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skip to main content search this blog a place where i turn my thoughts into words and share them with the world. home more… posts representing graphical model factors in numpy get link facebook twitter pinterest google+ email other apps february 02, 2018 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 can… 1 comment keep reading a streamlined workflow for scientific reproduciblility and efficiency get link facebook twitter pinterest google+ email other apps may 05, 2017 as an undergraduate researcher, i was often unnecessarily bogged down by my poor workflow. i consistently programmed "one off" scripts which sometimes turned into an important component of my research, and i was frequently frustrated by the way things were difficult to use. my workflow improved a little when i got to grad school, but i finally decided to come up with a workflow that i could be happy with early this semester. after collaborating with another student on a project and having a really hard time to integrate into his code, i decided i needed to come up with a good set of practices for my own sanity and for the sake of future collaborators as well. while there is an initial upfront cost of using good programming principles, the long term benefits will usually outweigh the initial costs by a lot. in this blog post, i'll summarize some key pieces to my new workflow. if you are struggling with some of the problems that i mentioned then i hope you will find … 21 comments keep reading kronecker product for matrices with special structure using linearoperators get link facebook twitter pinterest google+ email other apps february 27, 2017 i have been working with very large matrices for my research lately, and some of them are sparse, while others are dense but have some special structure that can be exploited to represent them efficiently. i have also been constructing enormous matrices from these smaller building blocks using the kronecker product. both numpy and scipy have functions built in to compute the kronecker product between two matrices, but they need the input matrices to be explicitly represented (as a dense numpy array or a sparse scipy matrix), and so they are not compatible with the special-structure matrices that i've been dealing with. i recently solved this problem, and that is the topic of this blog post. an example as an example to make things more concrete, consider the two matrices shown below: $$ a = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad b = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 &… 2 comments keep reading modeling and predicting the outcome of a sequence of games get link facebook twitter pinterest google+ email other apps february 18, 2017 here's something i though of today that i think is pretty interesting. consider the following toy problem: suppose alice and bob play a sequence of games, and the probability that alice wins game $i$ is given by $p_i$. further, assume $p_1 \sim u[0,1]$ is uniform and that $ p_{i+1} \sim beta(\alpha p_i + \beta, \alpha (1-p_i) + \beta) $ for known $ (\alpha, \beta) $. we are unable to directly observe $p_i$, but we do observe the outcome of each game $ o_i $ for $ 1 \leq i \leq k $. the task is to estimate $ p_{k+1} $ given the evidence $ o_1, \dots, o_k $. the figure below shows how alice's win probability might evolve over time. this particular figure was generated with $ \alpha = 1000 $ and $ \beta = 5 $. we can model this situation with a modified hidden markov model. the graph below shows how the random variables are related in the form of a directed graphical model: there are very efficient algorithms for doing inference and learning on hidden markov models, but t… 3 comments keep reading efficiently remove duplicate rows from a 2d numpy array get link facebook twitter pinterest google+ email other apps january 22, 2017 suppose you have a 2d numpy array and you want to remove duplicate rows (or columns). in this blog post, i'll show you a trick you can use to do this more efficiently than other proposed solutions. this algorithm is $ o(\max(n \log{n}, n m)) $ for an $ n \times m $ matrix, and works almost surely . by "almost surely" i mean that it is a randomized algorithm that works correctly with probability $1$. to find the unique rows of a matrix $a$, the algorithm works by generating a random vector $x$ of real numbers, computing the dot product $ y = a x $, then analyzing the unique elements of $y$. the indices of unique elements in $y$ is the same as the unique row indices in $a$ with probability $1$. below is an example to demonstrate how it works: > import numpy as np > a = np.random.randint(low=0, high=3, size=(10, 2)) > a array([[2, 1], [1, 2], [0, 0], [2, 2], [1, 2], [0, 0], [0, 2], [2, 1], [1, 0], [… 2 comments keep reading context-dependent pitch prediction with neural networks get link facebook twitter pinterest google+ email other apps january 02, 2017 this semester i took the machine learning class at umass, and for my final project i developed models for predicting characteristics of pitches based on the context that they were thrown in. the context consists of relevant and available information known before the pitch is thrown, such as the identities of the batter and pitcher, the batter height and stance, the pitcher handedness, the current count, and many others. in particular, i looked into modeling the distribution over pitch types and locations. this problem is challenging primarily because for a particular context (a specific setting of the different features that make up the context), there is usually very little data available for pitches thrown in that context. in general, for each context feature we include, we have to split up our data into $k$, groups where $k$ is the number of values that the context feature can take on (for batter stance it is $k=2$, for count it is $k=12$, etc). thus, developing models by part… 2 comments keep reading an interesting game get link facebook twitter pinterest google+ email other apps december 26, 2016 today i'm going to talk about an interesting game that i thought of, and in particular i'll show the probability that each player will win. while the rules of the game are not fair, the skill levels of each player are not necessarily the same either, so i formally define what i mean and find the conditions under which these factors balance out. without further ado, here is the problem statement: alice and bob play a game consisting of an indeterminate number of rounds. in each round, alice and bob compete to win the round, and thus score a point. alice is more skilled at this game and so she wins each round with probability $ p > 0.5 $ (independent of the score and other context), meaning bob wins each round with probability $ 1 - p $. the score starts at $ 0 - 0 $ and bob wins if he ever has more points than alice, and alice wins otherwise. what is the probability that bob wins this game? for what value of $ p $

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