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Methodology (stat.ME)

Mon, 01 May 2023

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1.Scaling of Piecewise Deterministic Monte Carlo for Anisotropic Targets

Authors:Joris Bierkens, Kengo Kamatani, Gareth O. Roberts

Abstract: Piecewise deterministic Markov processes (PDMPs) are a type of continuous-time Markov process that combine deterministic flows with jumps. Recently, PDMPs have garnered attention within the Monte Carlo community as a potential alternative to traditional Markov chain Monte Carlo (MCMC) methods. The Zig-Zag sampler and the Bouncy particle sampler are commonly used examples of the PDMP methodology which have also yielded impressive theoretical properties, but little is known about their robustness to extreme dependence or isotropy of the target density. It turns out that PDMPs may suffer from poor mixing due to anisotropy and this paper investigates this effect in detail in the stylised but important Gaussian case. To this end, we employ a multi-scale analysis framework in this paper. Our results show that when the Gaussian target distribution has two scales, of order $1$ and $\epsilon$, the computational cost of the Bouncy particle sampler is of order $\epsilon^{-1}$, and the computational cost of the Zig-Zag sampler is either $\epsilon^{-1}$ or $\epsilon^{-2}$, depending on the target distribution. In comparison, the cost of the traditional MCMC methods such as RWM or MALA is of order $\epsilon^{-2}$, at least when the dimensionality of the small component is more than $1$. Therefore, there is a robustness advantage to using PDMPs in this context.

2.On ordered beta distribution and the generalized incomplete beta function

Authors:Mayad Al-Saidi, Alexey Kuznetsov, Mikhail Nediak

Abstract: Motivated by applications in Bayesian analysis we introduce a multidimensional beta distribution in an ordered simplex. We study properties of this distribution and connect them with the generalized incomplete beta function. This function is crucial in applications of multidimensional beta distribution, thus we present two efficient numerical algorithms for computing the generalized incomplete beta function, one based on Taylor series expansion and another based on Chebyshev polynomials.

3.Bayesian system identification for structures considering spatial and temporal correlation

Authors:Ioannis Koune, Arpad Rozsas, Arthur Slobbe, Alice Cicirello

Abstract: The decreasing cost and improved sensor and monitoring system technology (e.g. fiber optics and strain gauges) have led to more measurements in close proximity to each other. When using such spatially dense measurement data in Bayesian system identification strategies, the correlation in the model prediction error can become significant. The widely adopted assumption of uncorrelated Gaussian error may lead to inaccurate parameter estimation and overconfident predictions, which may lead to sub-optimal decisions. This paper addresses the challenges of performing Bayesian system identification for structures when large datasets are used, considering both spatial and temporal dependencies in the model uncertainty. We present an approach to efficiently evaluate the log-likelihood function, and we utilize nested sampling to compute the evidence for Bayesian model selection. The approach is first demonstrated on a synthetic case and then applied to a (measured) real-world steel bridge. The results show that the assumption of dependence in the model prediction uncertainties is decisively supported by the data. The proposed developments enable the use of large datasets and accounting for the dependency when performing Bayesian system identification, even when a relatively large number of uncertain parameters is inferred.

4.DIF Analysis with Unknown Groups and Anchor Items

Authors:Gabriel Wallin, Yunxiao Chen, Irini Moustaki

Abstract: Measurement invariance across items is key to the validity of instruments like a survey questionnaire or an educational test. Differential item functioning (DIF) analysis is typically conducted to assess measurement invariance at the item level. Traditional DIF analysis methods require knowing the comparison groups (reference and focal groups) and anchor items (a subset of DIF-free items). Such prior knowledge may not always be available, and psychometric methods have been proposed for DIF analysis when one piece of information is unknown. More specifically, when the comparison groups are unknown while anchor items are known, latent DIF analysis methods have been proposed that estimate the unknown groups by latent classes. When anchor items are unknown while comparison groups are known, methods have also been proposed, typically under a sparsity assumption - the number of DIF items is not too large. However, there does not exist a method for DIF analysis when both pieces of information are unknown. This paper fills the gap. In the proposed method, we model the unknown groups by latent classes and introduce item-specific DIF parameters to capture the DIF effects. Assuming the number of DIF items is relatively small, an $L_1$-regularised estimator is proposed to simultaneously identify the latent classes and the DIF items. A computationally efficient Expectation-Maximisation (EM) algorithm is developed to solve the non-smooth optimisation problem for the regularised estimator. The performance of the proposed method is evaluated by simulation studies and an application to item response data from a real-world educational test

5.An unbiased non-parametric correlation estimator in the presence of ties

Authors:Landon Hurley

Abstract: An inner-product Hilbert space formulation of the Kemeny distance is defined over the domain of all permutations with ties upon the extended real line, and results in an unbiased minimum variance (Gauss-Markov) correlation estimator upon a homogeneous i.i.d. sample. In this work, we construct and prove the necessary requirements to extend this linear topology for both Spearman's \(\rho\) and Kendall's \(\tau_{b}\), showing both spaces to be both biased and inefficient upon practical data domains. A probability distribution is defined for the Kemeny \(\tau_{\kappa}\) estimator, and a Studentisation adjustment for finite samples is provided as well. This work allows for a general purpose linear model duality to be identified as a unique consistent solution to many biased and unbiased estimation scenarios.