The Tiger's Stripes
https://www.bowaggoner.com/blog/
A technical blog on math, computer science, and game theory.https://www.bowaggoner.com/blog/images/morphog.jpgThe Tiger's Stripes
https://www.bowaggoner.com/blog/
en2019-02-03Sub-Gamma Variables and Concentration
https://www.bowaggoner.com/blog/2019/02-03-sub-gamma-concentration/index.html
https://www.bowaggoner.com/blog/2019/02-03-sub-gamma-concentration/index.htmlFor variables that are usually small, subgaussian concentration bounds can often be loose. We can get better Bernstein bounds using the 'sub-gamma' approach.2019-02-03Measure Concentration II
https://www.bowaggoner.com/blog/2018/09-29-measure-concentration-ii/index.html
https://www.bowaggoner.com/blog/2018/09-29-measure-concentration-ii/index.htmlThis post will discuss how to get some key tail bounds from subgaussian assumptions, then introduce the wonderful world of martingale inequalities. This is a follow-up to posts on <a href='https://www.bowaggoner.com/blog/2017/10-07-measure-concentration/'>measure concentration</a> and <a href='https://bowaggoner.com/blog/2017/12-18-subgaussianity/'>subgaussian variables</a>.2018-09-29Prophet Inequalities
https://www.bowaggoner.com/blog/2018/08-25-prophet-inequalities/index.html
https://www.bowaggoner.com/blog/2018/08-25-prophet-inequalities/index.htmlSo-called 'Prophet Inequalities' are cute mathematical stopping-time theorems with interesting applications in mechanism design.
The problem is to choose when to stop and claim an arriving random variable, performing almost as well as if you had prophetic foresight.2018-08-25Prediction Markets II
https://www.bowaggoner.com/blog/2018/08-08-prediction-markets-ii/index.html
https://www.bowaggoner.com/blog/2018/08-08-prediction-markets-ii/index.htmlIn this follow-up on <a href='../../2017/10-03-prediction-markets/'>an introduction to prediction markets</a>, they are back and badder than ever with a more formal and general mathematical approach.2018-08-08Eliciting Means
https://www.bowaggoner.com/blog/2018/08-02-eliciting-means/index.html
https://www.bowaggoner.com/blog/2018/08-02-eliciting-means/index.htmlIt turns out that <em>Bregman divergences</em> generally characterize scoring rules to elicit an agent's expectation of a random variable.
This fact is important in machine learning and also has nice connections to the proper scoring rule characterization.
This is a continuation of the <a href='http://www.bowaggoner.com/blog/series.html#convexity-elicitation'>series on elicitation</a>.2018-08-02Weitzman's Pandora's Box Problem
https://www.bowaggoner.com/blog/2018/07-20-pandoras-box/index.html
https://www.bowaggoner.com/blog/2018/07-20-pandoras-box/index.htmlThe 'Pandora's Box' problem is a cool model of search for the best alternative under uncertainty with a neat and intuitive solution.
This post describes a somewhat different proof and intuition with a nice extension to matching.2018-07-20Hybrid Auction-Prediction Mechanisms
https://www.bowaggoner.com/blog/2018/06-23-hybrid-auction-prediction-mechanisms/index.html
https://www.bowaggoner.com/blog/2018/06-23-hybrid-auction-prediction-mechanisms/index.htmlThis post describes a mechanism for making a group decision (along with monetary transfers) based both on people's preferences and their predictions.2018-06-23The VCG Mechanism
https://www.bowaggoner.com/blog/2018/06-22-vcg/index.html
https://www.bowaggoner.com/blog/2018/06-22-vcg/index.htmlThis post reviews the famous Vickrey-Clarke-Groves mechanism: the canonical truthfulness-inducing auction (procedure involving monetary transfers) for making a group decision.2018-06-22Tight Bounds for Gaussian Tails and Hazard Rates
https://www.bowaggoner.com/blog/2018/03-17-gaussian-tails/index.html
https://www.bowaggoner.com/blog/2018/03-17-gaussian-tails/index.htmlThis post will show how to tightly bound the tail probabilities of the Gaussian distribution from both sides with a closed form.2018-03-17Eliciting Continuous Scalars
https://www.bowaggoner.com/blog/2018/03-16-eliciting-continuous-scalars/index.html
https://www.bowaggoner.com/blog/2018/03-16-eliciting-continuous-scalars/index.htmlContinuing the <a href='https://www.bowaggoner.com/blog/series.html#convexity-elicitation'.> series on elicitation</a>, we'll take a look at a special class of properties or statistics of distributions: those that are scalar real numbers (as opposed to vectors) and whose value is a continuous function of the distribution.2018-03-16Subgaussian Variables and Concentration
https://www.bowaggoner.com/blog/2017/12-18-subgaussianity/index.html
https://www.bowaggoner.com/blog/2017/12-18-subgaussianity/index.htmlIn this post we'll take a look at the definition of subgaussian random variables and see how those are used in measure concentration.2017-12-18Intro to Measure Concentration
https://www.bowaggoner.com/blog/2017/10-07-measure-concentration/index.html
https://www.bowaggoner.com/blog/2017/10-07-measure-concentration/index.htmlThis post will introduce the concept of 'tail bounds' or 'measure concentration' and cover the basics of Markov's, and Chebyshev's, and Chernoff-type bounds and how they are proven.2017-10-07Useful Bounds via Taylor's Theorem
https://www.bowaggoner.com/blog/2017/10-06-useful-bounds-taylors/index.html
https://www.bowaggoner.com/blog/2017/10-06-useful-bounds-taylors/index.htmlIn this post, we'll take a look at some common and useful bounds on the exponential and logarithm functions and see how they're derived using Taylor's Theorem.2017-10-06Prediction Markets
https://www.bowaggoner.com/blog/2017/10-03-prediction-markets/index.html
https://www.bowaggoner.com/blog/2017/10-03-prediction-markets/index.htmlThis post will introduce prediction markets based on cost functions and/or proper scoring rules. It will focus on intuition and background rather than technical details.2017-10-03Risk Aversion and Decisionmaking
https://www.bowaggoner.com/blog/2017/09-28-risk-aversion-and-decisionmaking/index.html
https://www.bowaggoner.com/blog/2017/09-28-risk-aversion-and-decisionmaking/index.htmlIn this post, I want to review the basics of risk aversion for you, then argue that risk-aversion naturally arises in any situation where agents face a decision under uncertainty.2017-09-28Eliciting Finite Properties
https://www.bowaggoner.com/blog/2017/04-22-finite-properties/index.html
https://www.bowaggoner.com/blog/2017/04-22-finite-properties/index.htmlIn this post we'll look at eliciting 'finite properties' of distributions in other words, multiple-choice questions about some unknown or future event.
This is a continuation of the <a href='http://www.bowaggoner.com/blog/series.html#convexity-elicitation'>series on elicitation</a>.2017-04-22Eliciting Properties of Distributions
https://www.bowaggoner.com/blog/2017/04-12-eliciting-properties/index.html
https://www.bowaggoner.com/blog/2017/04-12-eliciting-properties/index.htmlContinuing the <a href='http://www.bowaggoner.com/blog/series.html#convexity-elicitation'.> series on elicitation</a>, we'll take a look at <em>properties</em> or statistics of distributions: things like the mode, median, or variance.
This post will focus on the defining elicitation of properties and showing how the convex characterization of proper scoring rules extends. We'll look at more concrete cases and implications in later posts.2017-04-12k-Way Collisions of Balls in Bins
https://www.bowaggoner.com/blog/2017/01-06-collisions-in-bins/index.html
https://www.bowaggoner.com/blog/2017/01-06-collisions-in-bins/index.htmlA basic probability question is what happens when we draw i.i.d. samples from a distribution, often referred to as 'throwing balls into bins'.
Here I just want to show how the number of 'k-way collisions' can give a simple yet useful analysis for, especially, the size of the max-loaded bin.2017-01-06Convex Duality
https://www.bowaggoner.com/blog/2016/10-20-convex-duality/index.html
https://www.bowaggoner.com/blog/2016/10-20-convex-duality/index.htmlEvery convex function has a special relative (or perhaps 'evil twin') called its <em>conjugate</em> or <em>dual</em>.
In this post, we'll walk through the definition both formally and visually.2016-10-20Divergences and Value of Information
https://www.bowaggoner.com/blog/2016/10-07-value-divergences/index.html
https://www.bowaggoner.com/blog/2016/10-07-value-divergences/index.htmlThis is a follow-up to <a href='http://www.bowaggoner.com/blog/2016/09-24-generalized-entropies'>generalized entropies and the value of information</a>, where we discussed how value of information connects to generalized entropies. Here, we'll connect both to Bregman divergences, filling in a third side of the triangle.2016-10-07Risk Aversion and Max-Entropy
https://www.bowaggoner.com/blog/2016/10-02-risk-aversion-entropy/index.html
https://www.bowaggoner.com/blog/2016/10-02-risk-aversion-entropy/index.htmlI want to share a nice little problem that came up in discussion between <a href='http://yiling.seas.harvard.edu/'>Yiling Chen</a>, <a href='http://madhu.seas.harvard.edu/'>Madhu Sudan</a>, and myself.
It turns out to have a cute solution that connects the geometry of proper scoring rules with a 'max-entropy' rule.2016-10-02Generalized Entropies and the Value of Information
https://www.bowaggoner.com/blog/2016/09-24-generalized-entropies/index.html
https://www.bowaggoner.com/blog/2016/09-24-generalized-entropies/index.htmlIn this post, I'll discuss axioms for entropy functions that generalize Shannon entropy and connect them to the value of information for a rational decision maker.
We'll see that <a href='http://www.bowaggoner.com/blog/2016/09-20-convexity/'>convexity</a> turns out to be a natural and simple axiom that nicely links the two settings.2016-09-24Proper Scoring Rules
https://www.bowaggoner.com/blog/2016/09-22-proper-scoring-rules/index.html
https://www.bowaggoner.com/blog/2016/09-22-proper-scoring-rules/index.htmlHow can we elicit truthful predictions from strategic agents?
Proper scoring rules give a surprisingly complete and mathematically beautiful answer.
In this post, we'll work up to the classic characterization of all 'proper' (truthful) scoring rules in terms of convex functions of probability distributions.2016-09-22Convexity
https://www.bowaggoner.com/blog/2016/09-20-convexity/index.html
https://www.bowaggoner.com/blog/2016/09-20-convexity/index.htmlConvexity is a simple, intuitive, yet surprisingly powerful mathematical concept.
It shows up repeatedly in the study of efficient algorithms and in game theory.
The goal of this article to is to review the basics so we can put convexity to use later.2016-09-20