On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

Roy, Vivekananda; Hobert, James P.

doi:10.1016/j.jmva.2009.12.015

Cited by 23 publications

(30 citation statements)

References 25 publications

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“…Particular examples include: Gibbs sampling for hierarchical random effects models in [28]; van Dyk and Meng's algorithm for multivariate Student's t model [31]; Gibbs sampling for a family of Bayesian hierarchical general linear models in [24] (c.f. also [25]); block Gibbs sampling for Bayesian random effects models with improper priors [50]; Data Augmentation algorithm for Bayesian multivariate regression models with Student's t regression errors [47].…”

Section: Applicability Of the Resultsmentioning

confidence: 99%

Rigorous confidence bounds for MCMC under a geometric drift condition

Łatuszyński

Niemiro

2011

Journal of Complexity

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We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. We use these bounds to construct fixed-width nonasymptotic confidence intervals. For a possibly unbounded function $f:\stany \to R,$ let $I=\int_{\stany} f(x) \pi(x) dx$ be the value of interest and $\hat{I}_{t,n}=(1/n)\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory $n$ and burn-in time $t$ which ensure that $$P(|\hat{I}_{t,n}-I|\leq \varepsilon)\geq 1-\alpha.$$ The bounds depend only and explicitly on drift parameters, on the $V-$norm of $f,$ where $V$ is the drift function and on precision and confidence parameters $\varepsilon, \alpha.$ Next we analyse an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing Bayesian estimators in practically relevant models. We illustrate our bounds numerically in a simple example

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Section: Applicability Of the Resultsmentioning

confidence: 99%

Rigorous confidence bounds for MCMC under a geometric drift condition

Łatuszyński

Niemiro

2011

Journal of Complexity

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“…A formal derivation of both of these algorithms for the case a = 1 is provided in Roy and Hobert (2010), and the extension to general a is trivial. Also, standard arguments show that both Markov chains are Harris ergodic; that is, irreducible, aperiodic, and Harris recurrent.…”

Section: Drawmentioning

confidence: 99%

Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors

Jung

Hobert

2014

Statistics & Probability Letters

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“…Similarly,

\frac{1}{z_{i}} X^{T} Q^{- 1} X + \frac{1}{z_{i}} A^{- 1} - x_{i} x_{i}^{T} = \sum_{j \neq i} \frac{z_{j}}{z_{i}} x_{j} x_{j}^{T} + \frac{1}{z_{i}} A^{- 1}

is also positive definite. Another application of lemma 3 from Roy and Hobert yields

z_{i} x_{i}^{T} Ω x_{i} = x_{i}^{T} {(\frac{1}{z_{i}} X^{T} Q^{- 1} X + \frac{1}{z_{i}} A^{- 1})}^{- 1} x_{i} = x_{i}^{T} {(\sum_{j = 1}^{n} \frac{z_{j}}{z_{i}} x_{j} x_{j}^{T} + \frac{1}{z_{i}} A^{- 1})}^{- 1} x_{i} \leq 1 .

Thus,

d \sum i = <...>

…”

Section: Proof Of Propositionmentioning

confidence: 98%

“…We attack using an argument similar to one found in Roy and Hobert . The innermost integral can be expressed as

E [\sum_{i = 1}^{n} y_{i}^{T} \sum^{- 1} y_{i} - 2 \sum_{i = 1}^{n} x_{i}^{T} β \sum^{- 1} y_{i} + \sum_{i = 1}^{n} x_{i}^{T} β \sum^{- 1} β^{T} x_{i} | \sum, z, y],

where

β | \sum, z, y \sim N_{p, d} (μ, Ω, \sum)

.…”

Section: Proof Of Propositionmentioning

confidence: 99%

A note on the convergence rate of MCMC for robust Bayesian multivariate linear regression with proper priors

Backlund

Hobert

2020

Comp and Math Methods

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The multivariate linear regression model with errors from a scale mixture of Gaussian densities yields a complex likelihood function. Combining this likelihood with any nontrivial prior distribution leads to a highly intractable posterior density. If a conditionally conjugate prior is used, then there is a well known and easy‐to‐implement data augmentation (DA) algorithm available for exploring the posterior. Hobert et al recently showed that, under an improper conditionally conjugate prior (and weak regularity conditions), the Markov chain that drives the DA algorithm converges at a geometric rate. Unfortunately, the model studied by Hobert et al can only be used in situations where the X matrix has full column rank. In this note, analogous convergence rate results are established for a proper conditionally conjugate prior. An important advantage of using a proper prior is that, not only is the X matrix allowed to be column rank deficient, but it can also have more columns than rows, that is, our model is applicable in cases where p>n. This is an important extension in the era of big data.

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On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

Abstract: 2016-12-24T18:13:07

Cited by 23 publications

References 25 publications

Rigorous confidence bounds for MCMC under a geometric drift condition

Rigorous confidence bounds for MCMC under a geometric drift condition

Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors

A note on the convergence rate of MCMC for robust Bayesian multivariate linear regression with proper priors

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