This seems like a nice explanation of Bayes rule and of some parts of statistics that are often not well-presented and lead to confusion. But Bayesian inference is more than just Bayes rule. I dont see any discussion of the idea of probability as subjective belief in there, which is the core of a Bayesian view of probability. Bayes rule is just the machinery for updating beliefs, and as important as that is, frequentists use it too. Bayes rule only gains the huge significance when you have committed to the idea of probability as expressing degree of belief, and which makes you free to put probability distributions over parameters

Of course the best way to align your aim and intended aim is feedback. If you have an audience at your target level you can talk to, you can find out via practice what works with them. This is an advantages teachers often have.

Very true, Robin.

Anonreader: Agreed, if I was rewriting it today I'd call it "An Intuitive Explanation of Bayes's Rule". Again, I overshot the mark - I thought that from understanding a single Bayesian update, the Bayesian Way would automatically follow.

when you have committed to the idea of probability as expressing degree of belief

Do you folks think that probability is properly understood only as an expression of degree of belief (and that, say, the other ideas about it are confused, circular, or whatever), or do you recognize the others while primarily being interested in degree of belief?

For my part, while I recognize the usefulness of employing the probability calculus to deal with degree of belief, I don't think it's the only use of the probability calculus. For example, I don't think we gain much by insisting on an interpretation of Buffon's needle in terms of degree of belief. I think it's comprehensible in the context of something like Buffon's needle to derive probability from geometric symmetries without routing through the notion of probability as degree of belief.

In teaching or lecturing, I notice my own tendency, which I have since tried to correct, to avoid going over anything I had known for a long time and only mention the latest tidbit I'd learned. I attributed that to my own desire to avoid boredom and seek out important new insights. Also, through the well-known cognitive bias of projecting one's own mental state on others, I was subconsciously trying not to bore the students.

Oops! (that last post was not intended to test anyone's psychic ability) The problem of Bayesian reasoning is in the setting of prior probability. There is some self correction built in, so it is a better system than most (or any other if you prefer), but a particular problem raises its ugly that is relevant to overcoming bias. Suppose I want to discuss a particular phenomena or idea with a Bayesian. Suppose this Bayesian has set the prior probability of this phenomena or idea at zero. What would be the proper gradient to approach the subject in such a case?

that last post

I found it to be a model of brevity.

Douglas, if your interlocutor is a really consistent Bayesian and has a probability estimate of exactly zero for whatever-it-is then I advise you to talk to someone else instead. If (as is more commonly the case) their prior is merely very small, then what you need to do is to present them with evidence that brings their posterior probability high enough for them to think it worthy of further discussion.

This is in fact exactly the problem you face when discussing anything with anyone (Bayesian or not) who finds your position or some part of it wildly improbable. At least a Bayesian is (in principle) committed to taking appropriate note of evidence.

Constant, I think even the strictest subjective Bayesian should be happy to agree that in the presence of suitable symmetries the only sensible prior may be determined by those symmetries, and that in such cases you can save some mental effort by just talking about the symmetries. Just as you can talk about the axioms of number theory or logic or whatever even if you think they're really empirical generalizations rather than descriptions of Platonic Mathematical Reality, and usually when doing mathematics it's appropriate to do so.

the only sensible prior may be determined by those symmetries, and that in such cases you can save some mental effort by just talking about the symmetries

Okay, but that's really just treating other views of probability as convenient fictions, which isn't quite what I was hoping for.

I do not believe that the subjective interpretation of probability is a good match for the application of probability to the analysis of events that have already occurred. Conduct an experiment, observe (say) a normal distribution in some variable. That we subjectively expected a normal distribution prior to the run of the experiment, or that we now expect it in future runs, is all well and good, but it is not our expectation that accounts for the actual appearance of the normal distribution itself. If anything accounts for the actual appearance of a normal distribution, it is some property of the experimental setup, rather than an expectation of ours.

I am not denying that you can go ahead and talk about our degree of belief and the unique rational way to update that degree of belief. I think that's a perfectly legitimate topic. What I find unconvincing is the idea is that that's all that we can profitably apply the probability calculus to. I did google "bayesian quantum" and didn't find anything that answered my concern, though I did find assertions of the very position that I have a problem with. I don't deny that you can approach quantum mechanics from a bayesian perspective - obviously your beliefs can be informed by quantum mechanics and you can have beliefs about the results of experiments and so on; I simply think it leaves something out, because at the end we have not only our own belief about the result of the next quantum experiment to arrive at rationally based on our priors in combination with experiments we have observed, but also actual observed frequencies of past quantum experiments, and these are out there, they are not subjective, and the mathematical theory of probability strongly recommends itself as a tool in the analysis of the observed frequencies.

You thought elementary school students wouldn't be completely overwhelmed by a question like this?:

"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?

What do you think the answer is? If you haven't encountered this kind of problem before, please take a moment to come up with your own answer before continuing."

Seriously, even down to the use of big words like "mammographies".

Douglas writes: Suppose I want to discuss a particular phenomena or idea with a Bayesian. Suppose this Bayesian has set the prior probability of this phenomena or idea at zero. What would be the proper gradient to approach the subject in such a case?

I would ask them for their records or proof. If one is a consistent Bayesian who expects to model reality with any accuracy, the only probabilities it makes sense to set as zero or one are empirical facts specificied at a particular point in space-time (such as: "I made X observation of Y on Z equipment at W time") or statements within a formal logical system (which are dependent on assumptions and can be proved from those assumptions).

Even those kinds of statements are probably not legitimate candidates for zero/one probability, since there is always some probability, however minuscule that we have misremembered, misconstrued the evidence or missed a flaw in our proof. But I believe these are the only kinds of statements which can, even in principle have probabilities of zero or 1.

All other statements run up against possibilities for error that seem (at least to my understanding) to be embedded in the very nature of reality.

I took a glance at http://www.english.bham.ac.uk/who/myversion.htm ...

Wow, that's dense writing. I could follow it, but it took effort and I didn't care to put the kind of energy into studying it as one does with college textbooks.

The English link didn't seem particularly bad to me - its flaw was that it was deliberately too difficult for its subject matter, not that it was difficult in an absolute sense.

Josh, I don't think I'd have had any trouble following that problem at age 7, which is when I was taught to solve systems of equations.

Self-anchoring. I adjusted because I knew other grade-schoolers weren't as good at math, but, of course, I vastly underadjusted. I really had absolutely no concept of what it meant to not be good at math until a couple of years ago, and I probably still have no concept of it.

Constant,

I can't speak for other Bayesians, but I prefer to use the idea of Bayesian probabilities encode "states of information" as opposed to "degrees of belief". To me, overcoming bias means making sure your beliefs reflect the actual information available to you, so I prefer to use a phrase which directs attention to that information immediately. This isn't my idea; it's one of the key ideas put forward by E. T. Jaynes. To get a sense of how this works in a geometric problem similar to the Buffon needle problem, I recommend Jaynes's paper The Well-Posed Problem.

Constant- thank-you. Micheal- I was indocrinated into Bayes many years ago. I agree that the probability 0 is not a rational one. (Who would have guessed in 1900 that the things that seemed most certain were wrong?) Or perhaps I should say that the probability 0 (or 1) is not a scientific attitude- science is based on looking to know- the assumption on probability 0 is the excuse to not look. I'm thinking that is a difference between religion and science- science has to be wrong (so that it can advance) whereas religion has to be right (to be worthy of total faith). Hmmm, I like that.

Eliezer,

You are aware that you are exceptional, aren't you? If I remember correctly, that question is more difficult than anything on the math SAT on which very few high schoolers get perfect scores. I agree with you about not understanding what it means to be "bad" at math. This makes it difficult to assess when one simple problem is more difficult than another. I may be wrong, but that question certainly feels like it would be difficult for most high schoolers let alone elementary schoolers.

Not only is that question definitely harder than anything on the SAT, but being asked to solve a problem before being taught how to do it would be nearly unheard of in any of my math classes. (I also notice "the usual precedence rules" are mentioned further down the page - I don't recall seeing those in school until 6th grade, in an advanced class.)

An important transformation of probabilities is the log odds, the logarithm of the odds ratio of the probability, log(p/(1-p)). This has the advantage that you can simply add the log-likelihood to your log-odds prior to get the revised probability. Bayes becomes addition.

In the log odds, 1 comes out as positive infinity and 0 comes out as negative infinity.

Negative and positive infinity are not real numbers, and 0 and 1 are not probabilities.

Eliezer, you are right. One word for probability one is "certainty" and a word for probability zero is "impossible". Could we say then that with the exception of a situation of perfect, complete knowledge of conditions (a situation that may or may not actually exist in reality) that the Bayesian worldveiw would not include those words?

"One word for probability one is "certainty" and a word for probability zero is "impossible"." I think you should be cautious about using these words like this, at least if you might be talking about uncountable probability spaces. Using your definitions it is certain that (say) a normally distributed variable takes a value in the real numbers but impossible for it to take any such value.

I hope this isn't unfairly pedantic to point out. I can see that one could argue that for decision making in the real world you only assign probabilities to finitely many outcomes.

Matt- I'm seldom careful. The advantages of being carefree are too numerous to list, but one of the disadvantages is that I have to admit mistakes. You are not being overly pedantic. I'd probably make a lousy Bayesian. (I wonder what the prior probability of that is?) By the way, what do you think of a decision making protocol that assumes that the data gathering is random?

This explains why so many text books are so badly written. The authors were aiming too high.

Oh bugger. I just got linked here while preparing a talk for next Tuesday.

I found a comment on HN adding more evidence to this phenomenon:

I recall reading (I'd point to the book with a link if I could remember in what book I read this) that mathematicians who occasionally write expository articles on mathematics for the general public are often told by their professional colleagues, fellow research mathematicians, "Hey, I really liked your article [name of popular article] and I got a lot out of reading it." The book claimed that if mathematicians made a conscious effort to write understandably to members of the general public, their mathematics research would have more influence on other research mathematicians. That sounds like an important experiment to try for an early-career mathematician.

"Ten Lessons I wish I Had Been Taught", Gian-Carlo Rota 1997:

4. You are more likely to be remembered by your expository work

Let us look at two examples, beginning with Hilbert. When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert's name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry and for his text on integral equations. The term "Hilbert space" was introduced by Stone and von Neumann in recognition of Hilbert's textbook on integral equations, in which the word "spectrum" was first defined at least twenty years before the discovery of quantum mechanics. Hilbert's textbook on integral equations is in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.

Similarly, Hilbert's Foundations of Geometry, the book that made Hilbert's name a household word among mathematicians, contains little original work, and reaps the harvest of the work of several geometers, such as Kohn, Schur (not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri and several other Italians. Again, Hilbert's Zahlbericht, a fundamental contribution that revolutionized the field of number theory, was originally a survey that Hilbert was commissioned to write for publication in the Bulletin of the German Mathematical Society.

William Feller is another example. Feller is remembered as the author of the most successful treatise on probability ever written. Few probabilists of our day are able to cite more than a couple of Feller's research papers; most mathematicians are not even aware that Feller had a previous life in convex geometry.

Allow me to digress with a personal reminiscence. I sometimes publish in a branch of philosophy called phenomenology. After publishing my first paper in this subject, I felt deeply hurt when, at a meeting of the Society for Phenomenology and Existential Philosophy, I was rudely told in no uncertain terms that everything I wrote in my paper was well known. This scenario occurred more than once, and I was eventually forced to reconsider my publishing standards in phenomenology.

It so happens that the fundamental treatises of phenomenology are written in thick, heavy philosophical German. Tradition demands that no examples ever be given of what one is talking about. One day I decided, not without serious misgivings, to publish a paper that was essentially an updating of some paragraphs from a book by Edmund Husserl, with a few examples added. While I was waiting for the worst at the next meeting of the Society for Phenomenology and Existential Philosophy, a prominent phenomenologist rushed towards me with a smile on his face. He was full of praise for my paper, and he strongly encouraged me to further develop the novel and original ideas presented in it.

"Ten Lessons for the Survival of a Mathematics Department"%20ten%20lessons%20for%20the%20survival%20of%20a%20mathematics%20department.pdf):

6. Write expository papers

When I was in graduate school, one of my teachers told me, "When you write a research paper, you are afraid that your result might already be known; but when you write an expository paper, you discover that nothing is known."

Not only is it good for you to write an expository paper once in a while, but such writing is essential for the survival of mathematics. Look at the most influential writings in mathematics of the last hundred years. At least half of them would have to be classified as expository. Let me put it to you in the P.R. language that you detest. It is not enough for you (or anyone) to have a good product to sell; you must package it right and advertise it properly. Otherwise you will go out of business.

Now don't tell me that you are a pure mathematician and therefore that you are above and beyond such lowly details. It is the results of pure mathematics and not of applied mathematics that are most sought-after by physicists and engineers (and soon, we hope, by biologists as well). Let us do our best to make our results available to them in a language they can understand. If we don't, they will some day no longer believe we have any new results, and they will cut off our research funds. Remember, they are the ones who control the purse strings since we mathematicians have always proven ourselves inept in all political and financial matters.

An Intuitive Explanation of Bayesian Reasoning

Correct link: https://www.yudkowsky.net/rational/bayes