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Comprehensive, covering all major areas of the topic without significant omissions or inaccuracies.
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summarized, in ways appropriate for the most likely audiences.
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Quality scale Tagged Start All Creators Eliezer Yudkowsky,
Eric Bruylant Likes Alexei Andreev,
Eric Bruylant Related 'Rationality' of voting in elections "A single vote is very unlikely to swing the election, so your vote is unlikely to have an effect" versus "Many people similar to you are making a similar decision about whether to vote." - Eliezer Yudkowsky 99LDT x 1CDT oneshot PD tournament as arguable counterexample to LDT doing better than CDT Arguendo, if 99 LDT agents and 1 CDT agent are facing off in a one-shot Prisoner's Dilemma tournament, the CDT agent does better on a problem that CDT considers 'fair'. - Eliezer Yudkowsky A whirlwind tour A rapid tour of Eric's thoughts on the accelerator project. - Eric Bruylant Absent-Minded Driver dilemma A road contains two identical intersections. An absent-minded driver wants to turn right at the second intersection. "With what probability should the driver turn right?" argue decision theorists. - Eliezer Yudkowsky Accelerator Project The Accelerator Project aims to create a low-cost environment which facilitates rapid personal growt… - Eric Bruylant Arbital Arbital is the place for crowdsourced, intuitive math explanations. - Alexei Andreev Arbital lens A lens is a page that presents another page's content from a different angle. - Alexei Andreev Arbital: Google Maps for knowledge Take your understanding from where it is to where it wants to be. - Alexei Andreev Arbital: learning from Wikipedia How is Arbital different from Wikipedia? - Alexei Andreev Associative operation An **associative operation** $\bullet : X \times X \to X$ is a binary operation such that for all $x… - Nate Soares Associativity: Examples Yes: [Addition], [multiplication], string concatenation. No: [subtraction], [division], a Function … - Nate Soares Associativity: Intuition Associative functions can be interpreted as families of functions that reduce lists down to a singl… - Nate Soares 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: Beginner's guide Beginner's guide to learning about Bayes' rule. - Alexei Andreev Bayes' rule: Functional form Bayes' rule for to continuous variables. - 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: Probability form The original formulation of Bayes' rule. - Nate Soares 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 Belief revision as probability elimination Update your beliefs by throwing away large chunks of probability mass. - Eliezer Yudkowsky Bit The term "bit" refers to different concepts in different fields. The common theme across all the us… - Nate Soares Coherent decisions imply consistent utilities Why do we all use the 'expected utility' formalism? Because any behavior that can't be viewed from that perspective, must be qualitatively self-defeating (in various mathy ways). - Eliezer Yudkowsky Commutativity: Intuition We can think of commutativity either as an artifact of notation, or as a symmetry in the output of a… - Nate Soares Contributing to Arbital Want to help Arbital become awesome? - Eric Bruylant Cyclic Group Intro (Math 0) A finite cyclic group is a little bit like a clock. - Mark Chimes Death in Damascus Death tells you that It is coming for you tomorrow. You can stay in Damascus or flee to Aleppo. Whichever decision you actually make is the wrong one. This gives some decision theories trouble. - Eliezer Yudkowsky Derivative How things change - Michael Cohen 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 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? - Nate Soares 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 Fractional digits When $b$ and $x$ are integers, $\log_b(x)$ has a few good interpretations. It's roughly the length o… - Nate Soares Frequency diagrams: A first look at Bayes The most straightforward visualization of Bayes' rule. - Nate Soares Group The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common. - Nate Soares High-speed intro to Bayes's rule A high-speed introduction to Bayes's Rule on one page, for the impatient and mathematically adept. - Eliezer Yudkowsky Interest in mathematical foundations in Bayesianism "Want" this requisite if you prefer to see extra information about the mathematical foundations in Bayesianism. - Eliezer Yudkowsky Introduction to Bayes' rule: Odds form Bayes' rule is simple, if you think in terms of relative odds. - Eliezer Yudkowsky Introductory guide to logarithms Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mention… - Nate Soares Isomorphism: Intro (Math 0) Things which are basically the same, except for some stuff you don't care about. - Mark Chimes Join and meet Let $\langle P, \leq \rangle$ be a poset, and let $S \subseteq P$. The **join** of $S$ in $P$, deno… - Kevin Clancy Life in logspace The log lattice hints at the reason that engineers, scientists, and AI researchers find logarithms s… - Nate Soares Log as generalized length To estimate the log (base 10) of a number, count how many digits it has. - Nate Soares 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… - Nate Soares Mathematical induction Proving a statement about all positive integers by knocking them down like dominoes. - Douglas Weathers 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:
$$\… - Nate Soares 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 Probability distribution: Motivated definition People keep writing things like P(sick)=0.3. What does this mean, on a technical level? - Nate Soares Probability notation for Bayes' rule: Intro (Math 1) How to read, and identify, the probabilities in Bayesian problems. - Eliezer Yudkowsky Project outline: Intro to the Universal Property Outline detailing all the work required for a proposed Arbital Project - Eric Rogstad Proof of Bayes' rule Proofs of Bayes' rule, with graphics - Eliezer Yudkowsky Proof of Bayes' rule: Probability form Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_… - Nate Soares Proof of Rice's theorem A standalone proof of Rice's theorem, including one surprising lemma. - Patrick Stevens Properties of the logarithm - $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ for any $b$, this is the defining characteristic of … - Nate Soares Rice's Theorem Rice's Theorem tells us that if we want to determine pretty much anything about the behaviour of an arbitrary computer program, we can't in general do better than just running it. - Patrick Stevens Shift towards the hypothesis of least surprise When you see new evidence, ask: which hypothesis is *least surprised?* - Nate Soares 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 The End (of the basic log tutorial) That concludes our introductory tutorial on logarithms! You have made it to the end.
Throughout thi… - Nate Soares The characteristic of the logarithm Any time you find an output that adds whenever the input multiplies, you're probably looking at a (… - Nate Soares The log lattice Log as the change in the cost of communicating and other pages give physical interpretations of what… - Nate Soares The missing step between Zero and Hero Creating a space for high potential people grow and improve the world at scale. - Eric Bruylant Uncountability: Intuitive Intro Are all sizes of infinity the same? What does "the same" even mean here? - Jason Gross Universal property of the empty set The empty set can be characterised by how it interacts with other sets, rather than by any explicit property of the empty set itself. - Patrick Stevens Universal property of the product The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on. - Patrick Stevens Utility function The only coherent way of wanting things is to assign consistent relative scores to outcomes. - 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 Welcome to Arbital Front page explaining what Arbital is all about. - Alexei Andreev What is a logarithm? Logarithms are a group of functions that take a number as input and produce another number. There i… - Nate Soares Reverse Related Start This page gives a basic overview of the topic, but may be missing important information or have stylistic issues. If you're able to, please help expand or improve it! - Eliezer Yudkowsky