arXiv subjects

S. A. Sisson

Publications and source records attributed to S. A. Sisson.

At least 19 recordsLinked to original sources

arXiv2025

On the inverse of covariance matrices for unbalanced crossed designs

This paper addresses a long-standing open problem in the analysis of linear mixed models with crossed random effects under unbalanced designs: how to find an analytic expression for the inverse of $\mathbf{V}$, the covariance matrix of the observed response. The inverse matrix $\mathbf{V}^{-1}$ is required for likelihood-based estimation and inference. However, for unbalanced crossed designs, $\mathbf{V}$ is dense and the lack of a closed-form representation for $\mathbf{V}^{-1}$, until now, has made using likelihood-based methods computationally challenging and difficult to analyse mathematically. We use the Khatri--Rao product to represent $\mathbf{V}$ and then to construct a modified covariance matrix whose inverse admits an exact spectral decomposition. Building on this construction, we obtain an elegant and simple approximation to $\mathbf{V}^{-1}$ for asymptotic unbalanced designs. For non-asymptotic settings, we derive an accurate and interpretable approximation under mildly unbalanced data and establish an exact inverse representation as a low-rank correction to this approximation, applicable to arbitrary degrees of unbalance. Simulation studies demonstrate the accuracy, stability, and computational tractability of the proposed framework.

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arXiv2025

Asymptotics for EBLUPs within crossed mixed effect models

In this article, we derive the joint asymptotic distribution of empirical best linear unbiased predictors (EBLUPs) for individual and cell-level random effects in a crossed mixed effect model. Under mild conditions (which include moment conditions instead of normality for the random effects and model errors), we demonstrate that as the sizes of rows, columns, and, when we include interactions, cells simultaneously increase to infinity, the distribution of the differences between the EBLUPs and the random effects satisfy central limit theorems. These central limit theorems mean the EBLUPs asymptotically follow the convolution of the true random effect distribution and a normal distribution. Moreover, our results enable simple asymptotic approximations and estimators for the mean squared error (MSE) of the EBLUPs, which in turn facilitates the construction of asymptotic prediction intervals for the unobserved random effects. We show in simulations that our simple estimator of the MSE of the EBLUPs works very well in finite samples. Finally, we illustrate the use of the asymptotic prediction intervals with an analysis of movie rating data.

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arXiv2024

Increasing dimension asymptotics for two-way crossed mixed effect models

This paper presents asymptotic results for the maximum likelihood and restricted maximum likelihood (REML) estimators within a two-way crossed mixed effect model as the sizes of the rows, columns, and cells tend to infinity. Under very mild conditions which do not require the assumption of normality, the estimators are proven to be asymptotically normal, possessing a structured covariance matrix. The growth rate for the number of rows, columns, and cells is unrestricted, whether considered pairwise or collectively.

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arXiv2024

Model-Free Local Recalibration of Neural Networks

Artificial neural networks (ANNs) are highly flexible predictive models. However, reliably quantifying uncertainty for their predictions is a continuing challenge. There has been much recent work on "recalibration" of predictive distributions for ANNs, so that forecast probabilities for events of interest are consistent with certain frequency evaluations of them. Uncalibrated probabilistic forecasts are of limited use for many important decision-making tasks. To address this issue, we propose a localized recalibration of ANN predictive distributions using the dimension-reduced representation of the input provided by the ANN hidden layers. Our novel method draws inspiration from recalibration techniques used in the literature on approximate Bayesian computation and likelihood-free inference methods. Most existing calibration methods for ANNs can be thought of as calibrating either on the input layer, which is difficult when the input is high-dimensional, or the output layer, which may not be sufficiently flexible. Through a simulation study, we demonstrate that our method has good performance compared to alternative approaches, and explore the benefits that can be achieved by localizing the calibration based on different layers of the network. Finally, we apply our proposed method to a diamond price prediction problem, demonstrating the potential of our approach to improve prediction and uncertainty quantification in real-world applications.

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arXiv2020

High-dimensional inference using the extremal skew-$t$ process

Max-stable processes are a popular tool for the study of environmental extremes, and the extremal skew-$t$ process is a general model that allows for a flexible extremal dependence structure. For inference on max-stable processes with high-dimensional data, exact likelihood-based estimation is computationally intractable. Composite likelihoods, using lower dimensional components, and Stephenson-Tawn likelihoods, using occurrence times of maxima, are both attractive methods to circumvent this issue for moderate dimensions. In this article we establish the theoretical formulae for simulations of and inference for the extremal skew-$t$ process. We also incorporate the Stephenson-Tawn concept into the composite likelihood framework, giving greater statistical and computational efficiency for higher-order composite likelihoods. We compare 2-way (pairwise), 3-way (triplewise), 4-way, 5-way and 10-way composite likelihoods for models of up to 100 dimensions. Furthermore, we propose cdf approximations for the Stephenson-Tawn likelihood function, leading to large computational gains, and enabling accurate fitting of models in large dimensions in only a few minutes. We illustrate our methodology with an application to a 90-dimensional temperature dataset from Melbourne, Australia.

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arXiv2019

Likelihood-free approximate Gibbs sampling

Likelihood-free methods such as approximate Bayesian computation (ABC) have extended the reach of statistical inference to problems with computationally intractable likelihoods. Such approaches perform well for small-to-moderate dimensional problems, but suffer a curse of dimensionality in the number of model parameters. We introduce a likelihood-free approximate Gibbs sampler that naturally circumvents the dimensionality issue by focusing on lower-dimensional conditional distributions. These distributions are estimated by flexible regression models either before the sampler is run, or adaptively during sampler implementation. As a result, and in comparison to Metropolis-Hastings based approaches, we are able to fit substantially more challenging statistical models than would otherwise be possible. We demonstrate the sampler's performance via two simulated examples, and a real analysis of Airbnb rental prices using a intractable high-dimensional multivariate non-linear state space model containing 13,140 parameters, which presents a real challenge to standard ABC techniques.

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arXiv2018

Overview of Approximate Bayesian Computation

This Chapter, "Overview of Approximate Bayesian Computation", is to appear as the first chapter in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and concepts behind ABC methods with many examples and illustrations.

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arXiv2018

High-dimensional ABC

This Chapter, "High-dimensional ABC", is to appear in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and concepts behind extending ABC methods to higher dimensions, with supporting examples and illustrations.

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arXiv2018

ABC Samplers

This Chapter, "ABC Samplers", is to appear in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and algorithms used to sample from the ABC approximation to the posterior distribution, including methods based on rejection/importance sampling, MCMC and sequential Monte Carlo.

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arXiv2017

Recalibration: A post-processing method for approximate Bayesian computation

A new recalibration post-processing method is presented to improve the quality of the posterior approximation when using Approximate Bayesian Computation (ABC) algorithms. Recalibration may be used in conjunction with existing post-processing methods, such as regression-adjustments. In addition, this work extends and strengthens the links between ABC and indirect inference algorithms, allowing more extensive use of misspecified auxiliary models in the ABC context. The method is illustrated using simulated examples to demonstrate the effects of recalibration under various conditions, and through an application to an analysis of stereological extremes both with and without the use of auxiliary models. Code to implement recalibration post-processing is available in the R package, abctools.

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arXiv2014

Bayesian threshold selection for extremal models using measures of surprise

Statistical extreme value theory is concerned with the use of asymptotically motivated models to describe the extreme values of a process. A number of commonly used models are valid for observed data that exceed some high threshold. However, in practice a suitable threshold is unknown and must be determined for each analysis. While there are many threshold selection methods for univariate extremes, there are relatively few that can be applied in the multivariate setting. In addition, there are only a few Bayesian-based methods, which are naturally attractive in the modelling of extremes due to data scarcity. The use of Bayesian measures of surprise to determine suitable thresholds for extreme value models is proposed. Such measures quantify the level of support for the proposed extremal model and threshold, without the need to specify any model alternatives. This approach is easily implemented for both univariate and multivariate extremes.

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arXiv2014

Functional regression approximate Bayesian computation for Gaussian process density estimation

We propose a novel Bayesian nonparametric method for hierarchical modelling on a set of related density functions, where grouped data in the form of samples from each density function are available. Borrowing strength across the groups is a major challenge in this context. To address this problem, we introduce a hierarchically structured prior, defined over a set of univariate density functions, using convenient transformations of Gaussian processes. Inference is performed through approximate Bayesian computation (ABC), via a novel functional regression adjustment. The performance of the proposed method is illustrated via a simulation study and an analysis of rural high school exam performance in Brazil.

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arXiv2013

Simultaneous adjustment of bias and coverage probabilities for confidence intervals

A new method is proposed for the correction of confidence intervals when the original interval does not have the correct nominal coverage probabilities in the frequentist sense. The proposed method is general and does not require any distributional assumptions. It can be applied to both frequentist and Bayesian inference where interval estimates are desired. We provide theoretical results for the consistency of the proposed estimator, and give two complex examples, on confidence interval correction for composite likelihood estimators and in approximate Bayesian computation (ABC), to demonstrate the wide applicability of the new method. Comparison is made with the double-bootstrap and other methods of improving confidence interval coverage.

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arXiv2013

A Comparative Review of Dimension Reduction Methods in Approximate Bayesian Computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.

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arXiv2013

Diagnostic tools of approximate Bayesian computation using the coverage property

Approximate Bayesian computation (ABC) is an approach for sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples, (m,θ,y), from the prior, and then accepting as samples from the approximate posterior, those pairs (m,θ) for which y, or a summary of y, is "close" to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter, ε. Appropriate choice of εis important to producing a good quality approximation. This paper proposes diagnostic tools for the choice of εbased on assessing the coverage property, which asserts that credible intervals have the correct coverage levels. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re-analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. R code implementing the proposed methodology is freely available in the package "abc."

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arXiv2012

Approximate Bayesian computation and Bayes linear analysis: Towards high-dimensional ABC

Bayes linear analysis and approximate Bayesian computation (ABC) are techniques commonly used in the Bayesian analysis of complex models. In this article we connect these ideas by demonstrating that regression-adjustment ABC algorithms produce samples for which first and second order moment summaries approximate adjusted expectation and variance for a Bayes linear analysis. This gives regression-adjustment methods a useful interpretation and role in exploratory analysis in high-dimensional problems. As a result, we propose a new method for combining high-dimensional, regression-adjustment ABC with lower-dimensional approaches (such as using MCMC for ABC). This method first obtains a rough estimate of the joint posterior via regression-adjustment ABC, and then estimates each univariate marginal posterior distribution separately in a lower-dimensional analysis. The marginal distributions of the initial estimate are then modified to equal the separately estimated marginals, thereby providing an improved estimate of the joint posterior. We illustrate this method with several examples. Supplementary materials for this article are available online.

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arXiv2012

Approximate Bayesian Computation via Regression Density Estimation

Approximate Bayesian computation (ABC) methods, which are applicable when the likelihood is difficult or impossible to calculate, are an active topic of current research. Most current ABC algorithms directly approximate the posterior distribution, but an alternative, less common strategy is to approximate the likelihood function. This has several advantages. First, in some problems, it is easier to approximate the likelihood than to approximate the posterior. Second, an approximation to the likelihood allows reference analyses to be constructed based solely on the likelihood. Third, it is straightforward to perform sensitivity analyses for several different choices of prior once an approximation to the likelihood is constructed, which needs to be done only once. The contribution of the present paper is to consider regression density estimation techniques to approximate the likelihood in the ABC setting. Our likelihood approximations build on recently developed marginal adaptation density estimators by extending them for conditional density estimation. Our approach facilitates reference Bayesian inference, as well as frequentist inference. The method is demonstrated via a challenging problem of inference for stereological extremes, where we perform both frequentist and Bayesian inference.

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arXiv2012

Discussions on Fernhead and Prangle (2012)

Two contributions to the discussion of Fearnhead P. and D. Prangle (2012). Constructing summary statistics for approximate Bayesian computation: Semi-automatic approx- imate Bayesian computation, J. Roy. Statist. Soc. B, 74 (3).

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