The Correct Birth/Death Jacobian for Mixture Models

Nick FotiUncategorizedLeave a Comment

Reversible jump Markov Chain Monte Carlo (RJMCMC) [1] is an extension of the Metropolis-Hastings algorithm that allows sampling from a distribution over models with potentially different numbers of parameters. In this post we are interested in determining the number of components to use when modeling data with a mixture model. The number of components corresponds to the dimension of the space we are walking through. The point of this post is to clear up a common error seen in the literature involving computing a Jacobian that arises in the algorithm.

The Fundamental Matrix of a Finite Markov Chain

Nick FotiProbability1 Comment

The purpose of this post is to present the very basics of potential theory for finite Markov chains. This post is by no means a complete presentation but rather aims to show that there are intuitive finite analogs of the potential kernels that arise when studying Markov chains on general state spaces. By presenting a piece of potential theory for Markov chains without the complications of measure theory I hope the reader will be able to appreciate the big picture of the general theory.

The Natural Gradient

Nick FotiComputation, Machine Learning, StatisticsLeave a Comment

A common activity in statistics and machine learning is optimization. For instance, finding maximum likelihood and maximum a posteriori estimates require maximizing the likilihood function and posterior distribution respectively. Another example, and the motivating example for this post, is using variational inference to approximate a posterior distribution. Suppose we are interested in a posterior distribution, , that we cannot compute analytically. We will approximate with the variational distribution that is parameterized by the variational parameters . Variational inference then proceeds to minimize the KL divergence from to , . The dominant assumption in machine learning for the form of is a product distribution, that is (where we assume there are variational parameters). It can be shown that minimizing is equivalent … Read More

Bayesian nonparametrics in the real world

Nick FotiMachine Learning, StatisticsLeave a Comment

Bayesian nonparametrics allow the contruction of statistical models whose complexity is determined by the observed data. This is accomplished by specifying priors over infinite dimensional distributions. The most widely used Bayesian nonparametric priors in machine learning are the Dirichlet process, the beta process and their corresponding marginal processes the Chinese restaurant process and the Indian buffet process respectively. The Dirichlet process provides a prior for the mixing measure of an infinite mixture model and the beta process can be used as a prior for feature popularity in a latent feature model. The hierarchical Dirichlet process (HDP) also appears frequently in machine learning underlying topic models¬†with an infinite number of topics. A main selling point of Bayesian nonparametrics has been that … Read More