Let a die be labeled with increasing positive integers \(a_1,\ldots , a_n\), and let the probability of getting \(a_i\) be \(p_i>0\). We start at 0 and roll the die, adding whatever number we get to the current total. If \({\rm Pr}(N)\) is the probability that at some point we achieve the sum \(N\), then \(\lim_{N \to \infty} {\rm Pr}(N)\) exists and equals \(1/\rm{E}(X)\) iff \((a_1, \ldots, a_n)…
Gambling to optimize your expected bankroll mean is extremely risky, as you wager your entire bankroll for any favorable gamble, making ruin almost inevitable. But what if, instead, we gambled not to maximize the expected bankroll mean, but the expected bankroll median? Let the probability of winning a favorable bet be \(p\), and the net odds be \(b\). That is, if we wager \(1\) unit and win, we…
In my previous post I looked at how a group of experts may be combined into a single, more powerful, classifier which I call NaiveBoost after the related AdaBoost . I'll illustrate how it can be used with a few examples. As before, we're faced with making a binary decision, which we can view as an unknown label \( L \in \{ +1, -1 \}\). Furthermore, the prior distribution on \( L \) is assumed to…
In many situations we're faced with multiple expert opinions. How should we combine them together into one opinion, hopefully better than any single opinion? I'll demonstrate the derivation of a classifier I'll call NaiveBoost. We'll start with a simple situation, and later gradually introduce more complexity. Let each expert state a yes or no opinion in response to a yes/no question (binary…
Here's a simplified version for optimal Kelly bets when you have multiple outcomes (e.g. horse races). The Smoczynski & Tomkins algorithm, which is explained here (or in the original paper): https://en.wikipedia.org/wiki/Kelly_criterion#Multiple_horses Let's say there's a wager that, for every $1 you bet, will return a profit of $b if you win. Let the probability of winning be \(p\), and losing be…
I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under…
IMO 1989 #6: A permutation \(\{x_1, x_2, \ldots , x_m\}\) of the set \(\{1, 2, \ldots , 2n\}\), where \(n\) is a positive integer, is said to have property \(P\) if \( | x_i - x_{i+1} | = n\) for at least one \(i\) in \(\{1, 2, ... , 2n-1\}\). Show that for each \(n\) there are more permutations with property \(P\) than without. Solution: We first observe that the expected number of pairs \(\{i,…
Let's say we have a game that can be reasonably modeled as two independent Poisson processes with team \(i\) having parameter \(\lambda_i\). If one team wins in regulation with team \(i\) scoring \(n_i\), then it's well-known we have the MLE estimate \(\hat{\lambda_i}=n_i\). But what if the game ends in a tie in regulation with each team scoring \(n\) goals and we have sudden-death overtime? How…
If you're not familiar with Kaggle , it's an organization dedicated to data science competitions to both provide ways for companies to potentially do analytics at less cost, as well as to identify talented data scientists. Competitions are scored using a variety of functions, and the most common for binary classification tasks with confidence is something called log-loss, which is essentially…
The Kelly Criterion is an alternative to standard utility theory, which seeks to maximize expected utility. Instead, the Kelly Criterion seeks to maximize expected growth . That is, if we start out with an initial bankroll \(B_0\), we seek to maximize \(\mathrm{E}[g(t)]\), where \(B_t = B_0\cdot e^{g(t)}\). As a simple example, consider the following choice. We can have a sure $3000, or we can…
Find four prime divisors < 100 for \(3^{32}-2^{32}\). Source: British Math Olympiad, 2006. This factors nicely as \(3^{32}-2^{32} = \left(3^{16}+2^{16}\right)\left(3^{16}-2^{16}\right)\), and we can continue factoring in this way to get \[3^{32}-2^{32} = \left(3^{16}+2^{16}\right)\left(3^8+2^8\right)\left(3^4+2^4\right)\left(3^2+2^2\right)\left(3^2-2^2\right).\]The final three terms are \(5, 13,…
Let \(\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}\), where \(a\) and \(b\) are integers. What is the greatest common divisor of \(b\) and \(81\)? Source: 2011-2012 SDML High School 2a, problem 15. Let \((1+\sqrt{2})^n = a_n + b_n \sqrt{2}\). I've thought about this some more, and there's a nice way to describe the highest power of \(3\) that divides \(b_n\). This is probably outside of the scope of…
What is the largest even integer that cannot be written as the sum of two odd composite numbers? Source: AIME 1984 , problem 14. Note \(24 = 3\cdot 3 + 3\cdot 5\), and so if \(2k\) has a representation as the sum of even multiples of 3 and 5, say \(2k = e_3\cdot 3 + e_5\cdot 5\), we get a representation of \(2k+24\) as a sum of odd composites via \(2k+24 = (3+e_3)\cdot 3 + (5+e_5)\cdot 5\). But by…
In a previous entry I demonstrated one simple way to estimate an exponent for the Pythagorean win expectation . Another nice consequence of a Pythagorean win expectation formula is that it also makes it simple to estimate the run value of a win in baseball, the point value of a win in basketball, the goal value of a win in hockey etc. Let our Pythagorean win expectation formula be \[…
Given the number of runs scored and runs allowed by a baseball team, what's a good estimate for that team's win fraction? Bill James famously came up with what he called the " Pythagorean expectation " \[w = \frac{R^2}{R^2 + A^2},\] which can also be written as \[w = \frac{{(R/A)}^2}{{(R/A)}^2 + 1}.\] More generally, if team \(i\) scores \(R_i\) and allows \(A_i\) runs, the Pythagorean estimate…
In the book "The Only Rule Is It Has to Work: Our Wild Experiment Building a New Kind of Baseball Team" , Ben Lindbergh and Sam Miller recount a grand adventure to take command of an independent league baseball team, with the vision of trying every idea, sane or crazy, in an attempt to achieve a winning edge. Five infielders, four outfielders, defensive shifts, optimizing lineups - everything. It…
Assuming two NBA teams of equal strength with \(t\) seconds remaining, what is a safe lead at a prescribed confidence level? Bill James has a safe lead formula for NCAA basketball , and the topic has been addressed by other researchers at various levels of complexity, e.g. Clauset, Kogan and Redner . I'll present a simple derivation. Start by observing there are about 50 scoring groups per team…
For years the kilogram has been defined with respect to a platinum and iridium cylinder, but this is now no longer the case . Here's a puzzle about kilograms that's easy to state and understand, but the answer is very, very surprising. I've always had a fascination with really large numbers. First 100 when I was really little, and as I got older and more sophisticated numbers like a googol and the…
The following math problem, which appeared on a Scottish maths paper, has been making the internet rounds . The first two parts require students to interpret the meaning of the components of the formula \(T(x) = 5 \sqrt{36+x^2} + 4(20-x) \), and the final "challenge" component involves finding the minimum of \( T(x) \) over \( 0 \leq x \leq 20 \). Usually this would require a differentiation, but…
When you start doing more advanced sports analytics you'll eventually starting working with what are known as hierarchical, nested or mixed effects models . These are models that contain both fixed and random effects . There are multiple ways of defining fixed vs random random effects , but one way I find particularly useful is that random effects are being "predicted" rather than "estimated", and…
Elo's rating system became famous from its use in chess, but it and variations are now used in sports like the NFL to eSports like League of Legends . It also was infamously used on various "Hot or Not" type websites, as shown in this scene from the movie "Social Network": Of course, there's a mistake in the formula in the movie! What is the Elo rating system? As originally proposed, it presumes…
One of the simplest and most common power ranking models is known as the Bradley-Terry-Luce model , which is equivalent to other famous models such the logistic model and the Elo rating system . I'll be referring to "teams" here, but of course the same ideas apply to any two-participant game. Let me clarify what I mean when I use the term "power ranking". A power ranking supplies not only a…
There's lot of confusion about how best to get started doing baseball analysis. It doesn't have to be difficult! You can start doing it right away, even if you don't know anything about R, Python, Ruby, SQL or machine learning (most GMs can't code). Learning these and other tools makes it easier and faster to do analysis, but they're only part of the process of constructing a well-reasoned…
Some potentially useful SQL resources - explanations, visualizations, exercises, games, classes. A Visual Explanation of SQL Joins Datamonkey Introduction to Database Management Systems SQL Island Adventure Game PostgreSQL Exercises Public Affairs Data Journalism SQL Teaching's GitHub repo (if you're curious) Stanford's Self-Paced Database MOOC Hackr.io's SQL Section (good to check occasionally)…
During a recent chat with basketball analyst Seth Partnow , he mentioned a topic that came up during a discussion at the recent MIT Sloan Sports Analytics Conference . Who controls the pace of a game more, the offense or defense? And what is the percentage of pace responsibility for each side? The analysts came up with a rough consensus opinion, but is there a way to answer this question…