Writing

pag- -
Summaries

Summaries

DJ Rich

This briefly summarizes each part of the series on Probabilistic Graphical Models.

Notation Guide

Notation Guide

DJ Rich

Notation can be confusing. This post is to address it directly.

Part 7: Structure Learning

Part 7: Structure Learning

DJ Rich

Structure learning precedes parameter learning, whereby a graph or similarly abstract structure must be learnd from data. Doing so presents a formidable integration, but with techniques and approximations, a fruitful search over structures can be performed. For theoretical reasons, the task is considerably easier in the Bayesian Network case than in the alternative.

Part 6: Learning Parameters of a Markov Network

Part 6: Learning Parameters of a Markov Network

DJ Rich

The theory of Markov Network parameter learning is intuitive and instructive, but it exposes an intractable normalizer, forbidding the task from reducing to easier ones. Ultimately, the task is hard.

Part 4: Monte Carlo Methods

Part 4: Monte Carlo Methods

DJ Rich

Monte Carlo methods answer the inference task with a set of samples, sampled approximately from the target distribution. In total, they provide a supremely general toolset. However, to use them requires a skill for managing complexities of distributional convergence and autocorrelation.

Part 3: Variational Inference

Part 3: Variational Inference

DJ Rich

Variational Inference, a category of approximate inference algorithms, achieves efficiency by restricting inference to a computationally friendly set of distributions. Using tools from information theory, we may find the distribution that best approximates results of exact inference.

Part 2: Exact Inference

Part 2: Exact Inference

DJ Rich

Given a Probabilistic Graphical Model, exact inference algorithms exploit factorization and caching to answer questions about the system it represents.

Part 1: An Introduction to Probabilistic Graphical Models

Part 1: An Introduction to Probabilistic Graphical Models

DJ Rich

Probabilistic Graphical Models are born from a remarkable synthesis of probability theory and graph theory. They are among our most powerful tools for managing nature's baffling mixture of uncertainty and complexity.

Bias-Variance Trade-Off

Bias-Variance Trade-Off

DJ Rich

The bias-variance trade-off is a rare insight into the challenge of generalization.

Information Theory and Entropy

Information Theory and Entropy

DJ Rich

Entropy and its related concepts quantify the otherwise abstract concept of information. A tour reveals its relationship to information, binary encodings and uncertainty. Most intuitively, we're left with a simple analogy to 2D areas.

Generalized Linear Models

Generalized Linear Models

DJ Rich

A Generalized Linear Model, if viewed without knowledge of their motivation, can be a confusing tool. It's easier to understand if seen as a two knob generalization of linear regression.

Jensen's Inequality

Jensen's Inequality

DJ Rich

A visual makes Jensen's Inequality obvious and intuitive.

The Fisher Information

The Fisher Information

DJ Rich

The Fisher Information quantifies the information an observation carries for a parameter. The quantification becomes intuitive once we see it measuring a certain geometric quality.

The Copula and 2008

The Copula and 2008

DJ Rich

The copula provides a clever means for mixing and matching a set of marginal distributions with the joint-only mechanism of a joint distribution. However, its elegance and utility have been a dangerous lure.

The Matrix Inversion Lemma

The Matrix Inversion Lemma

DJ Rich

The Matrix Inversion Lemma looks intimidating, but it is easy to know when it applies. Doing so offers considerable computational speed ups.

Bayesian Optimization

Bayesian Optimization

DJ Rich

When optimizing a slow-to-evaluate and non-differentiable function, one may think random sampling is the only option--a naive approach likely to disappoint. However, Bayesian optimization, a clever exploit of the function assumed smoothness, disconfirms these intuitions.

The Fundamental Law of Active Management

The Fundamental Law of Active Management

DJ Rich

The Fundamental Law of Active Management decomposes a well known summarizing metric of an investment strategy. The decomposition yields two dimensions along which all strategies may be judged and suggests avenues for improvement.

The Exponential Family

The Exponential Family

DJ Rich

The exponential family is a generalization of distributions, inclusive of many familiar ones plus a universe of others. The general form brings elegant properties, illuminating all distributions within. In this post, we discuss what it is, how it applies and some of its properties.

The Trace as a Measure of Complexity

The Trace as a Measure of Complexity

DJ Rich

For a class of models, the trace provides a measure of model complexity that's useful for managing the bias variance trade-off.

The Conjugate Gradient Method

The Conjugate Gradient Method

DJ Rich

Utilizing a geometric perspective, we find an efficient algorithm for solving a special kind of system of equations.