Ryan's Repository of Random Reflections

In this post, I will discuss my findings in terms of an optimal strategy for Farkle , which is a dice game of chance.  If you are unfamiliar with the game, I encourage you to skim the linked article so that you can better understand this blog post.  All of my findings are based on sound mathematical methods and a computer program was used to determine the optimal strategy.  Before I begin, let me first state the value of different dice rolls I assumed while developing the strategy. Each 1: 100 points Each 5: 50 points 3 1's: 1000 points 3 2's: 200 points ... 3 6's: 600 points 4 1's: 2000 points ... 4 6's 1200 points 5 1's 3000 points ... 5 6's 1800 points Straight 1-6 1500 points 3 Pairs (e.g., 22 33 44): 750 points Everything else is considered to be a Farkle (0 points). There are many variants to Farkle, but I like to play by this set of rules. The "game state" of Farkle roll can roughly be characterized by 2 values: the curre

In this blog post, I am going to talk about writing efficient numpy code in python. I do a good amount of numerical linear algebra for my research and personal projects, and I typically code in python with numpy (and spicy) because it is very easy to use and it is also very efficient when used correctly. Consider the following problem, which will serve as a concrete task to use as an example throughout this post. Suppose we have a (column) vector $x$ of length $n = 2^k$ and we want to compute $H_k x$ where $H_k$ is a “hierarchical” matrix with branching factor $2$, defined by $$ \begin{align*} H_0 &= \begin{bmatrix} 1 \end{bmatrix} \\ H_{k+1} &= \begin{bmatrix} 1 & 1 \\ H_k & 0 \\ 0 & H_k \end{bmatrix} \end{align*} $$ Where the top row is a vector of all ones and 0 denotes a matrix of zeros having the same size as $H_k$. For example, $H_2$ is a $7 \times 4$ matrix that looks like this: $$ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0

I've been working a bit with graphical models lately for my research.  For a while I was using a library called pgmpy for their implementations of factor arithmetic and inference algorithms.  However, as my research is progressing I am needing more control than what pgmpy offers, so I decided to re-implement and extend the algorithms that I needed.  If you are in a similar situation and are finding yourself implementing your own graphical models algorithms, this post is for you. In the setting I am working in, I have a Markov random field model where there are only a small number of variables (no more than a few dozen, often much less).  However, each variable can take on a possibly large number of categorical values and the graph may contain relatively large cliques that make exact inference computationally challenging (although still possible). Suppose the model has $d$ variables and variable $i$ can take on $n_i$ possible values.  Usually a factor defined over $k$ variables