# 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.

comment: (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.

# 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:

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!