arXiv subjects

Nicholas G. Tawn

Publications and source records attributed to Nicholas G. Tawn.

5 recordsLinked to original sources

arXiv2026

Annealed Leap-Point Sampler for Multimodal Target Distributions

In Bayesian statistics, exploring high-dimensional multimodal posterior distributions poses major challenges for existing MCMC approaches. This paper introduces the Annealed Leap-Point Sampler (ALPS), which augments the target distribution state space with modified annealed (cooled) distributions, in contrast to traditional tempering approaches. The coldest state is chosen such that its annealed density is well-approximated locally by a Laplace approximation. This allows for automated setup of a scalable mode-leaping independence sampler. ALPS requires an exploration component to search for the mode locations, which can either be run adaptively in parallel to improve these mode-jumping proposals, or else as a pre-computation step. A theoretical analysis shows that for a d-dimensional problem the coolest temperature level required only needs to be linear in dimension, $\mathcal{O}\left(d\right)$, implying that the number of iterations needed for ALPS to converge is $\mathcal{O}\left(d\right)$ (typically leading to overall complexity $\mathcal{O}\left(d^3\right)$ when computational cost per iteration is taken into account). ALPS is illustrated on several complex, multimodal distributions that arise from real-world applications. This includes a seemingly-unrelated regression (SUR) model of longitudinal data from U.S. manufacturing firms, as well as a spectral density model that is used in analytical chemistry for identification of molecular biomarkers.

stat.ME↗

arXiv2021

Skew Brownian Motion and Complexity of the ALPS Algorithm

Simulated tempering is a popular method of allowing MCMC algorithms to move between modes of a multimodal target density π. The paper [24] introduced the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper, we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that under appropriate assumptions, the ALPS algorithm mixes in time O(d[log(d)]^2 ) or O(d), depending on which version is used.

math.PR↗

arXiv2019

Weight-Preserving Simulated Tempering

Simulated tempering is popular method of allowing MCMC algorithms to move between modes of a multimodal target density π. One problem with simulated tempering for multimodal targets is that the weights of the various modes change for different inverse-temperature values, sometimes dramatically so. In this paper, we provide a fix to overcome this problem, by adjusting the mode weights to be preserved (i.e., constant) over different inverse-temperature settings. We then apply simulated tempering algorithms to multimodal targets using our mode weight correction. We present simulations in which our weight-preserving algorithm mixes between modes much more successfully than traditional tempering algorithms. We also prove a diffusion limit for an version of our algorithm, which shows that under appropriate assumptions, our algorithm mixes in time O(d [log d]^2).

stat.CO↗

arXiv2018

Optimal Temperature Spacing for Regionally Weight-preserving Tempering

Parallel tempering is popular method for allowing MCMC algorithms to properly explore a $d$-dimensional multimodal target density. One problem with traditional power-based parallel tempering for multimodal targets is that the proportion of probability mass associated to the modes can change for different inverse-temperature values, sometimes dramatically so. Complementary work by the authors proposes a novel solution involving auxiliary targets that preserve regional weight upon powering up the density. This paper attempts to address the question of how to choose the temperature spacings in an optimal way when using this type of weight-preserving approach.The problem is analysed in a tractable setting for computation of the expected squared jumping distance which can then be optimised with regards to a tuning parameter. The conclusion is that for an appropriately constructed regionally weight-preserved tempering algorithm targeting a $d$-dimensional target distribution, the consecutive temperature spacings should behave as $\mathcal{O}\left(d^{-1/2}\right)$ and this induces an optimal acceptance rate for temperature swap moves that lies in the interval $[0,0.234]$.

stat.CO↗

arXiv2018

Accelerating Parallel Tempering: Quantile Tempering Algorithm (QuanTA)

Using MCMC to sample from a target distribution, $π(x)$ on a $d$-dimensional state space can be a difficult and computationally expensive problem. Particularly when the target exhibits multimodality, then the traditional methods can fail to explore the entire state space and this results in a bias sample output. Methods to overcome this issue include the parallel tempering algorithm which utilises an augmented state space approach to help the Markov chain traverse regions of low probability density and reach other modes. This method suffers from the curse of dimensionality which dramatically slows the transfer of mixing information from the auxiliary targets to the target of interest as $d \rightarrow \infty$. This paper introduces a novel prototype algorithm, QuanTA, that uses a Gaussian motivated transformation in an attempt to accelerate the mixing through the temperature schedule of a parallel tempering algorithm. This new algorithm is accompanied by a comprehensive theoretical analysis quantifying the improved efficiency and scalability of the approach; concluding that under weak regularity conditions the new approach gives accelerated mixing through the temperature schedule. Empirical evidence of the effectiveness of this new algorithm is illustrated on canonical examples.

stat.ME↗