scholarly journals Convergence rates of the random scan Gibbs sampler under the Dobrushin’s uniqueness condition

2017 ◽  
Vol 22 (0) ◽  
Author(s):  
Neng-Yi Wang
Keyword(s):  
Gibbs Sampler ◽  
Convergence Rates ◽  
Uniqueness Condition

scholarly journals Scalable inference for crossed random effects models

Biometrika ◽  
2019 ◽  
Author(s):  
O Papaspiliopoulos ◽  
G O Roberts ◽  
G Zanella
Keyword(s):  
Random Effects ◽  
Gibbs Sampler ◽  
Convergence Rates ◽  
Simple Modification ◽  
Random Effects Models ◽  
Monte Carlo Algorithms ◽  
Alternative State ◽  
Crossed Random Effects ◽  
Scalable Inference ◽  
Modern Analysis

Summary We develop methodology and complexity theory for Markov chain Monte Carlo algorithms used in inference for crossed random effects models in modern analysis of variance. We consider a plain Gibbs sampler and propose a simple modification, referred to as a collapsed Gibbs sampler. Under some balancedness conditions on the data designs and assuming that precision hyperparameters are known, we demonstrate that the plain Gibbs sampler is not scalable, in the sense that its complexity is worse than proportional to the number of parameters and data, but the collapsed Gibbs sampler is scalable. In simulated and real datasets we show that the explicit convergence rates predicted by our theory closely match the computable, but nonexplicit rates in cases where the design assumptions are violated. We also show empirically that the collapsed Gibbs sampler extended to sample precision hyperparameters significantly outperforms alternative state-of-the-art algorithms.

scholarly journals Convergence rates for a hierarchical Gibbs sampler

Bernoulli ◽  
2017 ◽  
Vol 23 (1) ◽  
pp. 603-625
Author(s):  
Oliver Jovanovski ◽  
Neal Madras
Keyword(s):  
Gibbs Sampler ◽  
Convergence Rates

scholarly journals Boundary layer solutions to singularly perturbed quasilinear systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Valentin Butuzov ◽  
Nikolay Nefedov ◽  
Oleh Omel'chenko ◽  
Lutz Recke
Keyword(s):  
Boundary Layer ◽  
Convergence Rates ◽  
Singularly Perturbed ◽  
Uniqueness Condition ◽  
Scalar Case ◽  
Monotonicity Condition ◽  
Asymptotic Convergence ◽  
Local Uniqueness ◽  
Quasilinear Systems ◽  
Ode Systems

<p style='text-indent:20px;'>We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}</tex-math></inline-formula>. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.</p>

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