The Tiger's Stripes
http://www.bowaggoner.com/blog/
A blog on fun research and tools in computer science and game theory.http://www.bowaggoner.com/blog/images/morphog.jpgThe Tiger's Stripes
http://www.bowaggoner.com/blog/
en2018-03-17Tight Bounds for Gaussian Tails and Hazard Rates
http://www.bowaggoner.com/blog/2018/03-17-gaussian-tails/index.html
http://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
http://www.bowaggoner.com/blog/2018/03-16-eliciting-continuous-scalars/index.html
http://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
http://www.bowaggoner.com/blog/2017/12-18-subgaussianity/index.html
http://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
http://www.bowaggoner.com/blog/2017/10-07-measure-concentration/index.html
http://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
http://www.bowaggoner.com/blog/2017/10-06-useful-bounds-taylors/index.html
http://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
http://www.bowaggoner.com/blog/2017/10-03-prediction-markets/index.html
http://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
http://www.bowaggoner.com/blog/2017/09-28-risk-aversion-and-decisionmaking/index.html
http://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
http://www.bowaggoner.com/blog/2017/04-22-finite-properties/index.html
http://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
http://www.bowaggoner.com/blog/2017/04-12-eliciting-properties/index.html
http://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
http://www.bowaggoner.com/blog/2017/01-06-collisions-in-bins/index.html
http://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
http://www.bowaggoner.com/blog/2016/10-20-convex-duality/index.html
http://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
http://www.bowaggoner.com/blog/2016/10-07-value-divergences/index.html
http://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
http://www.bowaggoner.com/blog/2016/10-02-risk-aversion-entropy/index.html
http://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
http://www.bowaggoner.com/blog/2016/09-24-generalized-entropies/index.html
http://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
http://www.bowaggoner.com/blog/2016/09-22-proper-scoring-rules/index.html
http://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
http://www.bowaggoner.com/blog/2016/09-20-convexity/index.html
http://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