I created this site to write about things that I find interesting: probability & Bayesian inference, data visualization, puzzles & games, finance, and books.
This week's Riddler pits the army of the dead vs. the army of the living. As the two armies battle, any fallen soldiers from the living army rise to fight with the dead. How many soldiers would each side need to make it a fair fight?
The Riddler this week asks us about random points on the edge of a circle. Specifically, if we generate $n$ random points around the circumference of a circle, how likely are those points to fall on only one side?
Another weekly Riddler, this time with both an analytical and simulated solution!
I have a distinct memory of participating in my elementary school's spelling bee when I was in second grade. I was the unlikely runner-up, even though I was competing against children in third and fourth grade. What was the secret to my overperformance? Not my natural spelling ability, but rather the rules of the game - a participant is eliminated from the spelling bee after failing to spell a word correctly, which means that going last is an advantage. I was lucky enough to be the near the tail-end of the participants in my spelling bee, which surely improved my final ranking. This week's Riddler asks us to quantify that advantage.
This week's riddler asks us to simulate a game of baseball using rolls of a dice. To solve this problem, we're going to treat the game of Baseball like a markov chain. Under the simplified dice framework, we identify various states of the game, a set of transition probabilities to subsequent states, and associated payoffs (runs scored) when certain states are reached as a result of game events. Using this paradigm, we can simulate innings probabilistically, count the runs scored by each team, and determine the winner.
The Riddler Express this week asks us about collecting sets of cards. In particular, we're interested in collecting a complete set of 144 unique cards. We purchase one random card at a time for $5 each. How many purchases should we expect to make - and how much money should we expect to spend - in order to collect at least one of every card?
I've learned that there are many automatic differentiation libraries in the Python ecosystem. Often these libraries are also machine learning libraries, where automatic differentiation serves as a means to an end - for example in optimizing model parameters in a neural network. However, the autograd library might be one of the purest, "simplest" (relatively speaking) options out there. Its goal is to efficiently compute deriviatives of numpy code, and its API is as close to numpy as possible. This means it's easy to get started right away if you're comfortable using numpy. In particular autograd claims to be able to differentiate as many times as one likes, and I thought a great way to test this would be to apply the Taylor Series approximation to some interesting functions.
As a follow up to my prior article on Black-Scholes in PyTorch, I wanted to explore more complex applications of automatic differentiation. As I showed before, automatic differentiation can be used to calculate the sensitivities, or "greeks", of a stock option, even if we use monte carlo techniques to calculate option price. Many exotic options can only be priced using monte carlo techniques, so automatic differentiation may be able to provide more accurate sensitivities in less time than traditional methods.
I've been experimenting with several machine learning frameworks lately, including Tensorflow, PyTorch, and Chainer. I'm fascinated by the concept of automatic differentiation. It's incredible to me that these libraries can calculate millions of partial derivatives of virtually any function with only one extra pass through the code. Automatic differentiation is critical for deep learning models, but I wanted to see how it could be applied to value financial derivatives.
I wrote yesterday about tracking my steps with a Garmin watch. Perhaps to keep me motivated and active, Garmin provides a daily step goal that moves up or down based on my activity. I've always been curious about how this algorithm works, but I couldn't find any resources that described it. Let's see if I can reverse engineer it instead.