Nate Soares

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by Nate Soares Sep 4 2015

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Nate Soares

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A googol A pretty small large number.
A googolplex A moderately large number, as large numbers go.
Abelian group A group where the operation commutes. Named after Niels Henrik Abel.
Abstract algebra The study of groups, fields, vector spaces, arithmetics, algebras, and more.
Algebraic structure Roughly speaking, an algebraic structure is a set $X$, known as the underlying set, paired with a co…
Arity (of a function) The arity of a function is the number of parameters that it takes. For example, the function $f(a, b…
Associative operation An **associative operation** $\bullet : X \times X \to X$ is a binary operation such that for all $x…
Associativity vs commutativity Associativity and commutativity are often confused, because they are both constraints on how a funct…
Associativity: Examples Yes: [Addition], [multiplication], string concatenation. No: [subtraction], [division], a Function …
Associativity: Intuition Associative functions can be interpreted as families of functions that reduce lists down to a singl…
Bag In mathematics, a "bag" is an unordered list. A bag differs from a set in that it can contain the sa…
Bayes' rule: Definition Bayes' rule is the mathematics of probability theory governing how to update your beliefs in the lig…
Bayes' rule: Probability form The original formulation of Bayes' rule.
Binary function A binary function $f$ is a function of two inputs (i.e., a function with arity 2). For example, $+,$…
Bit The term "bit" refers to different concepts in different fields. The common theme across all the us…
Bit (abstract) An abstract bit is an element of the set $\mathbb B$, which has two elements. An abstract bit is to …
Bit (of data) A bit of data is the amount of data required to single out one message from a set of two. Equivalen…
Bit (of data): Examples In the game "20 questions", one player (the "leader") thinks of a concept, and the other players ask…
Blue oysters A probability problem about blue oysters.
Cartesian product The Cartesian product of two sets $A$ and $B,$ denoted $A \times B,$ is the set of all [ordered\_pai…
Ceiling The ceiling of a real number $x,$ denoted $\lceil x \rceil$ or sometimes $\operatorname{ceil}(x),$ i…
Closure A set $S$ is _closed_ under an operation $f$ if, whenever $f$ is fed elements of $S$, it produces an…
Codomain (of a function) The codomain $\operatorname{cod}(f)$ of a function $f : X \to Y$ is $Y$, the set of possible outputs…
Codomain vs image It is useful to distinguish codomain from image both (a) when the type of thing that the function pr…
Communication: magician example Imagine that you and I are both magicians, performing a trick where I think of a card from a deck of…
Commutative operation A commutative function $f$ is a function that takes multiple inputs from a set $X$ and produces an o…
Commutativity: Examples Yes: addition, multiplication, maximum, minimum, rock-paper-scissors. No: subtraction, division, st…
Commutativity: Intuition We can think of commutativity either as an artifact of notation, or as a symmetry in the output of a…
Compressing multiple messages How many bits of data does it take to encode an $n$-message? Naively, the answer is $\lceil \log_2(n…
Conditional probability: Refresher Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?
Correspondence visualizations for different interpretations of "probability" Let's say you have a model which says a particular coin is 70% likely to be heads. How should we as…
Corrigibility "I can't let you do that, Dave."
Data capacity The data capacity of an object is defined to be the Logarithm of the number of different distinguish…
Decit Decimal digit
Dependent messages can be encoded cheaply Say you want to transmit a 2-message, a 4-message, and a 256-message to somebody. For example, you m…
Digit wheel A mechanical device for storing a number from 0 to 9. ![](http://www.cl.cam.ac.uk/~djg11/howcompu…
Direct sum of vector spaces The direct sum of two vector spaces $U$ and $W,$ written $U \oplus W,$ is just the sum of $U$ and $W…
Domain (of a function) The domain $\operatorname{dom}(f)$ of a function $f : X \to Y$ is $X$, the set of valid inputs for t…
Emulating digits In general, given enough $n$-digits, you can emulate an $m$-digit, for any $m, n \in$ $\mathbb N$. I…
Encoding trits with GalCom bits There are $\log_2(3) \approx 1.585$ bits to a Trit. Why is it that particular value? Consider the Ga…
Example problem Tag for pages that provide an example problem referenced by a number of other pages. The summary of…
Exchange rates between digits In terms of data storage, if a coin is worth $1, a digit wheel is worth more than $3.32, but less than $3.33. Why?
Fractional bits It takes $\log_2(8) = 3$ bits of data to carry one message from a set of 8 possible messages. Simila…
Fractional bits: Digit usage interpretation It is 316, not 500, that requires about two and a half digits to write down. 500 requires nearly 2.7…
Fractional bits: Expected cost interpretation In the GalCom thought experiment, you regularly have to send large volumes of information through de…
Fractional digits When $b$ and $x$ are integers, $\log_b(x)$ has a few good interpretations. It's roughly the length o…
Frequency diagram Visualizing Bayes' rule by manipulating frequencies in large populations
Frequency diagrams: A first look at Bayes The most straightforward visualization of Bayes' rule.
Function Intuitively, a function $f$ is a procedure (or machine) that takes an input and performs some opera…
Function: Physical metaphor Many functions can be visualized as physical mechanisms of wheels and gears, that take their inputs …
GalCom In the GalCom thought experiment, you live in the future, and make your money by living in the Dene…
GalCom: Rules 1. It costs 1 galcoin per bit to reserve on-peak bits in advance. (Galcoins are very expensive.) 2. …
Generalized associative law Given an associative operator $\cdot$ and a list $[a, b, c, \ldots]$ of parameters, all ways of red…
Graham's number A fairly large number, as numbers go.
Group The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
Group theory What kinds of symmetry can an object have?
How many bits to a trit? $\log_2(3) \approx 1.585.$ This can be interpreted a few different ways: 1. If you multiply the nu…
Image (of a function) The image $\operatorname{im}(f)$ of a function $f : X \to Y$ is the set of all possible outputs of $…
Information Information is a measure of how much a message grants an observer the ability to predict the world.…
Information theory The study (and quantificaiton) of information, and its communication and storage.
Interpretations of "probability" What does it *mean* to say that a fair coin has a 50% probability of coming up heads?
Intradependent encoding An encoding $E(m)$ of a message $m$ is intradependent if the fact that $E(m)$ encodes $m$ can be de…
Intradependent encodings can be compressed Given an encoding scheme $E$ which gives an Intradependent encoding of a message $m,$ we can in prin…
Introductory guide to logarithms Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mention…
Life in logspace The log lattice hints at the reason that engineers, scientists, and AI researchers find logarithms s…
Likelihood "Likelihood", when speaking of Bayesian reasoning, denotes *the probability of an observation, sup…
Likelihood function Let's say you have a piece of evidence $e$ and a set of hypotheses $\mathcal H.$ Each $H_i \in \math…
Likelihood notation The likelihood of a piece of evidence $e$ according to a hypothesis $H,$ known as "the likelihood of…
Likelihood ratio Given a piece of evidence $e$ and two hypothsese $H_i$ and $H_j,$ the likelihood ratio between them…
Linear algebra The study of [linear\_transformation linear transformations] and vector spaces.
List A list is an ordered collection of objects, such as `[0, 1, 2, 3]` or `["red", "blue", 0, "shoe"]`. …
Log as generalized length To estimate the log (base 10) of a number, count how many digits it has.
Log as the change in the cost of communicating When interpreting logarithms as a generalization of the notion of "length" and as digit exchange rat…
Log base infinity There is no log base infinity, but if there were, it would send everything to zero
Logarithm The logarithm base $b$ of a number $n,$ written $\log_b(n),$ is the answer to the question "how man…
Logarithm base 1 There is no log base 1.
Logarithm tutorial overview The logarithm tutorial covers the following six subjects: 1. What are logarithms? 2. Logarithms as…
Logarithm: Examples $\log_{10}(100)=2.$ $\log_2(4)=2.$ $\log_2(3)\approx 1.58.$ (TODO)
Logarithm: Exercises Without using a calculator: What is $\log_{10}(4321)$? What integer is it larger than, what integer …
Logarithmic identities - [ Inversion of exponentials]: $b^{\log_b(n)} = \log_b(b^n) = n.$ - [ Log of 1 is 0]: $\log_b(1) …
Logarithms invert exponentials The function $\log_b(\cdot)$ inverts the function $b^{(\cdot)}.$ In other words, $\log_b(n) = x$ imp…
Monoid A monoid $M$ is a pair $(X, \diamond)$ where $X$ is a [set\_theory\_set set] and $\diamond$ is an [a…
Nate's ruminations These posts are a mirror of posts on the blog [MindingOurWay.com](mindingourway.com) which pertain t…
Needs lenses This page has only a technical introduction. If you're able to, please help by adding an intuitive explanation!
Non-standard terminology A tag for terminology that is Arbital-specific, Arbital-originated, or just not very common outside …
Odds form to probability form The odds form of Bayes' rule works for any two hypotheses $H_i$ and $H_j,$ and looks like this: $$\…
Odds: Introduction What's the difference between probabilities and odds? Why is a 20% probability of success equivalent to 1 : 4 odds favoring success?
Odds: Refresher A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).
Operator An operation $f$ on a set $S$ is a function that takes some values from $S$ and produces a new value…
Order of a group The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ an…
Ordinary claims require ordinary evidence Extraordinary claims require extraordinary evidence, but ordinary claims *don't*.
Probability distribution: Motivated definition People keep writing things like P(sick)=0.3. What does this mean, on a technical level?
Probability interpretations: Examples Consider evaluating, in June of 2016, the question: "What is the probability of Hillary Clinton wi…
Proof of Bayes' rule: Probability form Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_…
Properties of the logarithm - $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ for any $b$, this is the defining characteristic of …
Range (of a function) The "range" of a function is an ambiguous term that is generally used to refer to either the functio…
Replacing Guilt In my experience, many people are motivated primarily by either guilt, shame, or some combination of…
Report likelihoods not p-values: FAQ This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…
Report likelihoods, not p-values If scientists reported likelihood functions instead of p-values, this could help science avoid p-ha…
Ring A ring is a kind of Algebraic structure which we obtain by considering groups as being "things with…
Set An unordered collection of distinct objects.
Set builder notation $\{ 2n \mid n \in \mathbb N \}$ denotes the set of all even numbers, using set builder notation. Set…
Shannon The shannon (Sh) is a unit of Information. One shannon is the difference in [info\_entropy entropy] …
Shift towards the hypothesis of least surprise When you see new evidence, ask: which hypothesis is *least surprised?*
Size of a set
Sock-dresser search There's a 4/5 chance your socks are in one of your dresser's 8 drawers. You check 6 drawers at random. What's the probability they'll be in the next drawer you check?
Sparking widgets 10% of widgets are bad and 90% are good. 4% of good widgets emit sparks, and 12% of bad widgets emit…
String (of text) A string (of text) is a series of letters (often denoted by quote marks), such as `"abcd"` or `"hell…
Subspace A subspace $U=(F_U, V_U)$ of a Vector space $W=(F_W, V_W)$ is a vector space where $F_U = F_W$ and $…
Sum of vector spaces The sum of two vector spaces $U$ and $W,$ written $U + W,$ is a vector space where the set of vector…
The End (of the basic log tutorial) That concludes our introductory tutorial on logarithms! You have made it to the end. Throughout thi…
The Stamp Collector Once upon a time, a group of naïve philosophers found a robot that collected trinkets. Well, more sp…
The characteristic of the logarithm Any time you find an output that adds whenever the input multiplies, you're probably looking at a (…
The log lattice Log as the change in the cost of communicating and other pages give physical interpretations of what…
There is only one logarithm All logarithm functions are the same, up to a multiplicative constant.
Thought experiment Meta-tag for thought experiments.
Trit Trinary digit
Underlying set What do a Group, a Partially ordered set, and a [ topological space] have in common? Each is a Set …
Vector space A vector space is a field $F$ paired with a Group $V$ and a function $\cdot : F \times V \to V$ (cal…
What is a logarithm? Logarithms are a group of functions that take a number as input and produce another number. There i…
Why is log like length? If a number $x$ is $n$ digits long (in Decimal notation), then its logarithm (base 10) is between $n…
Why is the decimal expansion of log2(3) infinite? Because 2 and 3 are relatively prime.
You're allowed to fight for something The first sort of guilt I want to address is the listless guilt, that vague feeling one gets after p…
concat (function) The string concatenation function `concat` puts two strings together, i.e., `concat("one","two")="on…
n-digit An $n$-digit is a physical object that can be stably placed into any of $n$ distinguishable states. …
n-message A message singling out one thing from a set of $n$ is sometimes called an $n$-message. For example,…

group

Nate Soares

comment

(5) was intended to assume that $n \in \mathbb R^{\ge 1},$ or possibly $\in \mathbb R^{\ge 0}$ if yo…
:, not '
Added a verbalization of the image. Does that help?
Examples?
FYI, it's either "X is composed of Y" or "X comprises Y" (according to the dictionary definition, at…
Fixed, thanks.
Fixed, thanks.
Fixed, thanks.
Fixed. (Would be nice to have a way to resolve these comments.)
Fixed.
How are these changes? (starting at prop 5, through the end)
How's this?
I mean, I wasn't going to say it myself, but that was my general impression :-p (I'm also dubious o…
I recommend rethinking the magnet metaphor, on the grounds that it is physically wrong. If you have …
I suggest adding more negative examples. (It's hard to learn a concept from only positive examples.)
I suggest making it explicit that $P$ is a distribution over a (possibly infinite) set of variables …
I think we don't need *that* much handholding. You've got to let people figure out some of the conne…
I tweaked this to make it a bit clearer while also cleaning up the latex (I think it's standard to u…
If those are the only two options, then you've gone mad :-) L(H|e) is defined to be P(e|H) (which, …
If you're going to start using probability density functions instead of just probability functions, …
Made a minor edit. If you want anything more, you'll need to be more specific.
No (and it won't, until someone starts writing good explanations of radicals). I think it's fine to …
No. edited for clarity, see if that helps.
Sorry :-p /p/3vp seemed too general a name for me back when I was first writing this, though I think…
Suggestion: Mark this thread as an "editor only" comment.
What's with the %% marks? Can all \[square bracket thingies\] be turned into percent-demarcated bloc…
Yep. Fixed, thanks.
Yes :-p
Yes.
would add "by highlighting text and using the menu that appears on the margin" (or other text that b…
wut
yep, nice catch.
{{lowercase title}} is for things like eBay and iPod, IIRC -- which arbital should support eventuall…

Edited

wiki

AI alignment The great civilizational problem of creating artificially intelligent computer systems such that running them is a good idea. - Eliezer Yudkowsky
AI arms races AI arms races are bad - Eliezer Yudkowsky
Ackermann function The slowest-growing fast-growing function. - Alex Appel
Arbital Arbital is the place for crowdsourced, intuitive math explanations. - Alexei Andreev
Arbital Markdown All about Arbital's extended Markdown syntax. - Alexei Andreev
Arbital: Solving online explanations An explanation of Arbital's mid-term goals - Eliezer Yudkowsky
Author's guide to Arbital How to write intuitive, flexible content on Arbital. - Alexei Andreev
Bayes' rule Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation. - Eliezer Yudkowsky
Bayes' rule examples Interesting problems solvable by Bayes' rule - Eliezer Yudkowsky
Bayes' rule: Functional form Bayes' rule for to continuous variables. - Eliezer Yudkowsky
Bayes' rule: Guide The Arbital guide to Bayes' rule - Eliezer Yudkowsky
Bayes' rule: Log-odds form A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence. - Eliezer Yudkowsky
Bayes' rule: Odds form The simplest and most easily understandable form of Bayes' rule uses relative odds. - Eliezer Yudkowsky
Bayes' rule: Odds form (Intro, Math 1) Introduction to the odds form of Bayes' rule - Eliezer Yudkowsky
Bayes' rule: Proportional form The fastest way to say something both convincing and true about belief-updating. - Eliezer Yudkowsky
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.) - Eliezer Yudkowsky
Bayesian view of scientific virtues Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course. - Eliezer Yudkowsky
Behaviorist genie An advanced agent that's forbidden to model minds in too much detail. - Eliezer Yudkowsky
Belief revision as probability elimination Update your beliefs by throwing away large chunks of probability mass. - Eliezer Yudkowsky
Category theory How mathematical objects are related to others in the same category. - Mark Chimes
Central examples List of central examples in Value Alignment Theory domain. - Eliezer Yudkowsky
Conditional probability The notation for writing "The probability that someone has green eyes, if we know that they have red hair." - Eliezer Yudkowsky
Cosmic endowment The 'cosmic endowment' consists of all the stars that could be reached from probes originating on Earth; the sum of all matter and energy potentially available to be transformed into life and fun. - Eliezer Yudkowsky
Decimal notation The winning architecture for numerals - Michael Cohen
Diamond maximizer How would you build an agent that made as much diamond material as possible, given vast computing power but an otherwise rich and complicated environment? - Eliezer Yudkowsky
Diseasitis 20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What's the chance you have Diseasitis? - Eliezer Yudkowsky
Edge instantiation When you ask the AI to make people happy, and it tiles the universe with the smallest objects that can be happy. - Eliezer Yudkowsky
Eliezer's vision for Arbital Why are we building this? What's the goal? - Eric Bruylant
Exponential Any function that constantly gets larger as a proportion of itself. - Joe Zeng
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. - Eliezer Yudkowsky
Finishing your Bayesian path on Arbital The page that comes at the end of reading the Arbital Guide to Bayes' rule - Eliezer Yudkowsky
Inductive prior Some states of pre-observation belief can learn quickly; others never learn anything. An "inductive prior" is of the former type. - Eliezer Yudkowsky
Integer An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv… - Michael Cohen
Introduction to Bayes' rule: Odds form Bayes' rule is simple, if you think in terms of relative odds. - Eliezer Yudkowsky
Introductory Bayesian problems Bayesian problems to try to solve yourself, before beginning to learn about Bayes' rule. - Eliezer Yudkowsky
Likelihood functions, p-values, and the replication crisis What's the whole Bayesian-vs.-frequentist debate about? - Eliezer Yudkowsky
List: value-alignment subjects Bullet point list of core VAT subjects. - Eliezer Yudkowsky
Mathematics Mathematics is the study of numbers and other ideal objects that can be described by axioms. - Eliezer Yudkowsky
Methodology of unbounded analysis What we do and don't understand how to do, using unlimited computing power, is a critical distinction and important frontier. - Eliezer Yudkowsky
Mind projection fallacy Uncertainty is in the mind, not in the environment; a blank map does not correspond to a blank territory. In general, the territory may have a different ontology from the map. - Eliezer Yudkowsky
Mindcrime Might a machine intelligence contain vast numbers of unhappy conscious subprocesses? - Eliezer Yudkowsky
Natural number The numbers we use to count: 0, 1, 2, 3, ... - Jaime Sevilla Molina
Odds Odds express a relative probability. - Eliezer Yudkowsky
Ontology identification problem How do we link an agent's utility function to its model of the world, when we don't know what that model will look like? - Eliezer Yudkowsky
P vs NP: Arguments against P=NP Why we believe P and NP are different - Jaime Sevilla Molina
Partially ordered set A set endowed with a relation that is reflexive, transitive, and antisymmetric. - Kevin Clancy
Path: Multiple angles on Bayes's Rule A learning-path placeholder page for learning multiple angles on Bayes's Rule. - Eliezer Yudkowsky
Pivotal event Which types of AIs, if they work, can do things that drastically change the nature of the further game? - Eliezer Yudkowsky
Posterior probability What we believe, after seeing the evidence and doing a Bayesian update. - Eliezer Yudkowsky
Prior probability What we believed before seeing the evidence. - Eliezer Yudkowsky
Probability notation for Bayes' rule The probability notation used in Bayesian reasoning - Eliezer Yudkowsky
Probability notation for Bayes' rule: Intro (Math 1) How to read, and identify, the probabilities in Bayesian problems. - Eliezer Yudkowsky
Product (Category Theory) How a product is characterized rather than how it's constructed - Mark Chimes
Proof of Bayes' rule Proofs of Bayes' rule, with graphics - Eliezer Yudkowsky
Proof of Bayes' rule: Intro Proof of Bayes' rule, assuming you know the rule itself, and the notations for the quantities involved. - Eliezer Yudkowsky
Quine A computer program that prints (or does other computations to) its own source code, using indirect self-reference. - Patrick LaVictoire
Rational number The rational numbers are "fractions". - Patrick Stevens
Real number A **real number** is any number that can be used to represent a physical quantity. Intuitively, rea… - Michael Cohen
Relation A **relation** is a set of [tuple\_mathematics tuples], all of which have the same [tuple\_arity ar… - Kevin Clancy
Relative likelihood How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence. - Eliezer Yudkowsky
Solomonoff induction: Intro Dialogue (Math 2) An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math - Eliezer Yudkowsky
Some men just want to watch the world learn In which Arbital gets paths. - Alexei Andreev
Standard agent properties What's a Standard Agent, and what can it do? - Eliezer Yudkowsky
Strength of Bayesian evidence From a Bayesian standpoint, the strength of evidence can be identified with its likelihood ratio. - Eliezer Yudkowsky
Strictly confused A hypothesis is strictly confused by the raw data, if the hypothesis did much worse in predicting it than the hypothesis itself expected. - Eliezer Yudkowsky
Subjective probability Probability is in the mind, not in the environment. If you don't know whether a coin came up heads or tails, that's a fact about you, not a fact about the coin. - Eliezer Yudkowsky
Task-directed AGI An advanced AI that's meant to pursue a series of limited-scope goals given it by the user. In Bostrom's terminology, a Genie. - Eliezer Yudkowsky
The rocket alignment problem If people talked about the problem of space travel the way they talked about AI... - Eliezer Yudkowsky
The square root of 2 is irrational The number whose square is 2 can't be written is a quotient of natural numbers - Dylan Hendrickson
Vinge's Law You can't predict exactly what someone smarter than you would do, because if you could, you'd be that smart yourself. - Eliezer Yudkowsky
Vingean uncertainty You can't predict the exact actions of an agent smarter than you - so is there anything you _can_ say about them? - Eliezer Yudkowsky
Waterfall diagram Visualizing Bayes' rule as the mixing of probability streams. - Eliezer Yudkowsky
Waterfall diagrams and relative odds A way to visualize Bayes' rule that yields an easier way to solve some problems - Eliezer Yudkowsky

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