Math
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Jensen's inequality and probability - September 01, 2019
Jensen's inequality is a beautiful little workhorse lemma from convex geometry. I have stumbled into it an number of times recently as I have been reading and writing about statistics and information theory, so I decided to split it off into its own blog post. After introducing the inequality (and proving that it holds), I will give a few snappy applications to basic statistics, including the classical inequality between arithmetic, geometric, and harmonic means as well as a basic inequality relating the mean, median, and standard deviation.
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The Radon-Nikodym Theorem - May 27, 2019
The Radon-Nikodym theorem is a workhorse result in measure theory, with numerous applications to probability dynamics (such as the existence of conditional expectations and the existence of KL-divergence). I will give a simple proof using Hilbert spaces.
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Gibbs' Inequality - May 27, 2019
It is a fairly standard fact that relative entropy (KL-divergence) is positive definite, but I was unsatisfied with the proofs of this fact that I saw when I glanced through the literature. In this post I will provide a complete proof which works on a general probability space.
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Decimals are Hard - April 14, 2018
Many people find fractions confusing and difficult, and there is a tendency to dismiss them in favor of decimals in daily life. But answering even basic questions about decimals requires confronting serious philosophical questions and some unsolved problems in mathematics.
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Monads - September 13, 2017
Those who have some experience with adjunctions have probably already encountered monads and comonads, if not by name - they are endofunctors which measure how far off an adjunction is from being an equivalence. In this post we shall go through lots of examples and a little bit of theory; there isn't much computer science in this post, except possibly for some exercises on graphs at the end.
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Galois Connections - September 02, 2017
A Galois connection is an adjunction between partially ordered sets (regarded as categories); they arise naturally all over the place in mathematics, and more recently they have found interesting applications to computer science. This post covers the basic theory and explores some examples from graph theory and programming.
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Adjunctions - September 02, 2017
Adjunctions are among the most fundamental constructions in category theory, and in mathematics more broadly. In this post we shall go carefully through the definition and some basic examples. There isn't much computer science specifically in this post, save for a few remarks about the category of graphs at the end, but adjunctions will make crucial appearances in future posts.
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The Category of Graphs - May 03, 2017
This post explores graphs from a categorical perspective. The focus is on introducing some of the key ideas of category theory in a familiar setting; in future posts we may see how category can clarify some difficult constructions in graph theory.
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Category Theory References - April 28, 2017
This post contains a list of references for learning the basics of category theory and its applications to computer science.
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What's in a Mean? - March 11, 2017
The concept of an average is a bit more complicated than what we learn in school.
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Higher Homotopy Groups via Vector Calculus - December 09, 2014
Calculating higher homotopy groups of spheres is infamously hard. Here is an example which uses only undergraduate vector calculus.
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The Alternating Group is Simple III - April 19, 2014
A detailed proof (by induction) that $A_n$ is simple for $n > 5$.
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The Alternating Group is Simple II - April 19, 2014
A detailed proof that $A_5$ is simple.
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The Alternating Group is Simple I - April 19, 2014
They may be easy to define, but the alternating groups are key players in modern algebra.
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The Geometry and Statistics of Noodles - January 07, 2013
Buffon's needle experiment, a statistical technique for calculating $\pi$, hints at a deep interplay between geometry and probability.