arXiv subjects

Yeun Ji Jung

Publications and source records attributed to Yeun Ji Jung.

2 recordsLinked to original sources

arXiv2019

A Hybrid Scan Gibbs Sampler for Bayesian Models with Latent Variables

Gibbs sampling is a widely popular Markov chain Monte Carlo algorithm that can be used to analyze intractable posterior distributions associated with Bayesian hierarchical models. There are two standard versions of the Gibbs sampler: The systematic scan (SS) version, where all variables are updated at each iteration, and the random scan (RS) version, where a single, randomly selected variable is updated at each iteration. The literature comparing the theoretical properties of SS and RS Gibbs samplers is reviewed, and an alternative hybrid scan Gibbs sampler is introduced, which is particularly well suited to Bayesian models with latent variables. The word "hybrid" reflects the fact that the scan used within this algorithm has both systematic and random elements. Indeed, at each iteration, one updates the entire set of latent variables, along with a randomly chosen block of the remaining variables. The hybrid scan (HS) Gibbs sampler has important advantages over the two standard scan Gibbs samplers. Firstly, the HS algorithm is often easier to analyze from a theoretical standpoint. In particular, it can be much easier to establish the geometric ergodicity of a HS Gibbs Markov chain than to do the same for the corresponding SS and RS versions. Secondly, the sandwich methodology developed in Hobert and Marchev (2008), which is also reviewed, can be applied to the HS Gibbs algorithm (but not to the standard scan Gibbs samplers). It is shown that, under weak regularity conditions, adding sandwich steps to the HS Gibbs sampler always results in a theoretically superior algorithm. Three specific Bayesian hierarchical models of varying complexity are used to illustrate the results. One is a simple location-scale model for data from the Student's $t$ distribution, which is used as a pedagogical tool. The other two are sophisticated, yet practical Bayesian regression models.

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arXiv2016

Convergence Analysis of the Data Augmentation Algorithm for Bayesian Linear Regression with Non-Gaussian Errors

Gaussian errors are sometimes inappropriate in a multivariate linear regression setting because, for example, the data contain outliers. In such situations, it is often assumed that the error density is a scale mixture of multivariate normal densities that takes the form $f(\varepsilon) = \int_0^\infty |Σ|^{-\frac{1}{2}} u^{\frac{d}{2}} \, ϕ_d \big( Σ^{-\frac{1}{2}} \sqrt{u} \, \varepsilon \big) \, h(u) \, du$, where $d$ is the dimension of the response, $ϕ_d(\cdot)$ is the standard $d$-variate normal density, $Σ$ is an unknown $d \times d$ positive definite scale matrix, and $h(\cdot)$ is some fixed mixing density. Combining this alternative regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX-DA algorithm that can be used to explore this posterior. This paper provides conditions (on $h$) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo (MCMC) algorithms. These results are extremely important from a practical standpoint because geometric ergodicity guarantees the existence of the central limit theorems that form the basis of all the standard methods of calculating valid asymptotic standard errors for MCMC-based estimators. The main result is that, if $h$ converges to 0 at the origin at an appropriate rate, and $\int_0^\infty u^{\frac{d}{2}} \, h(u) \, du < \infty$, then the DA and Haar PX-DA Markov chains are both geometrically ergodic. This result is quite far-reaching. For example, it implies the geometric ergodicity of the DA and Haar PX-DA Markov chains whenever $h$ is generalized inverse Gaussian, log-normal, inverted gamma (with shape parameter larger than $d/2$), or Fréchet (with shape parameter larger than $d/2$).

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