# Bayes' rule

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.

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.

# Other articles and introductions

Children:

- Bayes' rule examples
Interesting problems solvable by Bayes’ rule

- Waterfall diagram
Visualizing Bayes’ rule as the mixing of probability streams.

- Bayes' rule: Odds form
The simplest and most easily understandable form of Bayes’ rule uses relative odds.

- Proof of Bayes' rule
Proofs of Bayes’ rule, with graphics

- Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.

- Probability notation for Bayes' rule
The probability notation used in Bayesian reasoning

- Bayes' rule: Vector form
For when you want to apply Bayes’ rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)

- Bayes' rule: Log-odds form
A simple transformation of Bayes’ rule reveals tools for measuring degree of belief, and strength of evidence.

- Bayes' rule: Functional form
Bayes’ rule for to continuous variables.

- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.

- Bayes' rule: Guide
The Arbital guide to Bayes’ rule

- Path: Multiple angles on Bayes's Rule
A learning-path placeholder page for learning multiple angles on Bayes’s Rule.

- Shift towards the hypothesis of least surprise
When you see new evidence, ask: which hypothesis is

*least surprised?* - Bayes' rule: Definition
- Bayes' rule: Probability form
The original formulation of Bayes’ rule.

- Frequency diagram
Visualizing Bayes’ rule by manipulating frequencies in large populations

- Bayes' rule: Beginner's guide
Beginner’s guide to learning about Bayes’ rule.

- High-speed intro to Bayes's rule
A high-speed introduction to Bayes’s Rule on one page, for the impatient and mathematically adept.

Parents:

- Bayesian reasoning
A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.

The user already knows they’re on Arbital. Why not just call it “Guide” and “introductions”?

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!