The Expectation-Maximization Algorithm

This post is going to talk about a widely used method to find the maximum likelihood (MLE) or maximum a posteriori (MAP) estimate of parameters in latent variable models called the Expectation-Maximization algorithm. You have probably heard about the most famous variant of this algorithm called the k-means algorithm for clustering. Even though it's so ubiquitous, whenever I've tried to understand why this algorithm works, I never quite got the intuition right. Now that I've taken the time to work through the math, I'm going to attempt to explain the algorithm hopefully with a bit more clarity. We'll start by going back to the basics with latent variable models and the likelihood functions, then moving on to showing the math with a simple Gaussian mixture model [1].

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A Probabilistic Interpretation of Regularization

This post is going to look at a probabilistic (Bayesian) interpretation of regularization. We'll take a look at both L1 and L2 regularization in the context of ordinary linear regression. The discussion will start off with a quick introduction to regularization, followed by a back-to-basics explanation starting with the maximum likelihood estimate (MLE), then on to the maximum a posteriori estimate (MAP), and finally playing around with priors to end up with L1 and L2 regularization.

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Beyond Collaborative Filtering

I wrote a couple of posts about some of the work on recommendation systems and collaborative filtering that we're doing at my job as a Data Scientist at Rubikloud:

Here's a blurb:

Here at Rubikloud, a big focus of our data science team is empowering retailers in delivering personalized one-to-one communications with their customers. A big aspect of personalization is recommending products and services that are tailored to a customer’s wants and needs. Naturally, recommendation systems are an active research area in machine learning with practical large scale deployments from companies such as Netflix and Spotify. In Part 1 of this series, I’ll describe the unique challenges that we have faced in building a retail specific product recommendation system and outline one of the main components of our recommendation system: a collaborative filtering algorithm. In Part 2, I’ll follow up with several useful applications of collaborative filtering and end by highlighting some of its limitations.

Hope you like it!

A Probabilistic View of Linear Regression

One thing that I always disliked about introductory material to linear regression is how randomness is explained. The explanations always seemed unintuitive because, as I have frequently seen it, they appear as an after thought rather than the central focus of the model. In this post, I'm going to try to take another approach to building an ordinary linear regression model starting from a probabilistic point of view (which is pretty much just a Bayesian view). After the general idea is established, I'll modify the model a bit and end up with a Poisson regression using the exact same principles showing how generalized linear models aren't any more complicated. Hopefully, this will help explain the "randomness" in linear regression in a more intuitive way.

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Normal Approximation to the Posterior Distribution

In this post, I'm going to write about how the ever versatile normal distribution can be used to approximate a Bayesian posterior distribution. Unlike some other normal approximations, this is not a direct application of the central limit theorem. The result has a straight forward proof using Laplace's Method whose main ideas I will attempt to present. I'll also simulate a simple scenario to see how it works in practice.

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I'm Brian Keng, a former academic, current data scientist and engineer. This is the place where I write about all things technical.

Twitter: @bjlkeng


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