The Correct Birth/Death Jacobian for Mixture Models

[latexpage] Reversible jump Markov Chain Monte Carlo (RJMCMC)  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 Foti Probability 1 Comment

[latexpage] 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.

[latexpage] 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, $p$, that we cannot compute analytically. We will approximate $p$ with the variational distribution $q(\phi)$ that is parameterized by the variational parameters $\phi$. Variational inference then proceeds to minimize the KL divergence from $q$ to $p$, $KL(q||p)$. The dominant assumption in machine learning for the form of $q$ is a product distribution, that is $q = \prod_k q_k(\phi_k)$ (where we assume there are $K$ …