This paper develops Bayesian sample size formulae for experiments comparing two groups. We assume the experimental data will be analysed in the Bayesian framework, where pre-experimental information from multiple sources can be represented into robust priors. In particular, such robust priors account for preliminary belief about the pairwise commensurability between parameters that underpin the historical and new experiments, to permit flexible borrowing of information. Averaged over the probability space of the new experimental data, appropriate sample sizes are found according to criteria that control certain aspects of the posterior distribution, such as the coverage probability or length of a defined density region. Our Bayesian methodology can be applied to circumstances where the common variance in the new experiment is known or unknown. Exact solutions are available based on most of the criteria considered for Bayesian sample size determination, while a search procedure is described in cases for which there are no closed-form expressions. We illustrate the application of our Bayesian sample size formulae in the setting of designing a clinical trial. Hypothetical data examples, motivated by a rare-disease trial with elicitation of expert prior opinion, and a comprehensive performance evaluation of the proposed methodology are presented.
In developing products for rare diseases, statistical challenges arise due to the limited number of patients available for participation in drug trials and other clinical research. Bayesian adaptive clinical trial designs offer the possibility of increased statistical efficiency, reduced development cost and ethical hazard prevention via their incorporation of evidence from external sources (historical data, expert opinions, and real-world evidence), and flexibility in the specification of interim looks. In this paper, we propose a novel Bayesian adaptive commensurate design that borrows adapt…
Basket trials are increasingly used for the simultaneous evaluation of a new treatment in various patient subgroups under one overarching protocol. We propose a Bayesian approach to sample size determination in basket trials that permit borrowing of information between commensurate subsets. Specifically, we consider a randomised basket trial design where patients are randomly assigned to the new treatment or a control within each trial subset (`subtrial' for short). Closed-form sample size formulae are derived to ensure each subtrial has a specified chance of correctly deciding whether the new…
Basket trials have emerged as a new class of efficient approaches in oncology to evaluate a new treatment in several patient subgroups simultaneously. In this paper, we extend the key ideas to disease areas outside of oncology, developing a robust Bayesian methodology for randomised, placebo-controlled basket trials with a continuous endpoint to enable borrowing of information across subtrials with similar treatment effects. After adjusting for covariates, information from a complementary subtrial can be represented into a commensurate prior for the parameter that underpins the subtrial under …
One of the main goals of sequential, multiple assignment, randomized trials (SMART) is to find the most efficacious design embedded dynamic treatment regimes. The analysis method known as multiple comparisons with the best (MCB) allows comparison between dynamic treatment regimes and identification of a set of optimal regimes in the frequentist setting for continuous outcomes, thereby, directly addressing the main goal of a SMART. In this paper, we develop a Bayesian generalization to MCB for SMARTs with binary outcomes. Furthermore, we show how to choose the sample size so that the inferior e…
Power and sample size analysis comprises a critical component of clinical trial study design. There is an extensive collection of methods addressing this problem from diverse perspectives. The Bayesian paradigm, in particular, has attracted noticeable attention and includes different perspectives for sample size determination. Building upon a cost-effectiveness analysis undertaken by O'Hagan and Stevens (2001) with different priors in the design and analysis stage, we develop a general Bayesian framework for simulation-based sample size determination that can be easily implemented on modest co…
Determination of posterior probability for go-no-go decision and predictive power are becoming increasingly common for resource optimization in clinical investigation. There are vast published literature on these topics; however, the terminologies are not consistently used across the literature. Further, there is a lack of consolidated presentation of various concepts of the probability of success. We attempted to fill this gap. This paper first provides a detailed derivation of these probability of success measures under the frequentist and Bayesian paradigms in a general setting. Subsequentl…
Battery impedance spectroscopy models are given by fractional order (FO) differential equations. In the discrete-time domain, they give rise to state-space models where the latent process is not Markovian. Parameter estimation for these models is therefore challenging, especially for non-commensurate FO models. In this paper, we propose a Bayesian approach to identify the parameters of generic FO systems. The computational challenge is tackled with particle Markov chain Monte Carlo methods, with an implementation specifically designed for the non-Markovian setting. The approach is then applied…
Measurement of operator quality performance has been challenging in the construction fabrication industry. Among various causes, the learning effect is a significant factor, which needs to be incorporated in achieving a reliable operator quality performance analysis. This research aims to enhance a previously developed operator quality performance measurement approach by incorporating the learning effect (i.e., work experience). To achieve this goal, the Plateau learning model is selected to quantitatively represent the relationship between quality performance and work experience through a bet…
This paper devises a fully Bayesian sample size determination method for hierarchical model-based small area estimation with a decision risk approach. A new loss function specified around a desired maximum posterior variance target implements conventional official statistics criteria of estimator reliability (coefficient of variation of up to 20 per cent). This approach comes with an efficient binary search algorithm identifying the minimum effective sample size needed to produce small area estimates under this threshold constraint. Traditional survey sampling design tools can then be used to …
The development of a new diagnostic test ideally follows a sequence of stages which, amongst other aims, evaluate technical performance. This includes an analytical validity study, a diagnostic accuracy study and an interventional clinical utility study. Current approaches to the design and analysis of the diagnostic accuracy study can suffer from prohibitively large sample sizes and interval estimates with undesirable properties. In this paper, we propose a novel Bayesian approach which takes advantage of information available from the analytical validity stage. We utilise assurance to calcul…
The Net Promoter Score is a simple measure used by several companies as indicator of customer loyalty. Studies that address the statistical properties of this measure are still scarce and none of them considered the sample size determination problem. We adopt a Bayesian approach to provide point and interval estimators for the Net Promoter Score and discuss the determination of the sample size. Computational tools were implemented to use this methodology in practice. An illustrative example with data from financial services is also presented.
Differential networks (DN) are important tools for modeling the changes in conditional dependencies between multiple samples. A Bayesian approach for estimating DNs, from the classical viewpoint, is introduced with a computationally efficient threshold selection for graphical model determination. The algorithm separately estimates the precision matrices of the DN using the Bayesian adaptive graphical lasso procedure. Synthetic experiments illustrate that the Bayesian DN performs exceptionally well in numerical accuracy and graphical structure determination in comparison to state-of-the-art met…
Consider the problem of modeling hysteresis for finite-state random walks using higher-order Markov chains. This Letter introduces a Bayesian framework to determine, from data, the number of prior states of recent history upon which a trajectory is statistically dependent. The general recommendation is to use leave-one-out cross validation, using an easily-computable formula that is provided in closed form. Importantly, Bayes factors using flat model priors are biased in favor of too-complex a model (more hysteresis) when a large amount of data is present and the Akaike information criterion (…
Bayesian optimization methods allocate limited sampling budgets to maximize expensive-to-evaluate functions. One-step-lookahead policies are often used, but computing optimal multi-step-lookahead policies remains a challenge. We consider a specialized Bayesian optimization problem: finding the superlevel set of an expensive one-dimensional function, with a Markov process prior. We compute the Bayes-optimal sampling policy efficiently, and characterize the suboptimality of one-step lookahead. Our numerical experiments demonstrate that the one-step lookahead policy is close to optimal in this pr…
We consider Bayesian sample size determination using a criterion that utilizes the first two moments of the expected posterior variance. We study the resulting sample size in dependence on the chosen prior and explore the success rate for bounding the posterior variance below a prescribed limit under the true sampling distribution. Compared with sample size determination based on the expected average of the posterior variance the proposed criterion leads to an increase in sample size and significantly improved success rates. Generic asymptotic properties are proven, such as an asymptotic expre…
Credit risk modelling is an integral part of the global financial system. While there has been great attention paid to neural network models for credit default prediction, such models often lack the required interpretation mechanisms and measures of the uncertainty around their predictions. This work develops and compares Bayesian Neural Networks(BNNs) for credit card default modelling. This includes a BNNs trained by Gaussian approximation and the first implementation of BNNs trained by Hybrid Monte Carlo(HMC) in credit risk modelling. The results on the Taiwan Credit Dataset show that BNNs w…
Sample size determination for a data set is an important statistical process for analyzing the data to an optimum level of accuracy and using minimum computational work. The applications of this process are credible in every domain which deals with large data sets and high computational work. This study uses Bayesian analysis for determination of minimum sample size of vibration signals to be considered for fault diagnosis of a bearing using pre-defined parameters such as the inverse standard probability and the acceptable margin of error. Thus an analytical formula for sample size determinati…
This introduction to Bayesian statistics presents the main concepts as well as the principal reasons advocated in favour of a Bayesian modelling. We cover the various approaches to prior determination as well as the basis asymptotic arguments in favour of using Bayes estimators. The testing aspects of Bayesian inference are also examined in details.
Determination of an appropriate warranty length for the lifetime of the product is an important issue to the manufacturer. In this article, optimal warranty length of the product for the combined free replacement and the pro-rata warranty policy is computed based on the Type-II unified hybrid censored data. A non-linear pro-rata warranty policy is proposed in this context. The optimal warranty length is obtained by maximizing an expected utility function. The expectation is taken with respect to the posterior predictive model for the time-to-failure data. It is observed that the non-linear pro…
Ensembling can improve the performance of Neural Networks, but existing approaches struggle when the architecture likelihood surface has dispersed, narrow peaks. Furthermore, existing methods construct equally weighted ensembles, and this is likely to be vulnerable to the failure modes of the weaker architectures. By viewing ensembling as approximately marginalising over architectures we construct ensembles using the tools of Bayesian Quadrature -- tools which are well suited to the exploration of likelihood surfaces with dispersed, narrow peaks. Additionally, the resulting ensembles consist o…
My dissertation revolves around Bayesian approaches towards constrained statistical inference in the factor analysis (FA) model. Two interconnected types of restricted-model selection are considered. These types have a natural connection to selection problems in the exploratory FA (EFA) and confirmatory FA (CFA) model and are termed Type I and Type II model selection. Type I constrained-model selection is taken to mean the determination of the appropriate dimensionality of a model. This type of constrained-model selection connects with EFA in the sense of selecting the optimal dimensionality o…
Technical University of Denmark (DTU)（丹麦技术大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Specifying utility functions is a key step towards applying the discrete choice framework for understanding the behaviour processes that govern user choices. However, identifying the utility function specifications that best model and explain the observed choices can be a very challenging and time-consuming task. This paper seeks to help modellers by leveraging the Bayesian framework and the concept of automatic relevance determination (ARD), in order to automatically determine an optimal utility function specification from an exponentially large set of possible specifications in a purely data…
CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank. In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian trea…
This paper deals with the Bayesian analysis of graphical models of marginal independence for three way contingency tables. We use a marginal log-linear parametrization, under which the model is defined through suitable zero-constraints on the interaction parameters calculated within marginal distributions. We undertake a comprehensive Bayesian analysis of these models, involving suitable choices of prior distributions, estimation, model determination, as well as the allied computational issues. The methodology is illustrated with reference to two real data sets.
We propose modified frequentist definition for the determination of confidence intervals for the case of Poisson statistics. Namely, we require that 1-β' \geq \sum_{n=o}^{n_{obs}+k} P(n|λ) \geq α'. We show that this definition is equivalent to the Bayesian method with prior π(λ) \sim λ^{k}. We also propose modified frequentist definition for the case of nonzero background.
This paper explores an approach to Bayesian sample size determination in clinical trials. The approach falls into the category of what is often called "proper Bayesian", in that it does not mix frequentist concepts with Bayesian ones. A criterion for a "successful trial" is defined in terms of a posterior probability, its probability is assessed using the marginal distribution of the data, and this probability forms the basis for choosing sample sizes. We illustrate with a standard problem in clinical trials, that of establishing superiority of a new drug over a control.
University of Sheffield（谢菲尔德大学）；KTH – Royal Institute of Technology（瑞典皇家理工学院）；Wellcome Trust Centre for Human Genetics（惠康信托人类遗传学中心）
In this paper we present a fully Bayesian latent variable model which exploits conditional nonlinear(in)-dependence structures to learn an efficient latent representation. The latent space is factorized to represent shared and private information from multiple views of the data. In contrast to previous approaches, we introduce a relaxation to the discrete segmentation and allow for a "softly" shared latent space. Further, Bayesian techniques allow us to automatically estimate the dimensionality of the latent spaces. The model is capable of capturing structure underlying extremely high dimensio…
We investigate the relation between frequentist and Bayesian approaches. Namely, we find the "frequentist" Bayes prior π_{f}(λ,x_{obs}) = -\frac{\int_{-\infty}^{x_{obs}}\frac{\partial f(x,λ)}{\partial λ}dx}{f(x_{obs},λ)} (here f(x,λ) is the probability density) for which the results of frequentist and Bayes approaches to the determination of confidence intervals coincide. In many cases (but not always) the "frequentist" prior which reproduces frequentist results coincides with the Jeffreys prior.
The main object of Bayesian statistical inference is the determination of posterior distributions. Sometimes these laws are given for quantities devoid of empirical value. This serious drawback vanishes when one confines oneself to considering a finite horizon framework. However, assuming infinite exchangeability gives rise to fairly tractable {\it a posteriori} quantities, which is very attractive in applications. Hence, with a view to a reconciliation between these two aspects of the Bayesian way of reasoning, in this paper we provide quantitative comparisons between posterior distributions …
This work considers methods for imposing sparsity in Bayesian regression with applications in nonlinear system identification. We first review automatic relevance determination (ARD) and analytically demonstrate the need to additional regularization or thresholding to achieve sparse models. We then discuss two classes of methods, regularization based and thresholding based, which build on ARD to learn parsimonious solutions to linear problems. In the case of orthogonal covariates, we analytically demonstrate favorable performance with regards to learning a small set of active terms in a linear…
Estimating the prevalence of a disease is necessary for evaluating and mitigating risks of its transmission within or between populations. Estimates that consider how prevalence changes with time provide more information about these risks but are difficult to obtain due to the necessary sampling intensity and commensurate testing costs. We propose pooling and jointly testing multiple samples to reduce testing costs and use a novel nonparametric, hierarchical Bayesian model to infer population prevalence from the pooled test results. This approach is shown to reduce uncertainty compared to indi…
A general shape identification inverse problem is studied in a Bayesian framework. This problem requires the determination of the unknown shape of a domain in the Euclidean space from finite-dimensional observation data with some Gaussian random noise. Then, the stability of posterior is studied for observation data. For each point of the space, the conditional probability that the point is included in the unknown domain given the observation data is considered. The stability is also studied for this probability distribution. As a model problem for our inverse problem, a heat inverse problem i…
An efficient algorithm for the determination of Bayesian optimal discriminating designs for competing regression models is developed, where the main focus is on models with general distributional assumptions beyond the "classical" case of normally distributed homoscedastic errors. For this purpose we consider a Bayesian version of the Kullback- Leibler (KL) optimality criterion introduced by López-Fidalgo et al. (2007). Discretizing the prior distribution leads to local KL-optimal discriminating design problems for a large number of competing models. All currently available methods either requ…
The determination of the shapes of mortality curves, the estimation and projection of mortality patterns over time, and the investigation of differences in mortality patterns across different small underdeveloped populations have received special attention in recent years. The challenges involved in this type of problems are the common sparsity and the unstable behavior of observed death counts in small areas (populations). These features impose many dificulties in the estimation of reasonable mortality schedules. In this chapter, we present a discussion about this problem and we introduce the…
This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation. The holding times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning …
We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcom…
A major issue of extreme value analysis is the determination of the shape parameter $ξ$ common to Generalized Extreme Value (GEV) and Generalized Pareto (GP) distributions, which drives the tail behavior, and is of major impact on the estimation of return levels and periods. Many practitioners make the choice of a Bayesian framework to conduct this assessment for accounting of parametric uncertainties, which are typically high in such analyses characterized by a low number of observations. Nonetheless, such approaches can provide large credibility domains for $ξ$, including negative and positi…
We consider a continuous time two-armed bandit problem in which incomes are described by Poissonian processes. We develop Bayesian approach with arbitrary prior distribution. We present two versions of recursive equation for determination of Bayesian piece-wise constant strategy and Bayesian risk and partial differential equation in the limiting case. Unlike the previously considered Bayesian settings our description uses current history of the process and not evolution of the posterior distribution.
Given the cost and duration of phase III and phase IV clinical trials, the development of statistical methods for go/no-go decisions is vital. In this paper, we introduce a Bayesian methodology to compute the probability of success based on the current data of a treatment regimen for the multivariate linear model. Our approach utilizes a Bayesian seemingly unrelated regression model, which allows for multiple endpoints to be modeled jointly even if the covariates between the endpoints are different. Correlations between endpoints are explicitly modeled. This Bayesian joint modeling approach un…
Akaike's Bayesian information criterion (ABIC) has been widely used in geophysical inversion and beyond. However, little has been done to investigate its statistical aspects. We present an alternative derivation of the marginal distribution of measurements, whose maximization directly leads to the invention of ABIC by Akaike. We show that ABIC is to statistically estimate the variance of measurements and the prior variance by maximizing the marginal distribution of measurements. The determination of the regularization parameter on the basis of ABIC is actually equivalent to estimating the rela…
University of California Santa Barbara（加州大学圣巴巴拉分校）；Department of Mathematics, University of California Santa Barbara（加州大学圣巴巴拉分校数学系）；Department of Electrical and Computer Engineering, University of California Santa Barbara（加州大学圣巴巴拉分校电气与计算机工程系）；Facebook AI Systems Hardware/Software Co-design（Facebook人工智能系统硬件/软件协同设计）
While post-training model compression can greatly reduce the inference cost of a deep neural network, uncompressed training still consumes a huge amount of hardware resources, run-time and energy. It is highly desirable to directly train a compact neural network from scratch with low memory and low computational cost. Low-rank tensor decomposition is one of the most effective approaches to reduce the memory and computing requirements of large-size neural networks. However, directly training a low-rank tensorized neural network is a very challenging task because it is hard to determine a proper…
We derive an exact and efficient Bayesian regression algorithm for piecewise constant functions of unknown segment number, boundary location, and levels. It works for any noise and segment level prior, e.g. Cauchy which can handle outliers. We derive simple but good estimates for the in-segment variance. We also propose a Bayesian regression curve as a better way of smoothing data without blurring boundaries. The Bayesian approach also allows straightforward determination of the evidence, break probabilities and error estimates, useful for model selection and significance and robustness studie…
Virginia Tech（弗吉尼亚理工学院暨州立大学）；Argonne National Laboratory（阿贡国家实验室）
As neural networks have begun performing increasingly critical tasks for society, ranging from driving cars to identifying candidates for drug development, the value of their ability to perform uncertainty quantification (UQ) in their predictions has risen commensurately. Permanent dropout, a popular method for neural network UQ, involves injecting stochasticity into the inference phase of the model and creating many predictions for each of the test data. This shifts the computational and energy burden of deep neural networks from the training phase to the inference phase. Recent work has demo…
Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR process…
We present a Bayesian data fusion method to approximate a posterior distribution from an ensemble of particle estimates that only have access to subsets of the data. Our approach relies on approximate probabilistic inference of model parameters through Monte Carlo methods, followed by an update and resample scheme related to multiple importance sampling to combine information from the initial estimates. We show the method is convergent in the particle limit and directly suited to application on multi-sensor data fusion problems by demonstrating efficacy on a multi-sensor Keplerian orbit determ…
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly but are only represented indirectly in form of a computer routine. Such a routine implements the transformation a data vector undergoes under matrix multiplication. While efficient probing routines to estimate a matrix's diagonal or trace, based solely on such computationally affordable matrix-vector multiplications, are well known and frequently used in signa…
