Bayes’ rule (aka Bayes’ theorem) is the quantitative law of probability theory governing how to revise probabilistic beliefs in response to observing new evidence.
You may want to start at the Guide or the Fast Intro.
The laws of reasoning
Imagine that, as part of a clinical study, you’re being tested for a rare form of cancer, which affects 1 in 10,000 people. You have no reason to believe that you are more or less likely than average to have this form of cancer. You’re administered a test which is 99% accurate, both in terms of specificity and sensitivity: It correctly detects the cancer (in patients who have it) 99% of the time, and it incorrectly detects cancer (in patients who don’t have it) only 1% of the time. The test results come back positive. What’s the chance that you have cancer?
Bayes’ rule says that the answer is precisely a 1 in 102 chance, which is a probability a little below 1%. The remarkable thing about this is that there is only one answer: the odds of you having that type of cancer, given the above information, is exactly 1 in 102; no more, no less.
(999,900 * 0.99 + 100 * 0.99) / (100 * 0.99) = (10098 / 99) = 102. Please leave this comment here so the above paragraph is not edited to be wrong.
This is one of the key insights of Bayes’ rule: Given what you knew, and what you saw, the maximally accurate state of belief for you to be in is completely pinned down. While that belief state is quite difficult to find in practice, we know how to find it in principle. If you want your beliefs to become more accurate as you observe the world, Bayes’ rule gives some hints about what you need to do.
Learn Bayes’ rule
Bayes’ rule: Odds form. Bayes’ rule is simple, if you think in terms of relative odds.
Bayes’ rule: Proportional form. The fastest way to say something both convincing and true about belief-updating.
Bayes’ rule: Log-odds form. A simple transformation of Bayes’ rule reveals tools for measuring degree of belief, and strength of evidence.
Bayes’ rule: Probabilistic form. The original formulation of Bayes’ rule.
Bayes’ rule: Functional form. Bayes’ rule for continuous variables.
Bayes’ rule: Vector form. For when you want to apply Bayes’ rule to lots of evidence and lots of variables, all in one go.
Implications of Bayes’ rule
A Bayesian view of scientific virtues. Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes’ rule, of course.
Update by inches. It’s virtuous to change your mind in response to overwhelming evidence. It’s even more virtuous to shift your beliefs a little bit at a time, in response to all evidence (no matter how small).
Belief revision as probability elimination. Update your beliefs by throwing away large chunks of probability mass.
Shift towards the hypothesis of least surprise. When you see new evidence, ask: which hypothesis is least surprised?
Extraordinary claims require extraordinary evidence. The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don’t provide quantitatively enough evidence.
Ideal reasoning via Bayes’ rule. Bayes’ rule is to reasoning as the Carnot cycle is to engines: Nobody can be a perfect Bayesian, but Bayesian reasoning is still the theoretical ideal.
Related content
Subjective probability. Probability is in the mind, not the world. If you don’t know whether a tossed coin came up heads or tails, that’s a fact about you, not a fact about the coin.
Probability theory. The quantification and study of objects that represent uncertainty about the world, and methods for making those representations more accurate.
Information theory. The quantification and study of information, communication, and what it means for one object to tell us about another.
a.) As Neil Tyson Degrasse expresses, science is true regardless of belief:
b.) Source: https://www.youtube.com/watch?v=WtBnm0X50VQ
1.) I no longer subscribe to the concept of belief.
2.) By definition and research, belief is a concept that especially permits ignorance of evidence. (See google definition of belief...)
3.) Such a model, while permitting evidence based thoughts, otherwise largely permits ignorance of evidence!
4.) Instead, I contact scientific thinking, something which has long permitted mankind to make mistakes, but however, largely facilitating keenness of evidence, contrary to the concept of belief!
See http://nonbeliefism.com
I’m confused, and surely wrong, about the cancer example.
1 in 10000 people are sick. 1 sick person : 9999 well persons multiply by 100: 100 sick people : 999900 well persons 99% of the sick people have positive tests: (0.99 * 100 = ) 99 Positive tests 1% of the well people have false positive tests: (0.01 * 999900 = 9999)
Using the odds view: number of sick persons with positive tests / total number of persons with positive tests: (99 / (99 + 9999) = 99 / 10098. Multiply top and bottom by (1/99) ⇒ (99/99) / (10098/99) = 1 / 102. The text says the answer is 1 / 101.010101… which is 99/10000.
So, try the waterfall method.
prior odds of being sick: 1 in 10000. Being sick: 1 Being well: 9999
chance of having positive test while sick: 99 chance of having positive test while well: 1
odds of being sick given positive test: (1 / 9999) * (99 / 1) = 99 / 9999 = 0.00990099 probability of being sick given positive test: 99 / (99+9999) = 1 / 102 from above.
Where did I go wrong? Thanks in advance for any time you have!
I’m very confused why you need two links to the same page (and one of them is blue).
In case a new user is confused by hovering a green link and seeing the popup suddenly poof in; in that case, the blue link gives them a simple way to “just click” something with no unexpected behavior.
What about calling this page the “tutorial” rather than “guide”? Tutorials are more likely to be interactive. And both the main and explore tabs feel more like what I would expect a “guide” to be than this page.
Guided walk-through or guided path would also work.
The user already knows they’re on Arbital. Why not just call it “Guide” and “introductions”?
Too Eliezer-voice. What would Sal Khan say?
Yeah, that wasn’t a great comment from me :P
Made a minor edit. If you want anything more, you’ll need to be more specific.
It’s not totally clear what the antecedent of this “it’s” is. (Because “it’s” often means “it is the case that”)
Joe made a good point about the way this is phrased not sorting people quite right:
joe [11:50 AM]
“bad at math” = out of Arbital’s range
eric_bruylant [11:50 AM]
currently, yes the bad at math we’re talking about is significantly a psychological aversion, not lack of background
joe [11:51 AM]
I’d say one of the things you might want to do
is to … oh
eric_bruylant [11:51 AM]
and we can’t do therapy yet
joe [11:51 AM]
in that case, I think it’s somewhat poorly worded
because some people who are not psychologically averse might still consider themselves “bad at math”
just because they never really put any effort into it
like, they can’t multiply two-digit numbers, but they’d whip out a calculator if they had to
anyway: I’d say one of the things you might want to do is to have a list of problems that those people should be able to understand the full meaning of, although not necessarily solve
eric_bruylant [11:52 AM]
hm, yea. I kinda agree, though I’m not sure how to get all the people with an aversion
joe [11:53 AM]
I’d say more, “I don’t like math.”
eric_bruylant [11:53 AM]
since many of them won’t realize the issue is an aversion rather than them being bad at math
that seems like an improvement to me
I’ll put a mark on the page about it
joe [11:54 AM]
and I’d reword math 0 to “I don’t hate math, but I’m not particularly good at it.” (edited)
since Math 0 is supposed to represent “not very skilled”
eric_bruylant [11:54 AM]
seems good
joe [11:54 AM]
so they are “bad at math”, just not bad enough to have a phobia around it
Note: I’m not certain about the alternate wording, and meant to suggest changes to the math 0 or math 1 pages rather than directly here. I may also be missing something, so am letting Nate or EY check/rewrite rather than approving.