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Bayes's Theorem: What's the Big Deal?

• njohnson41 11y ago
  Good article. I'm only a bit disappointed that the author seems not to realize that Bayes' theorem is just a simple consequence of probability theory, and should be attractive not because "maybe the brain is Bayesian", but because it is based on sound set-theoretic and analytic principles. If Bayes' theorem is false, so is probability theory, and so is nearly everything we know about probability. Edit: Here is a good explanation of the theorem that makes it visually clear how only set theory is involved in deriving it: https://oscarbonilla.com/2009/05/visualizing-bayes-theorem/ https://oscarbonilla.com/2009/05/visualizing-bayes-theorem/
  • gaur 11y ago
    I guess a hard-line frequentist (if such a person exists) would counter that you can't assign probabilities to hypotheses or fixed parameters. Then Bayes's theorem (and every other statement about probability) is true only when applied to statements about how often a certain event will occur. But of course, most people do assign probabilities to hypotheses and fixed parameters, even if only informally. Bayesian probability theory is an attempt to formalize that kind of intuitive reasoning.
    • lottin 11y ago
      I think it's strange this sudden comeback of a theory that was dismissed more than 70 years ago by Fisher and many others, but no one, as far as I know, cares to explain why Fisher was wrong and why the theory is right. It makes me very suspicious, to be honest.
      • dragandj 11y ago
        Actually, many Bayesian textbooks cover that. Try Jaynes - Probability Theory: The Logic of Science.
    • eli_gottlieb 11y ago
      >I guess a hard-line frequentist (if such a person exists) would counter that you can't assign probabilities to hypotheses or fixed parameters. Then Bayes's theorem (and every other statement about probability) is true only when applied to statements about how often a certain event will occur. If you model "thinking" and "believing" as sampling in probabilistic programs (which they do in some schools of cognitive science), then Bayes' Theorem becomes a theorem about how often certain execution traces occur when the sampling program is run with fresh randomness. You then need none of the weird metaphysics associated with "subjective Bayesianism".
    • nshepperd 11y ago
      I have heard of people genuinely saying such a thing. Fortunately, it is nothing but an empty redefinition of the word "probability". In fact, rational degrees of belief in hypotheses do follow the kolmogorov axioms (as shown by eg. Cox's theorem or the VNM theorem), and bayes theorem does therefore apply. Whether or not someone refuses to call that "probability" makes no difference.
  • lmm 11y ago
    All probabilistic tools are accurate in theory, otherwise we wouldn't use them. Bayes' theorem is no different from e.g. the t-test in that regard. The question of whether it's worth using Bayes explicitly rather than other tools is and should be a question of whether we find it aligns with our understanding and helps us think more clearly.
  • mikeash 11y ago
    Just because a theorem is true doesn't mean you can't misuse it or that you don't need to do some work to map it to reality. For example, the Banach-Tarski theorem is solid, but that doesn't mean you can start a business making golf balls by buying one and then endlessly replicating it.
    • jsprogrammer 11y ago
      Certainly. Just because you can name a theorem doesn't mean you can derive it either. The article had no actual computation or derivation of the theorem. Instead, it talked about beliefs and other things that don't really exist (in regards to the computation of a probability value).
  • sorokod 11y ago
    Given that p(A and B) = p(B and A) p(A and B ) = p(A) * p(B|A) we have: p(A) * p(B|A) = p(B) * p(A|B) p(B|A) = p(B) * p(A|B) / p(A)
• theseatoms 11y ago
  > The potential for Bayes abuse begins with P(B), your initial estimate of the probability of your belief, often called the “prior.” tldr; priors matter
  • lordCarbonFiber 11y ago
    That was my thought as well. Garbage in = garbage out, that's pretty standard in most fields. I really didn't like how the author treated the theorem as if it's some sort of magic, aside from something everyone that's taken a college prob/stat class has derived from first principles.
    • sago 11y ago
      The problem is that advocates do treat it as a magical thing. They extrapolate from the fact it is proven to the claim that all knowledge is Bayesian, to the implication that all Bayesian reasoning is knowledge. This fashion is why, for example, BT has been used to both prove the resurrection of Jesus, and to prove that Jesus didn't exist: both to a very high probability.
      • lordCarbonFiber 11y ago
        I'm almost sad I've never met one of these people in the wild. I'd really like to sit down and watch someone, with a straight face, try to say they both have a set probability for their belief on Jesus AND the probability that some vague ~evidence~ would exist given that belief.
  • sageikosa 11y ago
    This was Dawkins' assessment in the use of Bayesian Probabilities to "prove" the god hypothesis...basically the numbers being entered were complete and utter fabrications, making the mathematics pointless except that they lent an air of quantitative rigor.
• stuartaxelowen 11y ago
  Read through most of the article just for "people can abuse priors"? Come on. Anything, used wrongly, can promote superstition and pseudoscience.
  • astine 11y ago
    I think you're missing the broader argument, which is using 'mathy' concepts to dress up poor reasoning. Obviously priors matter, but what matters most of all is how good/complete your evidence is. Using a mathematical formula to lend credence to weak evidence (through liberal use of assumptions) is a hallmark of pseudoscience. The same could be said of many of the abuses of statistics and Bayes theorem is merely one good example of this.
    • Anderkent 11y ago
      Is using mathy concepts to dress up poor reasoning worse than not using anything to back up your reasoning? At least you can point out exactly what's wrong with the mathy reasoning. A colleague of mine says 'Sometimes pulling numbers out of your arse and using them to make a decision is better than pulling a decision out of your arse'
      • astine 11y ago
        "Is using mathy concepts to dress up poor reasoning worse than not using anything to back up your reasoning?" I believe so. If your belief is baseless, or based on flimsy evidence or simple bias, it's best if that's obvious. Dressing up weak reasoning to seem stronger is a form of lying. It's what we call sophistry. A big part of the problem is that for a lot of people don't understand the math well enough to point out what's wrong with it or have a bias towards explanations that seem complex or sophisticated but really aren't. It's true that sometimes we have to make a decision based on poor or no evidence but it should be clear that that is the case when that is the case. Dressing up the argument only obfuscates that.
        • TeMPOraL 11y ago
• cwyers 11y ago
  > In many cases, estimating the prior is just guesswork, allowing subjective factors to creep into your calculations. You might be guessing the probability of something that--unlike cancer—does not even exist, such as strings, multiverses, inflation or God. You might then cite dubious evidence to support your dubious belief. In this way, Bayes’ theorem can promote pseudoscience and superstition as well as reason. Oh please. You can do plenty of psuedoscience and superstition with good old frequentist statistics. And of all the people you could pick to represent Bayesian statistics, instead of I don't know, Andrew Gelman or someone, the author picks... Eliezer Yudkowsky? If nothing else, this provides inspiration for me to quit procrastinating on my "ASK ME ABOUT ROKO'S BASILISK" novelty t-shirt idea.
  • privong 11y ago
    > Oh please. You can do plenty of psuedoscience and superstition with good old frequentist statistics. That's not really the point. The article is simply saying that Bayesian methods are not a silver bullet; it's not saying that other methods of statistics are free from problems.
    • cwyers 11y ago
      > The article is simply saying that Bayesian methods are not a silver bullet But it's not really saying it in passing; the headline is taken from that paragraph. At that point it almost feels like a takedown of a strawman Bayesian.
  • entrode 11y ago
    I suspect Eliezer is targeted specifically due to his tongue-in-cheek presentation of understanding Bayesian statistics as being initiation into a cult. Also due to the author's familiarity with the topic likely specifically as a result of Eliezer's efforts to popularize the subject and his association to him resultantly.
    • eli_gottlieb 11y ago
      Sure, but who cares? There's an entire field of statistics called "Bayesian statistics", who do actual math and statistics and don't give a damn about novelty T-shirts.
• RogerL 11y ago
  I prefer to think of it in terms of the statistical inversion problem. That is, we have an event(s) that occur, which we may imperfectly understand. We take noisy measurements of that event. Clearly, the causal relationship is the events cause the measurements - a bad measurement does not cause the event to move. But, in practice all we have are measurements, and from that we want to find an optimal (or good) estimate for what the events were. Hence, inversion. Bayes formula expresses P(x|y) in terms of P(Y|x), so you can perform the inversion using bayes. In many fields establishing the prior is difficult, hence frequentist methods are popular. There are many techniques for the statistical inversion problem. Trying to track a ballistic object in a vacuum? Fit the measurements to a second order polynomial (parabola) and you are done (well, you have to decide least squares vs robust methods, but it is not such a hard problem in the scheme of things). Trying to track a manuevering jet, stock prices, or disease incidence rates. Now your model of the problem is much less clear. We model lack of information as random variables. It isn't "random" in the deterministic sense, just in the sense that we don't know. Establish a good probabilistic description of that lack of knowledge in your prior, and you are probably going to get good result: this jet fighter is probabilisticly either moving straight, performing a coordinated turn, or performing an uncoordinated turn. Use a Markov chain to model those likelihoods (e.g.), and you may end up with good results. But if your modeling of the prior is poor, well, good luck to you, your output is probably nonsense.
  • jsprogrammer 11y ago
    Most measurements do affect the event.
    • jamiek88 11y ago
      ALL measurements at a quantum level at least!
      • cbd1984 11y ago
        Believe it or not, the Heisenberg Uncertainty Principle has nothing to do with measurement.
• GavinB 11y ago
  This headline does not reflect the article and is needlessly inflammatory. This article is an explanation of Bayes theorem and overall very positive of it. The "used wrongly" quote is just stating that it's not immune to biases and error. Pretty much any tool "used wrongly" can cause errors. (In case it's changed, the headline currently reads "Bayes theorem used wrongly, can promote superstition and pseudoscience")
  • mfoy_ 11y ago
    I think the title isn't quite so off. The article is trying to point out that people should be wary of using it wrongly. Like other commentators have pointed out, the saying "garbage in garbage out" only helps when you stop to think whether or not you're putting garbage in.
  • deleted 11y ago
    [deleted]
• Laaw 11y ago
  I've been saying this for years, and this is a large reason why I find the LessWrong folks to be almost entirely full of it. Their inability to come up with accurate priors is completely lost on many of the folks who follow this kind of thinking. A couple of comments are saying, "no duh" to this article, but those folks likely don't realize quite how many other people are falling into this trap. "Garbage in, garbage out" is only good advice when the person you're saying it to realizes they're putting garbage in.
  • woodchuck64 11y ago
    > I've been saying this for years, and this is a large reason why I find the LessWrong folks to be almost entirely full of it. Their inability to come up with accurate priors is completely lost on many of the folks who follow this kind of thinking. I assume you're saying LessWrong folks are more prone to miscalculating priors than most. Could you give some examples of this?
    • jacobolus 11y ago
      The LessWrong folks aren’t obviously better or worse at calculating priors than anyone else. The “problem” is that their hobby is spending their free time considering outlandish scenarios, inventing arbitrary assumptions related to such scenarios, drawing questionable conclusions, and then convincing themselves that because they used logic and math, their analysis must be correct. Plenty of other folks who spend time on similar activities with a less pseudo-rigorous framing end up as conspiracy theorists or occultists; belief in AI overlords ruling humanity, the technological singularity, cryogenics, or impending 1000-year human lifespans is far from the kookiest thing people convince themselves about.
      • TeMPOraL 11y ago
        Oh come on. That's how you're supposed to use math. To aid your thinking. Without it, and considering "outlandish scenarios", we would not have any scientific progress. Also, this is one strange thing - any time someone asks us (the STEM crowd), "what will I ever use math for in my life?", the default answer seems to be, "it's about having more tools for thinking, and greater clarity of thought; it'll make you smarter". But then, some of us turn around and refuse to acknowledge that people who actually learn math and try to apply it may be getting those promised results. Whether it's LW people, programmers, engineers or scientists, the moment it matters, the default conclusion is that math gives nothing.
• PaulHoule 11y ago
  So can frequentism. Many investigators in parapsychology who were sincere and intelligent appear to have based their career on the incorrect use of frequentist statistics. And it's not just them. Ernerst Rutherford, who discovered the atomic nucleus, "If your experiment needs statistics, you ought to do a better experiment." In the 1990s I was a physics grad student and I think none of the professors had ever heard of the idea of a parameter estimator so we had a bunch of ad-hoc ways to fit power law coefficients that gave different answers and no way to judge goodness of fit that was thought out at all. One postdoc in my lab suffered through a difficult job market before finally, after a decade of anxiety and uncertainty, got a tenure track position and eventually wrote a paper on how to fit power law curves... in a statistics journal. And this was in a good department with people in which I was proud of both the teaching and research going on.
  • NN88 11y ago
    Rutherford should just be happy he discovered the last reasonably tangible piece of matter humanity will see for a long time, if ever.
  • 3pt14159 11y ago
    I'm a little confused. Are you saying that a tenure track prof wrote a paper on how to evauluate fitted power law curves? Was it something else besides least squares? Because I can't possibly see this getting accepted to a statistics journal.
    • PaulHoule 11y ago
      It is far beyond least squares as it was and is practiced There already was stuff in the stats literature in the 1990s that was much better but people in the physics community (such as myself) were not aware of that literature. On the other hand, stats people were not particular aware of the way power laws were occuring in physics. I saw things that did not add up ten years earlier and Mark Newman did too but we were both so caught up in the rat race, consensus reality, collective delusion, whatever you call it that I left physics before I could address the problem and he suffered through years of bullshit before he could find the time to do something about it. Watching Mark write great papers, write great book chapters and suffer from tremendous anxiety over his career was a big reason why I left.
• astine 11y ago
  I've had people seriously claim to me that using Bayes theorem to evaluate beliefs that one deals with in ones everyday life using evidence that one comes across in everyday life was likely a good idea and would reduce bias. I wish I'd had the presence of mind to point out that that did nothing to eliminate the selection bias of one's own experience. No mathematical formula can draw meaning out of weak or flawed evidence. Trying to do so is like trying to 'enhance' a blurry photo so that you can see details in the photo that didn't exist.
  • Anderkent 11y ago
    Even if you only get weak of flawed evidence, then you do what you can to make the best decisions given that evidence. No one tries to suggest you do actual bayesian calculations on everything you know, for the simplest reason that it's not computationally viable. But if your beliefs directly contradict bayes then you're doing something wrong - there's an inconsistency that's likely worth investigating, unless the matter is really minor. It's a sanity check for your decision making, not a constructive algorithm for the best decision.
• smcg 11y ago
  Kind of a clickbait title considering the article is a bit more nuanced than that.
• hyperpape 11y ago
  Interesting that they mention the medical case, when there's some psychological work around the idea that we should present these cases in terms of natural frequencies instead of Bayes' theorem. The natural frequencies approach is to say "if 10000 people take the test, 100 will have cancer. Of them, 99 will get an accurate positive test, and 1 will have a false negative test. Of the other 9900, 99 will receive a false positive, and 9801 will receive a correct negative. What are the odds that someone who has a positive test has cancer?" It turns out that doctors and other professionals whose core professional competency doesn't concern probability do terribly when presented with percentages and Bayes theorem, but can handle natural frequencies quite well (here's one quick summary: http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/?_r=0 http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/...). As is obvious, this isn't an argument that Bayes' theorem is wrong--it's a theorem after all. It's an argument about which types of reasoning people can be easily taught.
  • Anderkent 11y ago
    It's true that when the question is formed in frequentist terms, the answer is much more intuitive. But is that how the problem occurs in real life? The doctor doesn't see ten thousand people take a test; they see a person take a test, and get either a positive or negative result. The traditional way of forming the problem seems closer to actual experience: 'your patient tested positive. you know how accurate the test is, and how common the disease is; how likely is it that the result is a false positive?'
    • hyperpape 11y ago
      I'm not quite sure what you're saying. Doctors don't observe probabilities or enormous frequencies. Either way, there are good odds that this is information that someone is communicating to them, not the result of their personal experience.
      • Anderkent 11y ago
        Doctors observe a result of the test, and know the basic probabilities (in the example, 99% test accuracy, 1% of population have the disease). The problem is that they [often] draw incorrect conclusions from those observations (99% test accuracy and you tested positive? well then you likely - 99% - have the disease, right?). The question formed as 'your one patient tested positively' is more immediately relevant, I'd think. The correspondence with actual practice is obvious. The question formed as 'out of 10000 ...' could be remembered as a quirk of statistics, but not actually recalled when someone tests positively for cancer.
• exelius 11y ago
  Bayesian probability is catching on because the technology has finally caught up to the point where it's feasible. Thanks to the web and the proliferation of big data, we now have enough observations that Bayesian models can be trained (this was always the hard part). This doesn't mean we don't need smart people figuring out what things the model should and shouldn't look at; but it's at least possible to do today. It also doesn't hurt that the Bayes theorem is at its heart a map-reduce problem. Where once Bayes' theorem was considered a cumbersome artifact for "brute forcing" probability, it's now likely faster than competing methods of statistical analysis. We almost seem to be getting to the point in ad targeting where demographics are expressed in terms of a set of bayesian properties. You don't even care what the properties are; just that they're potentially more willing than average to use your product.
  • GFK_of_xmaspast 11y ago
    > It also doesn't hurt that the Bayes theorem is at its heart a map-reduce problem. ??? I'm not following. You could say that (numerical) integration is a map-reduce problem but that's pretty trivial.
• vezzy-fnord 11y ago
  If you have some time to spare, here's a great paper on a similar topic: the (mis)use of mathematics and in particular equilibrium methods in economics after Samuelson and Hicks ("What Went Wrong with Economics?", Peter J. Boettke, 1997): http://www.the-dissident.com/Boettke_CR.pdf http://www.the-dissident.com/Boettke_CR.pdf Pretty much every debate on economic policy between the left and right is full of fallacies emanating from this episode in history.
• RodericDay 11y ago
  I think Stephen Bond did some excellent takedowns of the identity politics that has arisen around Bayes' Theorem back in the day. I wonder where he's at these days. The Cult of Bayes' Theorem http://laurencetennant.com/bonds/cultofbayes.html http://laurencetennant.com/bonds/cultofbayes.html > One of Yudkowsky's constant refrains, appropriating language from Frank Herbert's Dune, is "Politics is the Mind-killer". Under this rallying cry, Lesswrong insiders attempt to purge discussions of any political opinions they disagree with. They strive to convince themselves and their followers that they are dealing in questions of pure, refined "rationality" with no political content. However, the version of "rationality" they preach is expressly politicised.
  • TeMPOraL 11y ago
    I've seen this type of writing before. It's a kind of twisted pseudo-criticism you write against a group you dislike. You can compose stuff like this against any group. It sounds believable from the outside, especially if you start sceptical to begin with. But take a closer look - it's actually full of ad-hominems, cherry-picking facts and presenting them in worst light possible. I've been a part of several groups that were targeted by such prose - first the religious group I grew up in, that is a minority in my country; then the school I went to. My university year used (a very lite version of) such criticism against another, so I've seen it from the other side as well. Hell, people write shit like this about HN! It's hard to defend against such criticism. You'll get boggled down in refuting specific accusations, but this is something you can never win. The only winning move is to ignore it completely. Personally, I shun and shame people who write such stuff, regardless of whether I agree or disagree with their victims. Dishonesty is a poison that destroys societies. TL;DR: this text is harmful, malicious bullshit. If it at least offended people with style, there would be something to save it.
    • RodericDay 11y ago
      I know you're a huge advocate for Lesswrong, but not "everybody" or "anyone" has quotes ripe for picking like Yudkowsky. Stephen Bond is not just throwing some opinion out there, he's backing it up with first-hand sources: Yudkowsky on his simplified views of why race gets brought up: > "Race adds extra controversy to everything; in that sense, it's obvious what difference skin colour makes politically". > "Group injustice has no existence apart from injustice to individuals. It's individuals who have brains to experience suffering. It's individuals who deserve, and often don't get, a fair chance at life. [...] Skin colour has nothing to do with it, nothing at all." Yudkowsky on our current societal structure, adulating the people who give him funding: > One of the major surprises I received when I moved out of childhood into the real world, was the degree to which the world is stratified by genuine competence. Yudkowsky writing short stories about a society where rape is legal, leaving himself ample room for plausible deniability, but putting it up "for debate": >> "No, us. The ones who remembered the ancient world. Back then we still had our hands on a large share of the capital and tremendous influence in the grant committees. When our children legalized rape, we thought that the Future had gone wrong." >> Akon's mouth hung open. "You were that prude?" (https://web.archive.org/web/20131206200429/http://lesswrong.com/lw/y8/interlude_with_the_confessor_48/ https://web.archive.org/web/20131206200429/http://lesswrong....) I think some well-meaning LessWrongers get caught in the crossfire, but I think the essay makes a very well grounded case for the blindspots "rationalists" have towards politics that suit the identity of people like Yudkowski.
• SFjulie1 11y ago
  http://elis.dvo.ru/~lab_11/bib/hershkowitz-nadal.pdf http://elis.dvo.ru/~lab_11/bib/hershkowitz-nadal.pdf I found a thesis on the topic. Seems the upper bound of estimation when using bayesian probability for estimating parameters with data gets overestimated when the size of the data grows. It does not makes sense to me, can someone explain this to me as if I was an idiot? PS one of the searcher in his lab used to say: if you try to find to hard a physical law, you might eventually find it. (quote from the thesis below) Most of the results concerning the behavior of the mutual information, observed for this particular family, are ‘‘universal,’’ in that they will be qualitatively the same for any problem that can be formulated as either a parameter estimation task or a neural coding task. .... Besides the asymptotic regime p large, N arbitrary, we have also considered the case of large N at any given value of a5p/N. In this regime we have both replica calculations and exact bounds, in particular, an upper bound for the class information and explicit upper and lower bounds for the mutual information obtained with the techniques of [7]. The results suggest that the replica symmetry ansatz give the correct solution. The lower bound is then quite good whereas the upper bound overestimate the mutual information by a factor that keeps increasing with the data size.
• cronjobber 11y ago
  Here's a proposal: Bayesian scientists shouldn't select their own prior. Instead publish how your results would update any prior, including the one picked by me, the reader. I certainly haven't thought this through, but maybe this would make science more modular: combine the updates from M studies and calculate the new, combined update. Statisticians, does this work?
  • XFrequentist 11y ago
    Yes - Laplace originally proposed this, it's a good approach (and incidentally the basis for Bayesian meta-analysis). Google "skeptical prior" for more.
  • Fomite 11y ago
    I had a professor in graduate school who suggested exactly this - each study should conduct a meta-analysis of all previous studies on the subject, use that estimate as their prior, update, and publish for the next study... The problem is, having tried it, this is much more difficult to do in practice.
  • arcanus 11y ago
    This does happen. I've had paper reviewers request results with an appropriately selected uninformative prior: https://en.wikipedia.org/wiki/Jeffreys_prior https://en.wikipedia.org/wiki/Jeffreys_prior Typically, if you are practitioner in the field, it is not too difficult to identify instances where the result is highly dependent on the choice of prior.
  • abecedarius 11y ago
    There was a lesswrong post on that back before they moved to lesswrong.com: http://www.overcomingbias.com/2009/02/share-likelihood-ratios-not-posterior-beliefs.html http://www.overcomingbias.com/2009/02/share-likelihood-ratio...
  • conjectures 11y ago
    On the second point. It's called meta-analysis and is an important area of research. However it's very far from operating automagically and requires substantial manual input. On the first point this is equivalent to contracting a builder to build a house in any style. Choosing a prior isn't always just about picking a wide band around 3 or a narrow band around 2. This is like the builder offering a choice of curtain colours from a swatch. Making a commitment to implement any prior could be totally. Like the overcommitted builder being asked to reconstruct R'lyeh.
• such_a_casual 11y ago
  So let me get this straight, statistics can be misused? I almost stopped reading as soon as I saw the image from the big bang theory. (That show presents nerds the way other people want to see them. Not the way nerds actually are.) It appears that the author of this article is just a journalist: http://www.johnhorgan.org/ http://www.johnhorgan.org/ I don't see anything in there about him ever being a scientist, a statistician, or anyone who would actually use this theorem. In fact, he even went to school for journalism. I really don't understand where this author is coming from when he says things like, "I conveniently decided that Bayes was a passing fad." Why should we care about his opinion on the matter? Is he reporting to us what Bay Theorem is and its significance, or is he giving us his uniformed opinion on the matter?
• sago 11y ago
  The current fashion for BT really bugs me. BT inverts conditional probabilities. If you can estimate P(E), P(H) and P(E|H) better than P(H|E) it will give you a better result. It is one of many probability identities. But someone it has become 'the one', as if, say P(H|E) = P(H&E)/P(E) isn't much use, but put two of those together: world changing. I've seen so much crap come out of this fad. My particular favourite is in theology. William Lane Craig has demonstrated that Jesus raised from the dead, to a high probability. Richard Carrier has shown that there was no historical Jesus. Funny how few people ever run BT and find it contradicts their views. I think part of the problem comes from a lack of understanding of the difference between frequentist and bayesian interpretations of probability. I've yet to see these folks show BT working in anything but frequentist data. And then they'll switch and use it to demonstrate why their Bayesian situation is correct.
  • logfromblammo 11y ago
    Bayes does more than just invert the conditional. Of P(x), P(y), P(x|y), and P(y|x), if you know any three, then Bayes will give you the fourth. It's just an equation. Garbage inputs will yield garbage outputs. In the realm of theology, it is of no more use than Pascal's Wager as an expected value calculation. All the input values are made up, so the output value is equally fabricated. If you're using it on real, verifiable statistics, such as verified spam in an e-mail corpus, you can use Bayes to make a classifier to automatically identify spam to a high degree of accuracy. But if you are estimating for any of the three numbers you need to know, the fourth that you calculate will also be suspect.
  • rcthompson 11y ago
    Of course two people can come to different conclusions based on a Bayesian analysis of the same question, if their priors are different. The benefit of Bayes' Theorem is to make explicit the dependency of the result on the prior.
• logfromblammo 11y ago
  They were doing great until they got up to the re-test, claiming that the second positive gives you 99% certainty you have cancer. That only works if the second test is completely independent from the first. If you repeat the first test a second time, only for those who get a positive result on the first, the same condition that caused a false positive on the first can cause a false positive on the second. In reality, a cheap blood or urine test is likely to be followed by a more expensive test on a second portion of the same sample, then by an even more expensive tissue biopsy. Redoing an identical test only reduces random errors. It does not address the diagnostic bias of the test itself. For instance, a pregnancy test detects hCG, from the placenta of a developing fetus. A man with various types of cancer or liver disease may also produce hCG, and can therefore produce a false positive for every test of that type, no matter how many repetitions. This does not give him greater confidence that he is pregnant! Understanding Bayes also requires an understanding of event independence! For truly independent events, P(x|y) = P(x) and P(y|x) = P(y).
• jedberg 11y ago
  > Cognitive scientists conjecture that our brains incorporate Bayesian algorithms as they perceive, deliberate, decide. As a Cognitive scientist myself, this amused me. The reason being because in the 1950s, Cognitive scientists thought that the brain worked like telephone switching equipment. Basically, we fit our current model of cognition to the most popular model of computing at the time. Looks like the trend hasn't stopped (although to be fair we we were talking about the Baysian model of cognition 20 years ago, so at least that one lasted a while).
• philh 11y ago
  I feel like there are multiple questions getting conflated: * Does Bayes help honest enquirers to find the truth? (Relative to what? At what skill level?) * Does Bayes help bullshitters to hide the truth? (Relative to what? At what skill level of bullshitter and mark?) Re the second, I think it's difficult to use Bayes to bullshit someone who understands Bayes as well as you do.
• darawk 11y ago
  > If you get tested again, you can reduce your uncertainty I've always been bothered by statements like this about medical tests. This assumes that false positives are statistically independent. But isn't it more likely in general that false positives would be highly correlated in individuals, test administrators, or labs? E.g. If the same person takes the same test from the same doctor and sends it to the same lab, it seems extremely unlikely that the results will be independent. And to me at least, it seems highly likely that a false positive will correlate with some aspect of the individual's biological (e.g. some similar substance in the blood to what's being tested), and as such even using a different doctor/lab would not be all that likely to ameliorate this issue.
  • snowwrestler 11y ago
    These are questions that themselves would need to be answered with scientific research. In the absence of real empirical data, I'd be uncomfortable with saying it is "more likely in general that false positives would be highly correlated." Without data, we don't know how likely it is.
    • mhb 11y ago
      Which is why the author of the article needs to mention that the second test is independent of the first.
  • sophacles 11y ago
    I think this is a good question - I hope someone who actually knows about this stuff chimes in with some answers. In the mean time as a thought experiment, going with a blood test example: * levels of the compound being tested may rise or fall naturally, and the test checks for a certain concentration or above. * the patient may have the condition, but not have had it long enough for markers to have risen to "trigger" levels. * The patient may not exactly follow pre-draw instructions on eating etc, skewing results * in the case of false negative: the person's immune response may be temporarily suppressing the marker * in the case of something like cell counts - this sample could just be a random local variation * The tech or doctor or whoever could randomly make a mistake on one sample, but not on each sample. And so on. My devil's advocate point here is that the patient, doctor, lab, and so on are not deterministic code - there are a lot of random inputs in the entire process chain.
• lqdc13 11y ago
  I just love how some guy, who by his own admission read a little Wikipedia on the topic, is critiquing a statistical method. It would make for a much stronger argument if he actually showed some numbers where people are getting the priors wrong. That is, how often people get the priors wrong and the probability of mistake if they used a different strategy more commonly used in the field.
• haberman 11y ago
  I am only just learning about this stuff, but there are several things in this article that seem incorrectly explained. Conceptual clarity is paramount to me, so it drove me a little crazy! > Bayes’ theorem is a method for calculating the validity of beliefs (hypotheses, claims, propositions) based on the best available evidence (observations, data, information). Bayes theorem is a statement about probability, not "validity." This description makes it sound like Bayes theorem is a function BT(belief, evidence) = validity of belief. But it's not like that at all. Probability is a way of measuring uncertainty. Things are uncertain for two main reasons: either we can't observe them directly ("do I have this disease or not?") or they haven't happened yet ("what side will this coin flip land on?"). (If you believe in a deterministic universe, the second is just a special case of the first.) The "beliefs" (aka priors/posteriors) in Bayes theorem are statements of probability. To use the article's example, if it is claimed that 1% of the population has a certain disease, your "belief" or "prior" is that P(I have the disease) = 0.01. The article seems to get confused and think that the "belief" here is "I have the disease." Bayes theorem doesn't tell you about "the probability that a belief is true" like the article says, the belief is a probability. It also doesn't tell you if your belief is "valid." Bayes theorem takes your existing belief about the probability of something and gives you a new probability that incorporates some evidence you observed.
• maddilove 11y ago
  I thank you gays for fucking my cat shity she love some big black dick
• tzs 11y ago
  Referring to an appearance of Bayes' theorem on Sheldon's whiteboard: > Bayes’ theorem has become so popular that it even made a guest appearance on the hit CBS show Big Bang Theory That's not a sign that it has become popular, any more than the appearance of the standard human pedigree notation used by genetic counselors on the same whiteboard indicates that standard human pedigree notation has become popular among the general public. It's simply a sign that BBT takes care to make the whiteboards generally scientifically sensible. They have a UCLA physicist [1] consultant who works with the producers, writers, prop people, set decorators and others to try to make things scientifically accurate. In this particular episode, both Bayes' theorem and the genetic information are there because Sheldon was trying to figure out his chances of surviving until technology reaches the point that he can transfer his mind to an AI and what he would have to do to improve those chances. So both those things were on the whiteboard because they are things that would be perfectly reasonable to find on the whiteboard of someone doing that. They consulting physicist had a blog [2] where he covered most episodes and talked about the whiteboards and other scientific content. Here is the entry for the aforementioned episode [3]. [1] http://bigbangtheory.wikia.com/wiki/David_Saltzberg http://bigbangtheory.wikia.com/wiki/David_Saltzberg [2] https://thebigblogtheory.wordpress.com https://thebigblogtheory.wordpress.com [3] https://thebigblogtheory.wordpress.com/2010/09/30/s04e02-the-cruciferous-vegetable-amplification/ https://thebigblogtheory.wordpress.com/2010/09/30/s04e02-the...
  • n4r9 11y ago
    P(popular | appeared on BBT) < 1
• Houshalter 11y ago
  The fundamental insight of bayesian reasoning is that prior probabilities can matter a lot in some situations, even when the evidence seems fairly strong. Look at scientific papers. Usually there is something called a p-value attached to it, which is usually around 0.05. This means there is only a 5% chance that the same result could have been produced by random chance. How strong is that evidence, really? What if you started with a prior probability of 1 in 1,000 that it's actually true. That is, before you saw the paper, you would estimate there's only a 0.001% chance it's true. Out of 1,000 similar studies, 49 will be false and yet return a <0.05 p value. Only 1 will publish actually true results.
• SandroG 11y ago
  Bayes in action: The author of this article had a set of prior beliefs about the Bayes's Theorem, which evolved after he reviewed more evidence.
• ssanders82 11y ago
  Since we're on the subject, can anyone point me in the direction of how to account for correlated inputs? Without adjusting them, it can possibly give nonsensical probabilities (>1) but my math isn't good enough to decipher the few academic texts I've seen regarding this situation. A (very simple) example: I trade stocks. My starting point is that I think a stock has a 50% chance of rising next year. Then I want to do a Bayesian iteration with the stock's P/E ratio based on historical data for stocks with similar P/E ratios. Then I want to also incorporate the P/E ratio of the industry the stock is in. Obviously these two inputs are correlated and if you have enough correlated variables, the whole thing breaks down because the simple theorem only works if all the inputs are independent of each other.
• tonybeltramelli 11y ago
  Same topic but differently presented by Quanta Magazine https://www.quantamagazine.org/20151216-physicists-and-philosophers-debate-the-boundaries-of-science/ https://www.quantamagazine.org/20151216-physicists-and-philo...
• deleted 11y ago
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• auvrw 11y ago
  modal logic is an equally valid way to get at a lot of the quandaries associated with bayesian ideas. bayes' thm by itself is totally not a "big deal." the idea that every probability has an associated prior, even if it's not explicitly written in the notation (so think "prior of prior") is an interesting attempt to cope with uncertainty in a rigorous way. i do agree that in some places in the sciences, while "the numbers don't lie," the stats can be misleading. still, i can understand why it's useful to make some statements with statistics in order to quickly arrive at some first-order approximations.
• pdkl95 11y ago
  > Bayesians claim that their methods can help scientists overcome confirmation bias The claim isn't that Bayesianism somehow prevents biases. Using a Bayesian approach is important in science because frequentism answers the wrong question[1]. Many scientists operate as if the confidence interval is a Bayesian credible region, but it demonstrably is not ... I think the reason this mistake is so common is that in many simple cases ... the confidence interval and the credible region happen to coincide. Frequentism, in this case, correctly answers the question you ask, but only because of the happy accident that Bayesianism gives the same result for that problem. [1] http://jakevdp.github.io/blog/2014/06/12/frequentism-and-bayesianism-3-confidence-credibility/ http://jakevdp.github.io/blog/2014/06/12/frequentism-and-bay... (I'm referencing part 3 for the discussion of why frequentism is inappropriate in science, but I recommend reading the series from the beginning)
• yzmtf2008 11y ago
  To me, the single most easy to understand form of Bayes' Theorem is: P(A|B) * P(B) = P(A∩B) = P(B|A) * P(A) An intuitive explanation to the equation: The (possibility of events A and B both happening) equals the (possibility of A happening when B is happening) * the (possibility of B happening), which equals to the (possibility of B happening when A is happening) * the (possibility of A happening). Combining that with a Venn diagram: {A (A}∩{B) B} Since P(B|A) means the chance of event B happening when A is happening, in this case which indicates that both A and B are happening, it's really just the area of (A∩B) divided by (A), which can be translated to (P(A|B) * P(B)) / P(A).
• sixdimensional 11y ago
  I find Bayesian subjective probability even more interesting. It has been successfully applied in unique situations, such as during the cold war, by a panel of "experts" in the US, to locate Russian test fired rockets in the ocean.
• deleted 11y ago
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• mjgeddes 11y ago
  Bayesianism is a 'grand unified theory of reasoning' that all of science be should be based on assigning (and updating) probabilities for a list of possible outcomes; the probabilities are supposed to indicate your subjective degree of confidence that a given outcome will occur. Yudkowsky's 'Less Wrong' group of 'rationality' followers, aimed to try to force-fit all of science into the Bayesian framework. And of course it doesn't work at all. I think Andrew Gelman's criticisms are right on the mark. Probability theory was designed for reasoning about external observations - sensory data. (for example, "a coin has a 50% chance of coming up heads"). In terms of predicting things in the external world, it works very well. Where it breaks down is when you try to apply it to reasoning about your own internal thought processes. It was never intended to do this. As Gelman correctly points out, it is simply invalid to try to assign probabilities to mathematical statements or theories, for instance. You see 'Less Wrong' followers wasting years of their lives engaging in the most unbelievable and ludicrous intellectual contortions to try to force-fit all of science into Bayesianism. Go to the 'Less Wrong' blog and you can read reams and reams of these massively complicated and contorted ideas, including such hilarious nonsense as 'Updateless decision theory' and 'Timeless decision theory'. --- David Deutsch in his superb books, 'The Fabric Of Theory' and 'The Beginning Of Infinity', argued for a different theory of reasoning than Bayesianism. Deutsch (correctly in my view) pointed out that real science is not based on probabilistic predictions, but on explanations. So real science is better thought of as the growth or integration of knowledge, rather than probability calculations. In terms of dealing with internal models or hypothesis, I think the correct solution is not to assign probabilities, but rather to assign a 'conceptual coherence' value, so for instance rather than say 'outcome x has probability y' (where x is a hypothesis) you should say 'concept x has conceptual coherence value y' Conceptual coherence is the degree with which a hypothesis is integrated with the rest of your world-model, and I think it accurately captures in mathematical terms the ideas that Deutsch was trying express. Probabilities should be viewed as just special cases of conceptual coherence (in the cases of outcomes where you are dealing with external observations or sensory data, Bayesianism is perfectly valid). Then all of the problems with probability go away, and none of the massively complicated theories expounded on 'Less Wrong' are necessary ;)