idea of Bayesian inference) have firmly cemented the name Bayes to the idea of quantifying psychological uncertainty and correctly updating on the evidence. The Bayesian interpretation is what links these theorems to actual practice.
Bayes himself may not have been a 'Bayesian' in the modern sense, just as Plato wasn't a 'platonist' as most people use the term today. But the names have stuck, and 'Laplacian' or 'Ramseyan' wouldn't have quite the same ring.
I like Laplacian as a name better, but it's already a thing.
If I were to pretend that I'm a mainstream frequentist and consider "quantifying psychological uncertainty" to be subjective mumbo-jumbo with no place anywhere near real science :-D I would NOT have serious disagreements with e.g. Vaniver's list. Sure, I would quibble about accents, importances, and priorities, but there's nothing there that would be unacceptable from the mainstream point of view.
My biggest concern with the label 'Bayesianism' isn't that it's named after the Reverend, nor that it's too mainstream. It's that it's really ambiguous.
For example, when Yvain speaks of philosophical Bayesianism, he means something extremely modest -- the idea that we can successfully model the world without certainty. This view he contrasts, not with frequentism, but with Aristotelianism ('we need certainty to successfully model the world, but luckily we have certainty') and Anton-Wilsonism ('we need certainty to successfully model the world, but we lack certainty'). Frequentism isn't this view's foil, and this philosophical Bayesianism doesn't have any respectable rivals, though it certainly sees plenty of assaults from confused philosophers, anthropologists, and poets.
If frequentism and Bayesianism are just two ways of defining a word, then there's no substantive disagreement between them. Likewise, if they're just two different ways of doing statistics, then it's not clear that any philosophical disagreement is at work; I might not do Bayesian statistics because I lack skill with R, or because I've never heard about it, or because it's not the norm in my department.
There's a substantive disagreement if Bayesianism means 'it would be useful to use more Bayesian statistics in science', and if frequentism means 'no it wouldn't!'. But this methodological Bayesianism is distinct from Yvain's philosophical Bayesianism, and both of those are distinct from what we might call 'Bayesian rationalism', the suite of mantras, heuristics, and exercises rationalists use to improve their probabilistic reasoning. (Or the community that deems such practices useful.) Viewing the latter as an ideology or philosophy is probably a bad idea, since the question of which of these tricks are useful should be relatively easy to answer empirically.
Yes, it is my understanding that epistemologists usually call the set of ideas Yvain is referring to "probabilism" and indeed, it is far more vague and modest than what they call Bayesianism (which is more vague and modest still than the subjectively-objective Bayesianism that is affirmed often around these parts.).
BTW, I think this is precisely what Carnap was on about with his distinction between probability-1 and probability-2, neither of which did he think we should adopt to the exclusion of the other.
Err, actually, yes it is. The frequentist interpretation of probability makes the claim that probability theory can only be used in situations involving large numbers of repeatable trials, or selection from a large population. William Feller:
Or to quote from the essay coined the term frequentist:
Frequentism is only relevant to epistemological debates in a negative sense: unlike Aristotelianism and Anton-Wilsonism, which both present their own theories of epistemology, frequentism's relevance is almost only in claiming that Bayesianism is wrong. (Frequentism separately presents much more complicated and less obviously wrong claims within statistics and probability; these are not relevant, given that frequentism's sole relevance to epistemology is its claim that no theory of statistics and probability could be a suitable basis for an epistemology, since there are many events they simply don't apply to.)
(I agree that it would be useful to separate out the three versions of Bayesianism, whose claims, while related, do not need to all be true or false at the same time. However, all three are substantively opposed to one or both of the views labelled frequentist.)
Depends which frequentist you ask. From Aris Spanos's "A frequentist interpretation of probability for model-based inductive inference":
and
I don't understand what this "probability theory can only be used..." claim means. Are they saying that if you try to use probability theory to model anything else, your pencil will catch fire? Are they saying that if you model beliefs probabilistically, Math breaks? I need this claim to be unpacked. What do frequentists think is true about non-linguistic reality, that Bayesians deny?
Yes, but frequentists have zero problems with hypothetical trials or populations.
Do note that for most well-specified statistical problems the Bayesians and the frequentists will come to the same conclusions. Differently expressed, likely, but not contradicting each other.
I think they would have significant practical disagreement with #3, given the widespread use of NHST, but clever frequentists are as quick as anyone else to point out that NHST doesn't actually do what its users want it to do.
Hence the importance of the qualifier 'qualitative'; it seems to me that accents, importances, and priorities are worth discussing, especially if you're interested in changing System 1 thinking instead of System 2 thinking. The mainstream frequentist thinks that base rate neglect is a mistake, but the Bayesian both thinks that base rate neglect is a mistake and has organized his language to make that mistake obvious when it occurs. If you take revealed preferences seriously, it looks like the frequentist says base rate neglect is a mistake but the Bayesian lives that base rate neglect is a mistake.
Now, why Bayes specifically? I would be happy to point to Laplace instead of Bayes, personally, since Laplace seems to have been way smarter and a superior rationalist. But the trouble with naming methods of "thinking correctly" is that everyone wants to name their method "thinking correctly," and so you rapidly trip over each other. "Rationalism," for example, refers to a particular philosophical position which is very different from the modal position here at LW. Bayes is useful as a marker, but it is not necessary to come to those insights by way of Bayes.
(I will also note that not disagreeing with something and discovering something are very different thresholds. If someone has a perspective which allows them to generate novel, correct insights, that perspective is much more powerful than one which merely serves to verify that insights are correct.)
Yeah, I said if I were pretend to be a frequentist -- but that didn't involve suddenly becoming dumb :-)
I agree, but at this point context starts to matter a great deal. Are we talking about decision-making in regular life? Like, deciding which major to pick, who to date, what job offer to take? Or are we talking about some explicitly statistical environment where you try to build models, fit them, evaluate them, do out-of-sample forecasting, all that kind of things?
I think I would argue that recognizing biases (Tversky/Kahneman style) and trying to correct for them -- avoiding them altogether seems too high a threshold -- is different from what people call Bayesian approaches. The Bayesian way of updating on the evidence is part of "thinking correctly", but there is much, much more than just that.
At least one (and I think several) of biases identified by Tversky and Kahneman is "people do X, a Bayesian would do Y, thus people are wrong," so I think you're overstating the difference. (I don't know enough historical details to be sure, but I suspect Tversky and Kahneman might be an example of the Bayesian approach allowing someone to discover novel, correct insights.)
I agree, but it feels like we're disagreeing. It seems to me that a major Less Wrong project is "thinking correctly," and a major part of that project is "decision-making under uncertainty," and a major part of uncertainty is dealing with probabilities, and the Bayesian way of dealing with probabilities seems to be the best, especially if you want to use those probabilities for decision-making.
So it sounds to me like you're saying "we don't just need stats textbooks, we need Less Wrong." I agree; that's why I'm here as well as reading stats textbooks. But it also sounds to me like you're saying "why are you naming this Less Wrong stuff after a stats textbook?" The easy answer is that it's a historical accident, and it's too late to change it now. Another answer I like better is that much of the Less Wrong stuff comes from thinking about and taking seriously the stuff from the stats textbook, and so it makes sense to keep the name, even if we're moving to realms where the connection to stats isn't obvious.
I guess the distinction in my mind is that in a Bayesian approach one enumerates the various hypothesis ahead of time. This is in contrast to coming up with a single hypothesis and then adding in more refined versions based on results. There are trade-offs between the two. Once you get going with a Bayesian approach you are much better protected against bias; however if you are missing some hypothesis from your prior you don't find it.
Here are some specific responses to the 4 answers:
If you have a problem for which it is easy to enumerate the hypotheses, and have statistical data, then Bayes is great. If in addition you have a good prior probability distribution then you have the additional advantage that it is much easier to avoid bias. However if you find you are having to add in new hypotheses as you investigate then I would say you are using a hybrid method.
Even without Bayes one is supposed to specifically look for alternate hypothesis and search for the best answer.
On the Less Wrong welcome page the link next to the Bayesian link is a reference to the 2 4 6 experiment. I'd say this is an example of a problem poorly suited to Bayesian reasoning. It's not a statistical problem, and it's really hard to enumerate the prior for all rules for a list of 3 numbers ordered by simplicity. There's clearly a problem with confirmation bias, but I would say the thing to do is to step back and do some careful experimentation along traditional lines. Maybe Bayesian reasoning is helpful because it would encourage you to do that?
I would agree that a rationalist needs to be exposed to these concepts.
I wonder about this statement the most. It's hard to judge qualitative statements about probabilities. For example, I can say that I had a low prior belief in cryonics, and since reading articles here I have updated and now have a higher probability. I know I had some biases against the idea. However, I still don't agree and it's difficult to tell how much progress I've made in understanding the arguments.
Regarding Bayes, you might like my essay on the topic, especially if you have statistical training.
That paper did help crystallize some of my thoughts. At this point I'm more interested in wondering if I should be modifying how I think, as opposed to how to implement AI.
You are not alone in thinking the use of Bayes is overblown. It can;t be wrong, of course, but it can be impractical to use and in many real life situations we might not have specific enough knowledge to be able to use it. In fact, that's probably one of the biggest criticisms of lesswrong.