# probability theory Published articles for probability theory. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Earthmover Distance DevFeed: [Earthmover Distance]() Original publisher: [Read original article]() Published: 2018-03-05T08:00:15Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Code](), [Python](), [Library]() Tags: [code](), [earthmover-distance](), [gerrymandering](), [library](), [linear-programming](), [mathematics](), [optimal-transport](), [optimization](), [probability-theory](), [python](), [wasserstein-metric]() ### AI overview This tutorial explains how to measure distance between point sets with uncertain locations by treating each sample set as a discrete probability distribution. It formulates Earthmover distance as a linear program that minimizes the cost of moving probability mass between points. ### Source excerpt Problem: Compute distance between points with uncertain locations (given by samples, or differing observations, or clusters). For example, if I have the following three "points" in the plane, as indicated by their colors, which is closer, blue to green, or blue to red? It's not obvious, and there are multiple factors at work: the red points have fewer samples, but we can be more certain about the position; the blue points are less certain, but the closest non-blue point to a blue point is green; and the green points are equally plausibly "close to red" and "close to blue. ## Bayesian Ranking for Rated Items DevFeed: [Bayesian Ranking for Rated Items]() Original publisher: [Read original article]() Published: 2017-03-13T09:00:49Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [recommendation systems](), [Statistics](), [Python](), [Data Science]() Tags: [catalog](), [conjugate-prior](), [dirichlet-distribution](), [display](), [distribution](), [inference](), [mathematics](), [probability-theory](), [python](), [ranking](), [sorting](), [statistics]() ### AI overview This article explains how to rank items with discrete user ratings using a Bayesian model with a Dirichlet prior. It presents a Python solution for estimating an item's expected rating from observed ratings, prior beliefs, and a utility function. ### Source excerpt Problem: You have a catalog of items with discrete ratings (thumbs up/thumbs down, or 5-star ratings, etc.), and you want to display them in the "right" order. Solution: In Python ''' score: [int], [int], [float] -> float Return the expected value of the rating for an item with known ratings specified by `ratings`, prior belief specified by `rating_prior`, and a utility function specified by `rating_utility`, assuming the ratings are a multinomial distribution and the prior belief is a Dirichlet distribution. ## Evaluating Splatoon's Ranking System DevFeed: [Evaluating Splatoon's Ranking System]() Original publisher: [Read original article]() Author: Evan Miller Published: 2015-07-21T00:30:00Z Content type: article Language: en Sources: [Evan Miller]() Topics: [math](), [bug]() Tags: [analysis](), [bug](), [probability-theory]() ### AI overview Evan Miller analyzes Splatoon's Ranked Battle ranking system using probability theory and matrix math. He finds that if battles are decided purely by chance, nearly 75% of players eventually reach A−, A, or A+, including more than 36% at A+, revealing a subtle flaw in the system. ### Source excerpt Evaluating Splatoon's Ranking System -- Using probability theory to uncover a subtle bug in Splatoon. ## The Giant Component and Explosive Percolation DevFeed: [The Giant Component and Explosive Percolation]() Original publisher: [Read original article]() Published: 2015-02-02T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [Data analysis]() Tags: [big-o-notation](), [erdos-renyi](), [first-moment-method](), [giant-component](), [graph](), [graphs](), [inequality](), [markov](), [method-of-moments](), [network-science](), [percolation](), [probability-theory](), [random-graph](), [random-graphs](), [random-variables]() ### AI overview This article develops rigorous results about Erdős-Rényi random graphs, following a conjecture about connectivity at edge probability p = 5/n. It introduces threshold theorems and begins explaining the first moment method, Markov's inequality, and isolated vertices. ### Source excerpt Last time we left off with a tantalizing conjecture: a random graph with edge probability $ p = 5/n$ is almost surely a connected graph. We arrived at that conjecture from some ad-hoc data analysis, so let's go back and treat it with some more rigorous mathematical techniques. As we do, we'll discover some very interesting "threshold theorems" that essentially say a random graph will either certainly have a property, or it will certainly not have it. ## Martingales and the Optional Stopping Theorem DevFeed: [Martingales and the Optional Stopping Theorem]() Original publisher: [Read original article]() Published: 2014-03-03T10:00:38Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [math]() Tags: [2-sat](), [conditional-probability](), [expectation](), [gambling](), [martingales](), [mathematics](), [optional-stopping-theorem](), [primer](), [probability-theory](), [random](), [random-variables](), [randomized-algorithm](), [stochastic-processes](), [theory]() ### AI overview This primer introduces martingales as models of fair betting games and explains their connection to probability theory. It begins with a geometric-distribution exercise involving repeated die throws, then introduces the ABRACADABRA problem using a monkey typing random letters. ### Source excerpt This is a guest post by my colleague Adam Lelkes. The goal of this primer is to introduce an important and beautiful tool from probability theory, a model of fair betting games called martingales. In this post I will assume that the reader is familiar with the basics of probability theory. For those that need to refresh their knowledge, Jeremy's excellent primers (1, 2) are a good place to start. ## Probabilistic Bounds -- A Primer DevFeed: [Probabilistic Bounds -- A Primer]() Original publisher: [Read original article]() Published: 2013-04-15T11:14:32Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithms, Complexity](), [math](), [Learning](), [Streaming]() Tags: [algorithm](), [algorithms](), [chebyshev](), [chernoff](), [chernoff-bound](), [inequality](), [learning-theory](), [machine-learning](), [markov](), [mathematics](), [probabilistic](), [probabilistic-method](), [probability-theory](), [random-variables](), [streaming](), [streaming-algorithms](), [variance]() ### AI overview This tutorial introduces probabilistic bounds used in algorithm analysis, machine learning theory, randomized algorithms, and streaming algorithms. It focuses on the Chernoff bound and presents simpler bounds from Markov's and Chebyshev's inequalities, including short proofs. ### Source excerpt Probabilistic arguments are a key tool for the analysis of algorithms in machine learning theory and probability theory. They also assume a prominent role in the analysis of randomized and streaming algorithms, where one imposes a restriction on the amount of storage space an algorithm is allowed to use for its computations (usually sublinear in the size of the input). While a whole host of probabilistic arguments are used, one theorem in particular (or family of theorems) is ubiquitous: the Chernoff bound. ## Conditional (Partitioned) Probability -- A Primer DevFeed: [Conditional (Partitioned) Probability -- A Primer]() Original publisher: [Read original article]() Published: 2013-03-28T13:36:09Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [math](), [function]() Tags: [bayes-theorem](), [conditional-probability](), [mathematics](), [partitions](), [primer](), [probability-theory](), [random-variables](), [set](), [theory]() ### AI overview A mathematically rigorous primer on conditional probability. It reviews finite probability spaces, probability mass functions, events, random variables, and partitions as tools for decomposing events and variables and reasoning about total probability. ### Source excerpt One of the main areas of difficulty in elementary probability, and one that requires the highest levels of scrutiny and rigor, is conditional probability. The ideas are simple enough: that we assign probabilities relative to the occurrence of some event. But shrewd applications of conditional probability (and in particular, efficient ways to compute conditional probability) are key to successful applications of this subject. This is the basis for Nate Silver's success, the logical flaws of many a political pundit, and the ability for a robot to tell where it is in an environment. ## Probability Theory -- A Primer DevFeed: [Probability Theory -- A Primer]() Original publisher: [Read original article]() Published: 2013-01-04T13:45:54Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [math](), [Statistics](), [Machine Learning & Artificial Intelligence]() Tags: [expectation](), [mathematics](), [primer](), [probabilistic](), [probability-theory](), [random-variables](), [set-theory](), [statistics](), [theory](), [variables](), [variance]() ### AI overview A primer on finite probability theory that introduces probability spaces, random variables, terminology, and basic results using naive set theory. It emphasizes mathematical formalism rather than real-world applications. ### Source excerpt It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables. ## Google's Page Rank--The Final Product DevFeed: [Google's Page Rank--The Final Product]() Original publisher: [Read original article]() Published: 2011-06-20T22:07:41Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Google](), [Matrix](), [math](), [Web]() Tags: [computational-complexity](), [eigenvalues](), [eigenvectors](), [google](), [mathematics](), [matrix](), [page-rank](), [probability-theory](), [programming](), [pseudocode](), [ranking](), [search-engine](), [web]() ### AI overview This article explains how PageRank addresses dangling nodes and non-unique rankings. It introduces positive column-stochastic matrices and the Perron-Frobenius Theorem, then uses random web surfing as a probability-based intuition for webpage importance. ### Source excerpt Dangling Nodes and Non-Uniqueness Recall where we left off last time. Given a web $ W$ with no dangling nodes, the link matrix for $ W$ has 1 as an eigenvalue, and if the corresponding eigenspace has dimension 1, then any associated eigenvector gives a ranking of the pages in $ W$ which is consistent with our goals. The first problem is that if there is a dangling node, our link matrix has a column of all zeros, and is no longer column-stochastic.