arXiv subjects

Bryant Davis

Publications and source records attributed to Bryant Davis.

3 recordsLinked to original sources

arXiv2021

Approximating the Spectral Gap of the P\'olya-Gamma Gibbs Sampler

The self-adjoint, positive Markov operator defined by the P\'olya-Gamma Gibbs sampler (under a proper normal prior) is shown to be trace-class, which implies that all non-zero elements of its spectrum are eigenvalues. Consequently, the spectral gap is $1-\lambda_*$, where $\lambda_* \in [0,1)$ is the second largest eigenvalue. A method of constructing an asymptotically valid confidence interval for an upper bound on $\lambda_*$ is developed by adapting the classical Monte Carlo technique of Qin et al. (2019) to the P\'olya-Gamma Gibbs sampler. The results are illustrated using the German credit data. It is also shown that, in general, uniform ergodicity does not imply the trace-class property, nor does the trace-class property imply uniform ergodicity.

math.ST↗

arXiv2020

On the convergence complexity of Gibbs samplers for a family of simple Bayesian random effects models

The emergence of big data has led to so-called convergence complexity analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms behave as the sample size, $n$, and/or the number of parameters, $p$, in the underlying data set increase. This type of analysis is often quite challenging, in part because existing results for fixed $n$ and $p$ are simply not sharp enough to yield good asymptotic results. One of the first convergence complexity results for an MCMC algorithm on a continuous state space is due to Yang and Rosenthal (2019), who established a mixing time result for a Gibbs sampler (for a simple Bayesian random effects model) that was introduced and studied by Rosenthal (1996). The asymptotic behavior of the spectral gap of this Gibbs sampler is, however, still unknown. We use a recently developed simulation technique (Qin et. al., 2019) to provide substantial numerical evidence that the gap is bounded away from 0 as $n \rightarrow \infty$. We also establish a pair of rigorous convergence complexity results for two different Gibbs samplers associated with a generalization of the random effects model considered by Rosenthal (1996). Our results show that, under strong regularity conditions, the spectral gaps of these Gibbs samplers converge to 1 as the sample size increases.

math.ST↗

arXiv2015

The density of primes dividing a term in the Somos-5 sequence

The Somos-5 sequence is defined by $a_{0} = a_{1} = a_{2} = a_{3} = a_{4} = 1$ and $a_{m} = \frac{a_{m-1} a_{m-4} + a_{m-2} a_{m-3}}{a_{m-5}}$ for $m \geq 5$. We relate the arithmetic of the Somos-5 sequence to the elliptic curve $E : y^{2} + xy = x^{3} + x^{2} - 2x$ and use properties of Galois representations attached to $E$ to prove the density of primes $p$ dividing some term in the Somos-5 sequence is equal to $\frac{5087}{10752}$.

math.NT↗