Prior

Our (potentially rich or complex) state of knowledge and *propensity to learn,* before seeing the evidence, expressed as a [1zj probability function].  This is a deeper and more general concept than '[-1rm]'.  A prior probability is like guessing the chance that it will be cloudy outside, in advance of looking out a window.  The more general notion of a Bayesian prior would include probability distributions that answered the question, "*Suppose* I saw the Sun rising on 999 successive days; would I afterwards think the probability of the Sun rising on the next day was more like 1000/1001, 1/2, or 1 - 10^-6?"  In a sense, a baby can be said to have a 'prior' before it opens its eyes, and then to develop a model of the world by updating on the evidence it sees after that point.  The baby's 'prior' expresses not just its current ignorance, but the different kinds of worlds the baby would end up believing in, depending on what sensory evidence they saw over the rest of their lives.  Key subconcepts include [219 ignorance priors] and [21b inductive priors], and key examples are [21c Laplace's Rule of Succession] and [11w Solomonoff induction].


Aaron Gertler

[summary: Nothing here yet.]

Automatically generated page for "Aaron Gertler" user.
If you are the owner, click [here to edit](https://arbital.com/edit/2d8).


Introduction to Effective Altruism

Effective altruism (EA) means using evidence and reason to take actions that help others as much as possible.

The world has a lot of terrible problems, but we can't work on them all at once. And we don't agree on what the worst problems are.

Despite this, most people want to "make the world a better place". What should those people actually be doing?

Some of the questions EA sets out to answer:

 - Is it better to help people in your own country, or abroad?
 - Is it better to take a high-paying job and give money to charity, or to work for a charity directly?
 - Is it better to support an intervention with a lot of strong evidence, or to support research on a promising new intervention?

The answers to these kinds of questions will be different for each person. What you should do depends on who you are: What are your skills? What are your values? How much time do you have?

But no matter who you are, effective altruism can help you figure out how to improve the world. To learn more, follow the links below.

##More on EA##

Are you a high school student? Are you a college student?

Do you work for a nonprofit or a business with a social mission?

Are you religious?

Which of the following issues most concerns you? (Politics, poverty, health, education, animals, environment, far future)

Are you interested in donating? In direct work?

Are you financially comfortable?

##Common Questions##

 - What beliefs do all EAs have in common?
 - What are the different causes EAs support?
 - What are some actions EAs take to improve the world?
 - What makes EA different from other ways of helping people?
 - What are some common criticisms of EA, and how do EAs respond?
 - What is the philosophical justification for EA?

Here are some really good articles from outside Arbital:

 - [What is Effective Altruism?](https://whatiseffectivealtruism.com) (The best general introduction to the topic.)

**This page is a work in progress. If you have questions, or you'd like to see a topic added, please contact the author: aaronlgertler (at) gmail.com.**


Arbital

[summary: Arbital is a platform for finding, reading, and creating crowdsourced, intuitive explanations for <s>everything</s> math. It is fundamentally collaborative, which allows everyone to contribute and share their expertise. To see what the platform can do, check out [1zq Arbital's guide to Bayes' rule]. To help out with the content, take a look at [4d6].]

Arbital is a platform for finding, reading, and creating crowdsourced, intuitive explanations for <s>everything</s> math. Several existing websites ([Wikipedia](https://www.wikipedia.org/), [MathOverflow](http://mathoverflow.net/), and [Wolfram MathWorld](http://mathworld.wolfram.com/) among others) already work well as references, but it's hard to learn from them unless you are already familiar with the subject. Other websites, like [Khan Academy](https://www.khanacademy.org/math) and [Better Explained](http://betterexplained.com/), have some great explanations (at least for K-12 material) but they are limited by the fact that they are closed platforms. Arbital is fundamentally a collaborative platform, which allows everyone to contribute explanations, add examples, and share their expertise.

## Ideal explanation platform

What would the ideal platform for (math) explanations look like? For example, if you wanted to learn Bayes' theorem, what would you want to see behind [1zq this link]?

The ideal platform would use each reader's existing knowledge, expertise, and learning preferences to generate a tailored explanation. If they were comfortable with mathematical notation, the ideal platform would show them equations instead of wordy explanations. If they were missing certain prerequisites, the ideal platform would make it easy to catch up on those. If they already knew half the explanation, the ideal platform would skip past it. If one explanation didn't work for them, the ideal platform would offer another.

The ideal platform would be interactive, as if you were studying with a personal tutor. Asking for a reminder about what a [-3rb] is would be as easy as hovering your mouse over a link. It would be easy to ask questions and instantly get detailed answers. It would be easy to speed up an explanation or slow it down.

The ideal platform would have intuitive, memorable, mind-blowing explanations that reliably produce that magical "click" feeling. The one you get when something finally makes perfect sense.
%%note:
A few examples:
[What if? - Relativistic baseball](http://what-if.xkcd.com/1/)
[Wait Buy Why - Fermi paradox](http://waitbutwhy.com/2014/05/fermi-paradox.html)
[Better Explained - Intuitive trigonometry](http://betterexplained.com/articles/intuitive-trigonometry/)
[Scott Aarson - Who can name the bigger number?](http://www.scottaaronson.com/writings/bignumbers.html)
And videos too:
[Vi Hart - Spirals and Fibonacci numbers](https://www.youtube.com/watch?v=ahXIMUkSXX0)
[US auto industry - How differential steering works](https://www.youtube.com/watch?v=yYAw79386WI)
%%
It would be easy to give feedback to authors to help them incrementally improve each page to make that happen.

That ideal platform doesn't exist yet, but we're building it, and it's called [58p Arbital]. What you see before you is a beta version, and we're going to continue working hard to improve it and make our vision a reality.

## Enthusiastic community and effective tools

Our top priority is to grow our community of like-minded people who love helping others learn. We are avid learners ourselves, and some of the most interesting things we've learned are the ones we've taught ourselves outside the context of classrooms and courses. We like explaining things and sharing our knowledge with others. If you look at the success of Wikipedia and MathOverflow, it's clear that communities with a shared mission can produce amazing, useful, and lasting resources that benefit the whole world.

We aim to amplify each author's effort by building a new set of tools tailored specifically for reading and collaboratively creating online explanations. We've already implemented a [14q number of useful features]. We also aim to build an ecosystem of modular pages that will make it easy for authors to reuse existing content and create explanations that can build on each other. We want to empower our authors to write in a way that works best for them, while preserving overall quality.

## Future of Arbital

Currently, we are focused on math explanations. Eventually, we plan to move beyond math to computer science, physics, statistics, economics, health, e-sports, and everything else. One day, you'll be able to use Arbital to get to the very frontier of human knowledge. Then, we'll extend the platform into one that can foster discussions and help all humanity push that frontier forward.

If you want to see that day come sooner, [4d6 you can help]! You can contribute individual pages, incrementally improve existing pages by proposing edits, and craft entire explanation [1rt paths]. You can also read, learn, and provide feedback. Each improvement you make helps the entire platform.

Let's work together to build [58p the best explanation platform this world has ever seen!]


Start

Start-Class pages are visibly incomplete, or need significant work to be useful.  This covers pages missing essential information, pages without coherent structure, pages lacking [1kl summaries] (or first paragraphs that serve as a summary), pages with sufficiently severe prose or stylistic issues, and any other problem that prevents pages from being useful to readers or fitting into their domains.  Start-Class pages have significant content, but are still in the category "should be seen by editors" rather than "should be seen by readers."

Contrast:

- [72], for tiny pages containing only a paragraph or a couple of sentences of text.
- [4gs], for pages which only contain a formal definition and don't try to explain the topic.

**Also** use the tag [4v] if the page is being actively edited.

**[4yg Quality scale]**

* [4ym]
* [4gs]
* [72]
* [3rk]
* [4y7]
* [4yd]
* [4yf]
* [4yl]


Strictly factual question

A *strict question of fact* has an answer determined solely by the state of the material universe, and sufficiently straightforward math, in a clearly understandable way - none of our uncertainty about this question is about definitions, values, or viewpoints; we are just wondering which quarks go where.

Questions of strict fact:

- What happens if I press the giant red button?(\*)
- Did Sally claim more charitable donations as an income tax deduction than Bob?
- Are there unicorns anywhere on Earth?
- Will Peano arithmetic prove a contradiction if I search all the proofs less than a billion steps long?

Questions not yet of strict fact:

- Should I press the giant red button?
- Is Sally a more charitable person than Bob?
- Do unicorns exist?
- Will Zermelo-Frankel set theory prove a contradiction if I search for one forever?

Why the second group aren't yet questions of strict fact:

- Because the notion of *should* has not yet been fully specified or determined.
- Because we haven't said exactly what it means to be charitable.
- Because "exist" is a much more ambiguous notion than "exist inside our galaxy".  For example, there could be unicorns $10^{1,000,000}$ lightyears away, or inside some other mathematical universe that has as much "existence" as this one does.  The notion of "X exists" is less firmly nailed down than "X is inside our closet".
- Because the notion of 'forever' is more mathematically fraught than 'for ten to the billionth power years'.  For example, nailing down exactly which infinity we're talking about can't be done in a system of first-order logic.

(\*)  At least, this is a straightforward question so long as we don't poke too hard at the nature of [ what-if counterfactuals].  In most real-life situations, the question "What happens if I turn on this blender with a fork inside?" is something that has a sufficiently straightforward material answer for us to say that it's just a question of material facts.


Sample space

[summary: A sample space is the [3jz set] $\Omega$ of possible ways that a part of the world might be, as far as you know.]

Motivation
===

If we are uncertain about some part of the world, then there are multiple different things that might happen there, as far as we know.
The sample space for a part of the world is the set of things that could possibly happen  there. We can use a sample space to reason using [1bv probability theory], so that we assign probabilities to [event_probability events] in a consistent way.

Definition
===

A sample space is a [3jz set], usually denoted $\Omega$.

[todo: examples]
[todo: further points]


Probability distribution (countable sample space)

[summary: A probability distribution on a countable [3t4] $\Omega$ is a [3jy] $\mathbb{P}: \Omega \to [0,1]$ such that $\sum_{\omega \in \Omega} \mathbb{P}(\omega) = 1$.]

Definition
===

A probability distribution on a countable [3t4] $\Omega$ is a [3jy] $\mathbb{P}: \Omega \to [0,1]$ such that $\sum_{\omega \in \Omega} \mathbb{P}(\omega) = 1$.

Intuition
=====

We express a belief that "$x\in \Omega$ happens with probability $r$" by setting $\mathbb{P}(x) = r$. So a probability distribution divides up our anticipation of what will happen, out of the set $\Omega$ of things that might possibly happen.

[todo: examples, futher points]


Proving too much

An argument "proves too much" when you can just as easily or naturally extend it to prove false things as true things.  For example, the argument "Nobody should engage in genetic engineering because Nazis" can just as easily be extended to say "Nobody should eat vegetables because Hitler ate vegetables", so the general rule *proves too much.*

See also:

- http://en.wikipedia.org/wiki/Proving_too_much
- http://slatestarcodex.com/2013/04/13/proving-too-much/


Fallacies

A "fallacy" is a mode of thought that is asserted (by calling it a "fallacy") to lead us easily into error, or to not usefully contribute to the project of distinguishing truth from falsehood.  For example, Argumentum Ad Hitlerum is a 'fallacy' because while Hitler did a number of bad things, he also ate vegetables, so "Hitler did X" is not a reliable means of arguing that nobody should do X.


Sample spaces are too large

[summary: Sample spaces enumerate everything that could possibly happen, so that we can reason consistently under uncertainty. But this enumeration usually enumerates too many things, making it difficult to do computations using the raw sample space.]

When we are reasoning about an uncertain world using [1bv probability theory], we have a [3t4 sample space] of possible ways the world could be, and we assign
probabilities to outcomes in the sample space with a [3tb probability distribution]. Unfortunately, often the sample space (which is supposed to contain every possible way the world might turn out) is very large and possibly infinite.

For example, if we are thinking about a box with some particles in it, our sample space is the enormous set of all possible arrangements of the particles.
To reason usefully about the box, we'd have to do computations using a gigantic table specifying a separate number for every single distinct arrangement of particles.
Two arrangements that are pretty much the same, as far as we care, take up just as much of our computational resources as two importantly different arrangements.
So, even though we could in principle reason consistently under uncertainty just using a probability distribution over a sample space, we cannot do so easily,
because we are trying to keep track of too much.

[todo: examples]


Uncountable sample spaces are way too large

[summary:
If the sample space $\Omega$ is [2w0 uncountable], then in general we can't even define a probability distribution over $\Omega$ in the same way we defined
[3tb probability distributions] over countable sample spaces, i.e. by just assigning numbers to each point in the sample space.  Any function $f: \Omega \to [0,1]$ with $\sum_{\omega \in \Omega} f(\omega) = 1$ can only assign positive values to at most countably many elements of $\Omega$. ]

If the sample space $\Omega$ is [2w0 uncountable], then in general we can't even define a probability distribution over $\Omega$ in the same way we defined
[3tb probability distributions] over countable sample spaces, i.e. by just assigning numbers to each point in the sample space.  Any function $f: \Omega \to [0,1]$ with $\sum_{\omega \in \Omega} f(\omega) = 1$ can only assign positive values to at most countably many elements of $\Omega$. But this means we can't, for example, talk about a
[uniform_distribution uniform distribution] over the interval $[0,2]$, which intuitively should assign equal probability to everywhere in the interval.


Non-standard terminology

A tag for terminology that is Arbital-specific, Arbital-originated, or just not very common outside of Arbital.


Needs summary

By default, a page's [1kl summary] (that shows up in the popover when a user hovers over a [17f greenlink]) is the page's first paragraph. Sometimes the first paragraph is not well-suited for a popover summary. Add this tag to pages that need to have a custom summary.

Some pages may require an [5cb accessible], [4pt technical], or [4q2 brief] summary.


Psychologizing

"Psychologizing" is when we go past arguing that a position *is in fact* wrong, and start considering *what could have gone wrong in people's minds* to make them believe something so wrong.  Realistically, we can't entirely avoid psychologizing - sometimes cognitive biases are real and understanding how they potentially apply is important.  Nonetheless, since psychologizing is potentially a pernicious and poisonous activity, Arbital's discussion norms say to explicitly tag or greenlink with "[43h]" each place where you speculate about the psychology of how people could possibly be so wrong.  Again, psychologizing isn't necessarily wrong - sometimes people do make mistakes for systematic psychological reasons that are understandable - but it's dangerous, potentially degenerates extremely quickly, and deserves to be highlighted each and every time.

If you go off on an extended discourse about why somebody is mentally biased or thinking poorly, *before* you've argued on the object level that they are in fact mistaken about the subject matter, this is [Bulverism](https://en.wikipedia.org/wiki/Bulverism).  Bulverism is flatly not okay on Arbital.  If you want to speculate about what goes wrong in the minds of people who believe in UFOs, then a good way of doing this on Arbital would be to *first* link a page arguing that there are in fact no UFOs, and say "I think this page establishes there are no UFOs; if you're not familiar with the arguments, read that first.  Given that I think there are in fact no UFOs, I will now [43h psychologize] about why people would mistakenly believe in UFOs..."

If that clear label makes your page less snappy, persuasive, and fun to read, please consider that to be an *intended effect* of this rule.  Psychologizing *is* fun and that's why we want to take steps against it coming to dominate discussions here.


Bulverism

[Bulverism](https://en.wikipedia.org/wiki/Bulverism) is when you [43h explain what goes so horribly wrong in people's minds] when they believe X, before you've actually explained why X is wrong.  It can be seen as a subspecies of the fallacy of "[poisoning the well](https://en.wikipedia.org/wiki/Poisoning_the_well)".  **Bulverism is forbidden on Arbital.**  Every time you talk about what goes wrong in people's minds that makes them believe X, make sure you first discuss why X is, in fact, wrong, or route your readers to a page that does discuss it.  Also see the entry on [43h].


Square visualization of probabilities on two events

$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$$

[summary:
$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$$

We can represent a [joint_probability_distribution_on_event probability distribution] $\bP(A,B)$ over  two [event_probability events] $A$ and $B$ as a square:

We could also represent $\bP$ by [factoring_probability factoring], so using $\bP(A,B) = \bP(A)\; \bP(B \mid A)$ we'd make this picture:

]

Say we have two [event_probability events], $A$ and $B$, and a [joint_probability_distribution_on_event probability distribution] $\bP$ over whether or not they happen. We can represent $\bP$ as a square:

So for example, the [1rh probability] $\bP(A,B)$ of both $A$ and $B$ occurring is the ratio of \[the area of the dark red region\] to \[the area of the entire square\]:

Visualizing probabilities in a square is neat because we can draw simple pictures that highlight interesting facts about our probability distribution.

Below are some pictures illustrating:

* [4cf independent events] (What happens if the columns and the rows in our square *both* line up?)

* [marginal_probability marginal probabilities] (If we're looking at a square of probabilities, where's the probability $\bP(A)$ of $A$ or the probability $\bP(\neg B)$?)

* [1rj conditional probabilities] (Can we find in the square the probability $\bP(B \mid A)$ of $B$ if we condition on seeing $A$? What about the conditional probability $\bP(A \mid B)$?)

* [factoring_probability factoring a distribution] (Can we always write $\bP$ as a square? Why do the columns line up but not the rows?)

* the process of computing [1rh joint probabilities] from [factoring_probability factored probabilities]

Independent events
===

Here's a picture of the joint distribution of [4cf two independent events] $A$ and $B$:

Now the rows for $\bP(B)$ and  $\bP(\neg B)$ line up across the two columns. This is because $\bP(B \mid A) = \bP(B) = \bP(B \mid \neg A)$. When $A$ and $B$ are independent, updating on $A$ or $\neg A$ doesn't change the probability of $B$.

For more on this visualization of independent events, see the aptly named [4cl].

Marginal probabilities
===

We can see the [marginal_probability marginal probabilities] of $A$ and $B$ by looking at some of the blocks in our square. For example, to find the probability $\bP(\neg A)$ that $A$ doesn't occur, we just need to add up all the blocks where $\neg A$ happens: $\bP(\neg A) = \bP(\neg A, B) + \bP(\neg A, \neg B)$.

Here's the probability $\bP(A)$ of $A$, and the probability $\bP(\neg A)$ of $\neg A$:

Here's the probability $\bP(\neg B)$ of $\neg B$:

In these pictures we're dividing by the area of the whole square. Since the probability of anything at all happening is 1, we could just leave it out, but it'll be helpful for comparison while we think about conditionals next.

Conditional probabilities
===

We can start with some probability $\bP(B)$, and then *assume* that $A$ is true to get a [1rj conditional probability] $\bP(B \mid A)$ of $B$. Conditioning on $A$ being true is like restricting our whole attention to just the possible worlds where $A$ happens:

Then the conditional probability of $B$ given $A$ is the proportion of these $A$ worlds where $B$ also happens:

If instead we condition on $\neg A$, we get:

So our square visualization gives a nice way to see, at a glance, the conditional probabilities of $B$ given $A$ or given $\neg A$:

 We don't get such nice pictures for $\bP(A \mid B)$:

More on this next.

Factoring a distribution
===

Recall the square showing our joint distribution $\bP$:

Notice that in the above square, the reddish blocks for $\bP(A,B)$ and $\bP(A,\neg B)$ are the same width and form a column; and likewise the blueish blocks for $\bP(\neg A,B)$ and $\bP(\neg A,\neg B)$. This is because we chose to [factoring_probability factor] our probability distribution starting with $A$:

$$\bP(A,B) = \bP(A) \bP( B \mid A)\ .$$

Let's use the [event_variable_equivalence equivalence] between [event_probability events] and [binary_variable binary random variables], so if we say $\bP( B= \true \mid A= \false)$ we mean $\bP(B \mid \neg A)$. For any choice of truth values $t_A \in \{\true, \false\}$ and $t_B \in \{\true, \false\}$, we have

$$\bP(A = t_A,B= t_B) = \bP(A= t_A)\; \bP( B= t_B \mid A= t_A)\ .$$

The first factor $\bP(A = t_A)$ tells us how wide to make the red column $(\bP(A = \true))$ relative to the blue column $(\bP(A = \false))$. Then the second factor $\bP( B= t_B \mid A= t_A)$ tells us the proportions of dark $(B = \true)$ and light $(B = \false)$ within the column for $A = t_A$.

We could just as well have factored by $B$ first:

$$\bP(A = t_A,B= t_B) = \bP(B= t_B)\; \bP( A= t_A \mid B= t_b)\ .$$

Then we'd draw a picture like this:

By the way, earlier when we factored by $A$ first, we got simple pictures of the probabilities $\bP(B \mid A)$ for $B$ conditioned on $A$.  Now that we're factoring by $B$ first, we have simple pictures for the conditional probability $\bP(A \mid B)$:

and for the conditional probability $\bP(A \mid \neg B)$:

Computing joint probabilities from factored probabilities
===

Let's say we know the factored probabilities for $A$ and $B$, factoring by $A$. That is, we know $\bP(A = \true)$, and we also know $\bP(B = \true \mid A = \true)$ and $\bP(B = \true \mid A = \false)$. How can we recover the joint probability $\bP(A = t_A, B = t_B)$ that $A = t_A$ is the case and also $B = t_B$ is the case?

Since

$$\bP(B = \false \mid A = \true) = \frac{\bP(A = \true, B = \false)}{\bP(A = \true)}\ ,$$

we can multiply the prior $\bP(A)$ by the conditional $\bP(\neg B \mid A)$ to get the joint $\bP(A, \neg B)$:

$$\bP(A = \true)\; \bP(B = \false \mid A = \true) = \bP(A = \true, B = \false)\ .$$

If we do this at the same time for all the possible truth values $t_A$ and $t_B$, we get back the full joint distribution:

[todo: information theory. a couple things, then point to another page. eg show example when two things have lots of mutual info.]


Two independent events

$$
\newcommand{\bP}{\mathbb{P}}
$$

[summary:
$$
\newcommand{\bP}{\mathbb{P}}
$$
We say that two [event_probability events], $A$ and $B$, are *independent* when learning that $A$ has occurred does not change your probability that $B$ occurs. That is, $\bP(B \mid A) = \bP(B)$.
Another way to state independence is that $\bP(A,B) = \bP(A) \bP(B)$. ]

We say that two [event_probability events], $A$ and $B$, are *independent* when learning that $A$ has occurred does not change your probability that $B$ occurs. That is, $\bP(B \mid A) = \bP(B)$.
Equivalently,  $A$ and $B$ are independent if $\bP(A)$ doesn't change if you condition on $B$: $\bP(A \mid B) = \bP(A)$.

Another way to state independence is that $\bP(A,B) = \bP(A) \bP(B)$.

All these definitions are equivalent:

$$\bP(A,B) = \bP(A)\;  \bP(B \mid A)$$

by the [chain_rule_probability chain rule], so

$$\bP(A,B) = \bP(A)\;  \bP(B)\;\; \Leftrightarrow \;\; \bP(A)\;  \bP(B \mid A) = \bP(A)\;  \bP(B) \ ,$$

and similarly for $\bP(B)\;  \bP(A \mid B)$.


Two independent events: Square visualization

$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$$

[summary:
$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$$

Say $A$ and $B$ are independent [event_probability events], so $\bP(A, B) = \bP(A)\bP(B).$ Then we can draw their joint probability distribution using the using the [496 square visualization] of probabilities:

]

This is what independence looks like, using the [496 square visualization] of probabilities:

We can see that the [event_probability events] $A$ and $B$ don't interact; we say that $A$ and $B$ are *independent*. Whether we look at the whole square, or just the red part of
the square where $A$ is true, the probability of $B$ stays the same. In other words, $\bP(B \mid A) = \bP(B)$. That's what we mean by independence: the
probability of $B$ doesn't change if you condition on $A$.

Our square of probabilities can be generated by multiplying together the probability of $A$ and the probability of $B$:

This picture demonstrates another way to define what it means for $A$ and $B$ to be independent:

$$\bP(A, B) = \bP(A)\bP(B)\ .$$

In terms of factoring a joint distribution
--

Let's contrast independence with non-independence. Here's a picture of two ordinary, non-independent events $A$ and $B$:

(If the meaning of this picture isn't clear, take a look at [496].)

We have the red blocks for $\bP(A)$ and the blue blocks for $\bP(\neg A)$ lined up in columns. This means we've [factoring_probability factored] our
probability distribution using $A$ as the first factor:

$$\bP(A,B) = \bP(A) \bP(B \mid A)\ .$$

We could just as well have factored by $B$ first: $\bP(A,B) = \bP(B) \bP( A \mid B)\ .$ Then we'd draw a picture like this:

Now, here again is the picture of [4cf two independent events] $A$ and $B$:

In this picture, there's red and blue lined-up columns for $\bP(A)$ and $\bP(\neg A)$, and there's *also*  dark and light lined-up rows for $\bP(B)$ and
$\bP(\neg B)$. It looks like we somehow [factoring_probability factored] our probability distribution $\bP$ using both $A$ and
$B$ as the first factor.

In fact, this is exactly what happened: since $A$ and $B$ are [4cf independent], we have that $\bP(B \mid A) = \bP(B)$. So the diagram
above is actually factored according to $A$ first: $\bP(A,B) = \bP(A) \bP(B \mid A)$. It's just that $\bP(B \mid A)= \bP(B) = \bP(B \mid \neg A)$, since $B$
is independent from $A$. So we don't need to have different ratios of dark to light (a.k.a. conditional probabilities of $B$) in the left and right columns:

In this visualization, we can see what happens to the probability of $B$ when you condition on $A$ or on $\neg A$: it doesn't change at all. The ratio of
\[the area where $B$ happens\] to \[the whole area\], is the same as the ratio $\bP(B \mid A)$ where we only look at the area where $A$ happens, which is the
same as the ratio $\bP(B \mid \neg A)$ where we only look at the area where $\neg A$ happens. The fact that the probability of $B$ doesn't change when we
condition on $A$ is exactly what we mean when we say that $A$ and $B$ are independent.

The square diagram above is *also* factored according to $B$ first, using $\bP(A,B) = \bP(B) \bP(A \mid B)$. The red / blue ratios are the same in both rows
because $\bP(A \mid B) = \bP(A) = \bP(A \mid \neg B)$, since $A$ and $B$ are independent:

We couldn't do any of this stuff if the columns and rows didn't both line up. (Which is good, because then we'd have proved the false statement that any two
events are independent!)

In terms of multiplying marginal probabilities
---

Another way to say that $A$ and $B$ are independent variables %note:We're using  the [event_variable_equivalence equivalence] between [event_probability
events] and [binary_random_variable binary variables].% is that for any truth values $t_A,t_B \in \{\true, \false\},$

$$\bP(A = t_A, B= t_B) = \bP(A = t_A)\bP(B = t_B)\ .$$

So the [1rh joint probabilities] for $A$ and $B$ are computed by separately getting the probability of $A$ and the probability of $B$, and then
multiplying the two probabilities together. For example, say we want to compute the probability $\bP(A, \neg B) = \bP(A = \true, B = \false)$. We start with
the [marginal_probability marginal probability] of $A$:

and the probability of $\neg B$:

and then we multiply them:

We can get all the joint probabilities this way. So we can visualize the whole joint distribution as the thing that you get when you multiply two independent
probability distributions together. We just overlay the two distributions:

To be a little more mathematically elegant, we'd use the [topological_product topological product of two spaces] shown earlier to draw the joint distribution
as a product of the distributions of $A$ and $B$:


Square visualization of probabilities on two events: (example) Diseasitis

$$
\newcommand{\bP}{\mathbb{P}}
$$

[summary: $$ \newcommand{\bP}{\mathbb{P}} $$

We can visualize the [22s]:

Then we can see at a glance that the probability $\bP(D \mid B)$ of the patient having Diseasitis given that the tongue depressor is black, despite some intuitions, isn't all that big:

$$\bP(D \mid B) = \frac{\bP(B, D)}{ \bP(B, D) + \bP(B, \neg D)} = \frac{3}{7} $$

]

From the [22s]:

> You are screening a set of patients for a disease, which we'll call Diseasitis. Based on prior epidemiology, you expect that around 20% of the patients in the screening population will in fact have Diseasitis. You are testing for the presence of the disease using a tongue depressor containing a chemical strip. Among patients with Diseasitis, 90% turn the tongue depressor black. However, 30% of the patients without Diseasitis will also turn the tongue depressor black. Among all the patients with black tongue depressors, how many have Diseasitis?

It seems like, since Diseasitis very strongly predicts that the patient has a black tongue depressor, it should be the case that the [-1rj] $\bP( \text{Diseasitis} \mid \text{black tongue depressor})$ is big. But actually, it turns out that a patient with a black tongue depressor is more likely than not to be completely Diseasitis-free.

Can we see this fact at a glance? Below, we'll use the [-496] to draw pictures and use our visual intuition.

To introduce some notation: our [1rm prior probability] $\bP(D)$ that the patient has Diseasitis is $0.2$. We think that if the patient is sick $(D)$, then it's 90% likely that the tongue depressor will turn black $(B)$: we assign conditional probability $\bP(B \mid D) = 0.9$. We assign conditional probability $\bP(B \mid \neg D) = 0.3$ that the tongue depressor will be black even if the patient isn't sick. We want to know $\bP(D \mid B)$, the [1rp posterior probability] that the patient has Diseasitis given that we've seen a black tongue depressor.

If we wanted to, we could solve this problem precisely using [1lz]:

$$
\begin{align}
\bP(D \mid B) &= \frac{\bP(B \mid D) \bP(D)}{\bP(B)}\\
&= \frac{0.9 \times 0.2}{ \bP(B, D) + \bP(B, \neg D)}\\
&= \frac{0.18}{ \bP(D)\bP(B \mid D) + \bP(\neg D)\bP(B \mid \neg D)}\\
&= \frac{0.18}{ 0.18 + 0.24}\\
&= \frac{0.18}{ 0.42} = \frac{3}{7}  \approx 0.43\ .
\end{align}
$$

So even if we've seen a black tongue depressor, the patient is more likely to be healthy than not: $\bP(D \mid B) < \bP(\neg D \mid B) \approx 0.57$.

Now, this calculation might be enlightening if you are a real expert at [1lz]. A better calculation would probably be the [1x9 odds ratio form of Bayes's rule].

But either way, maybe there's still an intuition saying that, come on, if the tongue depressor is such a strong indicator of Diseasitis that $\bP(B \mid D) = 0.9$, it must be that $\bP(D \mid B) =big$.

Let's use the [496 square visualization of probabilities] to make it really visibly obvious that $\bP(D \mid B) < \bP(\neg D \mid B)$, and to figure out why $\bP(B \mid D) = big$ doesn't imply $\bP(D \mid B) =big$.

We start with the probability of $\bP(D)$ (so we're  [factoring_probability factoring] our probabilities by $D$ first):

Now let's break up the red column, where $D$ is true and the patient has Diseasitis, into a block for the probability $\bP(B \mid D)$ that $B$ is also true, and a block for the probability $\bP(\neg B \mid D)$ that $B$ is false.

>Among patients with Diseasitis, 90% turn the tongue depressor black.

That is, in 90% of the  outcomes where $D$ happens, $B$ also happens. So $0.9$ of the red column will be dark ($B$), and $0.1$ will be light:

>However, 30% of the patients without Diseasitis will also turn the tongue depressor black.

So we break up the blue $\neg D$ column by $\bP(B \mid \neg D) = 0.3$ and $\bP(\neg B \mid \neg D) = 0.7$:

Now we would like to know the probability $\bP(D \mid B)$ of Diseasitis once we've observed that the tongue depressor is black. Let's break up our diagram by whether or not $B$ happens:

Conditioning on $B$ is like only looking at the part of our distribution where $B$ happens. So the probability $\bP(D \mid B)$ of $D$  conditioned on $B$ is the proportion of that area where $D$ also happens:

Here we can see why $\bP(D \mid B)$ isn't all that big. It's true that $\bP(B,D)$ is big relative to $\bP(\neg B,D)$, since we know that $\bP(B \mid D)$ is big (patients with Diseasitis almost always have black tongue depressors):

But this ratio doesn't really matter if we want to know $\bP(D \mid B)$, the probability that a patient with a black tongue depressor has Diseasitis. What matters is that we also assign a reasonably high probability $\bP(B, \neg D)$ to the patient having a black tongue depressor but nevertheless *not* suffering from Diseasitis:

So even when we see a black tongue depressor, there's still a pretty high chance the patient is healthy anyway, and our [-1rp] $\bP(D\mid B)$ is not that high. Recall our square of probabilities:

When asked about $\bP(D\mid B)$, we think of the really high probability $\bP(B\mid D) = 0.9$:

Really, we should look at the part of our probability mass where $B$ happens, and see that a sizeable portion goes to places where $\neg D$ happens, and the patient is healthy:

Side note
---

The square visualization is very similar to [1x1 frequency diagrams], except we can just think in terms of probability mass rather than specifically frequency. Also, see that page for [1x1 waterfall diagrams], another way to visualize updating probabilities.


Emphemeral premises

A purported [43h psychological] fallacy wherein people confronted with an incongruent proposition X will say "Oh, not-X because Y", and perhaps a week later, say, "Oh, not-X because Z."  [Ephemeral Premises](https://www.facebook.com/yudkowsky/posts/10154268184344228) is when you make up assumptions to deflect X, don't write them down or list them anywhere, and since there's no central list, you might be making up different assumptions when talking to someone else.  It's a disposable premise being used as a parry in short-term combat, not part of a long-term edifice being made with consistent, well-defended materials.  Needs to be paired with understanding of the [multiple_stage_fallacy Multiple-Stage Fallacy] so that you don't fall for the reverse fallacy of making people list out lots of Ys and Zs to make arbitrary propositions sound very improbable.


Multiple stage fallacy

[summary:  The purported "Multiple-Stage Fallacy" is when you list multiple 'stages' that need to happen on the way to some final outcome, assign probabilities to each 'stage', multiply the probabilities together, and end up with a small final answer.  The alleged problem is that you can do this to almost *any* proposition by staring at it hard enough, including propositions that turn out to be true.

An alleged example would be that in August 2015, renowned forecaster Nate Silver wrote "[Trump's Six Stages of Doom](http://fivethirtyeight.com/features/donald-trumps-six-stages-of-doom/)" in which he gave Donald Trump a 2% chance of getting the Republican *nomination* (not the presidency), based on six stages each with at most a 50% probability of being passed.  In late March, Trump had passed the first four stages Nate listed, and prediction markets gave Trump a 75% chance of clinching the Republican nomination.  By Nate Silver's logic, Trump's probability of passing the remaining stages should have been $0.50 \cdot 0.50 = 0.25.$

The three main purported problems with the Multiple-Stage Fallacy are that:

- To get the final estimate, you logically need to multiply [1rj conditional probabilities], not unconditional probabilities.  Getting your mind to give good conditional probabilities isn't easy; Nate Silver might have failed to imagine how much he'd *actually* update after *actually* seeing Trump pass the first four stages.
- Often, people fail to consider disjunctive alternatives, or ways that not all the stages may need passing.
- People tend to assign middle-tending probabilities.   So if you list enough stages, you can drive the apparent probability of anything down to zero, *even if you seem to be soliciting probabilities from the reader.*]

In August 2015, renowned statistician and predictor Nate Silver wrote "[Trump's Six Stages of Doom](http://fivethirtyeight.com/features/donald-trumps-six-stages-of-doom/)" in which he gave Donald Trump a 2% chance of getting the Republican *nomination* (not the presidency).  Silver reasoned that Trump would need to pass through six stages to get the nomination, "Free-for-all", "Heightened scrutiny", "Iowa and New Hampshire", "Winnowing", "Delegate accumulation", and "Endgame."  Nate Silver argued that Trump had at best a 50% chance of passing each stage, implying a final nomination probability of at most 2%.

In late March, Trump had passed the first four stages, while prediction markets gave him a 75% chance of clinching the Republican nomination.  By Nate Silver's logic, Trump's probability of passing the remaining stages should have been $0.50 \cdot 0.50 = 0.25$ *conditional* on Trump passing the first four stages.

Nate Silver might not have been wrong to assign Trump a low advance probability of being nominated.  Many people were surprised by Trump's nomination.  But it seems more likely that Silver committed an error when he said, specifically, that if we'd observed Trump to pass the first four stages, this would probably be taking place in a world where Trump had a 50% probability of passing the two remaining stages.

The purported "Multiple-Stage Fallacy" is when you list multiple 'stages' that need to happen on the way to some final outcome, assign probabilities to each 'stage', multiply the probabilities together, and end up with a small final answer.  The alleged problem is that you can do this to almost any kind of proposition by staring at it hard enough, including things that actually happen.

On a probability-theoretic level, the three problems at work in the usual Multiple-Stage Fallacy are as follows:

- You need to multiply *conditional* probabilities rather than the absolute probabilities. When you're considering a later stage, you need to assume that the world was such that every prior stage went through. Nate Silver was [43h probably] trying to simulate his prior model of Trump accumulating enough delegates in March through June, not imagining his *updated* beliefs about Trump and the world after seeing Trump be victorious up to March.
  - Even if you're aware in principle that you need to use conditional probabilities - Nate Silver certainly knew about them - it's hard to update far *enough* when you imagine the pure hypothetical possibility that Trump wins stages 1-4 for some reason - compared to how much you actually update when you actually see Trump winning! (Some sort of reverse hindsight bias or something? We don't realize how much we'd need to update our current model if we were already that surprised?)
- Often, people neglect to consider disjunctive alternatives - there may be more than one way to reach a stage, so that not *all* the listed things need to happen. Trump accumulated enough delegates in Nate's fifth stage that there was no "Endgame" convention fight in the supposed sixth stage.
- People have tendencies to assign middle-tending probabilities. So if you list enough stages, you can drive the apparent probability of anything down to zero, *even if you seem to be soliciting probabilities from the reader.*
 - If you're a [motivated skeptic](https://wiki.lesswrong.com/wiki/Motivated_skepticism), you will be tempted to list more 'stages'.

The Multiple-Stage Fallacy is particularly dangerous for people who've read studies on the dangers of probabilistic overconfidence.  In late March, the 75% prediction-market probabilities must have corresponded to, e.g., something like an 80% chance of getting enough delegates and a 90% chance of passing the convention conditional on getting enough delegates.  Imagine how overconfident this might have sounded without the prediction market to establish a definite probability - "Oh, don't you know that what people assign 90% confidence doesn't usually happen 90% of the time?"

Instances of the Multiple-Stage Fallacy may also sound more persuasive to readers who've read about the [Conjunction Fallacy](http://lesswrong.com/lw/ji/conjunction_fallacy/).

Yudkowsky [argues](https://www.facebook.com/yudkowsky/posts/10154036150109228):

> If you're not willing to make "overconfident" probability assignments like those, then you can drive the apparent probability of anything down to zero by breaking it down into enough 'stages'. In fact, even if someone hasn't heard about overconfidence, people's probability assignments often trend toward the middle, so you can drive down their "personally assigned" probability of anything just by breaking it down into more stages...

> From beginning to end, I've never used this style of reasoning and I don't recommend that you do so either. \[Since\] even Nate Silver couldn't get away with it, I think we just shouldn't try. It's a doomed methodology.


Empirical probabilities are not exactly 0 or 1

[summary:  [Cromwell's Rule](https://en.wikipedia.org/wiki/Cromwell%27s_rule) in statistics forbids us to assign probabilities of exactly $0$ or $1$ to any empirical proposition - that is, it is always possible to be mistaken.

- Probabilities of exactly $0$ or $1$ correspond to infinite [1zh log odds], meaning that no finite amount of evidence can ever suffice to reach them, or overturn them.  Once you assign probability $0$ or $1,$ you can never change your mind.
- Sensible [4mr universal priors] never assign probability exactly $0$ or $1$ to any predicted future observation - their hypothesis space is always broad enough to include a scenario where the future is different from the past.]

[Cromwell's Rule](https://en.wikipedia.org/wiki/Cromwell%27s_rule) in statistics argues that no empirical proposition should be assigned a subjective probability of *exactly* $0$ or $1$ - it is always *possible* to be mistaken.  (Some argue that this rule should be generalized to logical facts as well.)

A probability of exactly $0$ or $1$ corresponds to infinite [1zh log odds], and would require infinitely [22x strong] evidence to reach starting from any finite [1rm prior].  To put it another way, if you don't start out infinitely certain of a fact before making any observations (before you were born), you won't reach infinite certainty after any finite number of observations involving finite probabilities.

All sensible [4mr universal priors] seem so far to have the property that they never assign probability exactly $0$ or $1$ to any predicted future observation, since their hypothesis space is always broad enough to include an imaginable state of affairs in which the future is different from the past.

If you did assign a probability of exactly $0$ or $1,$ you would be unable to [1ly update] no matter how much contrary evidence you observed.  [1rb Prior odds] of 0 : 1 (or 1 : 0), times any finite [1rq likelihood ratio], end up yielding 0 : 1 (or 1 : 0).

As Rafal Smigrodski put it:

> "I am not totally sure I have to be always unsure.  Maybe I could be legitimately sure about something.  But once I assign a probability of 1 to a proposition, I can never undo it.  No matter what I see or learn, I have to reject everything that disagrees with the axiom.  I don't like the idea of not being able to change my mind, ever."


Universal prior

A "universal" [27p prior] is a [3tb probability distribution] that assigns positive probability to *every thinkable hypothesis*, for some reasonable meaning of "every thinkable hypothesis".  A central example would be [11w Solomonoff induction], in which the observations are a sequence of bits, the universal prior is "every possible computer program that generates bit sequences", and every such computer program starts with a positive probability.


Guide

Meta tag for the start page of a [327 multi-page guide].


Work in progress

A meta tag for pages unfinished pages which an author is still making major changes to.

[todo: Define/explain guidelines for use.]


Subjective probability

[summary:  What does it *mean* to say that a fair coin has a 50% probability of landing heads?

Imagine flipping a coin and slapping it against your wrist. It's already landed either heads or tails. The fact that you don't know whether it's heads or tails is a fact about _you,_ not a fact about the coin. Ignorance is in the mind, not in the world.

Subjective probabilities are a tool for quantifying uncertainty about the world. In your mind's representation of the coin, you're unsure about which way the coin came up. Those probabilities, represented in your brain, are your subjective probabilities.

[1bv] is concerned with the formalization, study, and manipulation of subjective probabilities.

Several [7ry coherence theorems] suggest that classical probability is a *uniquely* good way to quantify subjective uncertainty.  This in turn means that while probabilities may be 'subjective' in the sense of existing in our minds rather than outside reality, the laws governing the manipulation and [1ly updating] of these probabilities are solid and determined.]

What does it *mean* to say that a flipped coin has a 50% probability of landing heads?

There are multiple ways to answer this question, depending on what you mean by "probability". This page discusses "subjective probabilities," which are a tool for quantifying your personal uncertainty about the world.

Imagine flipping a coin and slapping it against your wrist. It's already landed either heads or tails. The fact that you don't know whether it's heads or tails is a fact about _you,_ not a fact about the coin. Ignorance is in the mind, not in the world.

So your mind is representing the coin, and you're unsure about which way the coin came up. Those probabilities, represented in your brain, are your subjective probabilities. [1bv] is concerned with the formalization, study, and manipulation of subjective probabilities.

If probabilities are simply subjective mental states, what does it mean to say that probabilities are "good," "correct," "accurate," or "true"? The subjectivist answer, roughly, is that a probability distribution becomes more accurate as it puts more of its probability mass on the true possibility within the set of all possibilities it considers.  For more on this see [4yj].

Subjective probabilities, given even a small [ grain of truth], will become *more* accurate as they interact with reality and execute [1ly Bayesian updates].  Your subjective belief about "Is it cloudy today?" is materially represented in your brain, and becomes more accurate as you look up at the sky and causally interact with it: light from the clouds in the sky comes down, enters your retina, is transduced to nerve impulses, processed by your visual cortex, and then your subjective belief about whether it's cloudy becomes more accurate.

'Subjective probability' designates our view of probability as an epistemic state, something inherently in the mind, since reality itself is not uncertain.  It doesn't mean 'arbitrary probability' or 'probability that somebody just made up with no connection to reality'.  Your belief that it's cloudy outside (or sunny) is a belief, but not an arbitrary or made-up belief.  The same can be true about your statement that you think it's 90% likely to be sunny outside, because it was sunny this morning and it's summer, even though you're currently in an interior room and you haven't checked the weather.  The outdoors itself is not wavering between sunny and cloudy; but your *guess* that it's 9 times more likely to be sunny than cloudy is not ungrounded.

Several [7ry coherence theorems] suggest that classical probabilities are a *uniquely* good way of quantifying the relative credibility we attach to our guesses; e.g. that even if it's probably sunny, it's still *more* likely for it to be cloudy outside than for the Moon to be made of green cheese.  This in turn says that while the probabilities themselves may exist in our minds, the laws that govern the manipulation and updating of these probabilities are as solid as any other mathematical fact.

For an example of a solid law governing subjective probability, see Arbital's [1zq guide to Bayes' rule].


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A meta tag for pages which don't have [597 clickbait]. Clickbait style guidelines:

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Yes: "Period or other punctuation at the end."
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