# Bayes' rule: Guide

Bayes’ rule or Bayes’ theorem is the law of probability governing the strength of evidence—the rule saying how much to revise our probabilities (change our minds) when we learn a new fact or observe new evidence.

You may want to learn about Bayes’ rule if you are:

• A professional who uses statistics, such as a scientist or doctor;

• A computer programmer working in machine learning;

• A human being.

As Philip Tetlock found when studying “superforecasters”, people who were especially good at predicting future events:

The superforecasters are a numerate bunch: many know about Bayes’ theorem and could deploy it if they felt it was worth the trouble. But they rarely crunch the numbers so explicitly. What matters far more to the superforecasters than Bayes’ theorem is Bayes’ core insight of gradually getting closer to the truth by constantly updating in proportion to the weight of the evidence.

— Philip Tetlock and Dan Gardner, Superforecasting

## Learning Bayes’ rule

This guide to Bayes’ rule uses Arbital’s technology to allow for multiple flavors of introduction. They vary by technical level, speed, and topics covered. After you pick your path, remember that you can still switch between pages, in particular by using the “Say what?” and “Go faster” buttons.

Which case fits you best? a: I want to have a basic theoretical and practical understanding of the Bayes’ rule. -wants: [62d,Bayes’ Rule and its implications b: I can easily read algebra and don’t mind the explanation moving at a fast pace. Just give me the basics, quick! wants: Bayes’ Rule and its different forms -wants: Bayes’ Rule and its implications c: I want the basics, but I’m also interested in reading more about the theoretical implications and the reasons why Bayes’ rule is considered so important. wants: Bayes’ Rule and its implications -wants: Bayes’ Rule and its different forms d: I’d like to read everything! I want to have a deep theoretical and practical understanding of the Bayes’ rule. wants: Bayes’ Rule and its different forms,Bayes’ Rule and its implications ]

wants-requisite(Bayes’ Rule and its different forms): %%wants-requisite(Bayes’ Rule and its implications): %box: Your path will go over all forms of Baye’s Rule, along with developing deep appreciation for its scientific usefulness. Your path will contain 12 pages:

• Frequency diagrams: A first look at Bayes

• Waterfall diagrams and relative odds

• Introduction to Bayes’ rule: Odds form

• Bayes’ rule: Proportional form

• Extraordinary claims require extraordinary evidence

• Ordinary claims require ordinary evidence

• Bayes’ rule: Log-odds form

• Shift towards the hypothesis of least surprise

• Bayes’ rule: Vector form

• Belief revision as probability elimination

• Bayes’ rule: Probability form

• Bayesian view of scientific virtues %

%start-path(Comprehensive guide to Bayes’ Rule)% %% %%!wants-requisite(Bayes’ Rule and its implications): %box: No time to waste! Let’s plunge directly into a single-page abbreviated introduction to Bayes’ rule. % %% <div>

!wants-requisite(Bayes’ Rule and its different forms): %%wants-requisite(Bayes’ Rule and its implications): %box: Your path will teach you the basic odds form of Bayes’ rule at a reasonable pace and then delve into the deep mysteries of the Bayes’ Rule! Your path will contain 8 pages:

• Frequency diagrams: A first look at Bayes

• Waterfall diagrams and relative odds

• Introduction to Bayes’ rule: Odds form

• Belief revision as probability elimination

• Extraordinary claims require extraordinary evidence

• Ordinary claims require ordinary evidence

• Shift towards the hypothesis of least surprise

• Bayesian view of scientific virtues %

%start-path(Bayes’ Rule and its implications)% %%

%%!wants-requisite(Bayes’ Rule and its implications): %box: Your path will teach you the basic odds form of Bayes’ rule at a reasonable pace. It will contain 3 pages:

• Frequency diagrams: A first look at Bayes

• Waterfall diagrams and relative odds

• Introduction to Bayes’ rule: Odds form %

%start-path(Introduction to Bayes’ Rule odds form)% %% <div>

Children:

Parents:

• Bayes' rule

Bayes’ rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.

• Bayesian reasoning

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

• Joe made a good point about the way this is phrased not sorting people quite right:

joe AM
“bad at math” = out of Arbital’s range

eric_bruylant AM
currently, yes the bad at math we’re talking about is significantly a psychological aversion, not lack of background

joe AM
I’d say one of the things you might want to do

is to … oh

eric_bruylant AM
and we can’t do therapy yet

joe 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 AM
hm, yea. I kinda agree, though I’m not sure how to get all the people with an aversion

joe AM
I’d say more, “I don’t like math.”

eric_bruylant 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 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 AM
seems good

joe 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.

• 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.

• 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.

• a.) As Neil Tyson Degrasse expresses, science is true regardless of belief:

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!