Bayesian Approaches to Copula Modelling

A comprehensive comparison of copula models and multivariate normal distribution for geo-material parametric data

This case study is a based on measurements made approximately at 20cm lengths along a down-the-hole diamond drill core from a pyrite mine in South Australia. The measurements are the P-wave velocity, magnetic susceptibility and impedance. The trivariate distribution is modelled using Gaussian, Student-t and vine copulas and the results are compared in terms of goodness of fit and differences in extreme values from distributions obtained by simulation from the copulas. The vine copula provides the best fit for the variables. Trivariate linear spatial Gaussian, Student-t and vine copulas are used to predict magnetic susceptibility one step below the depth of the drill core. The vine copula allows for more detailed modelling of the error structure, and so provides more accurate 90% prediction intervals. The 90% prediction interval for the vine copula is wider than that for the Student-t copula, and both are wider than the interval obtained with the Gaussian copula. In general, copulas provides a more realistic modelling of geological variables and hence allows for accurate assessment of risk and uncertainty.

11018 Background: The conventional interpretation of superiority oncology trials relies on statistical significance defined by P values, which are frequently misinterpreted and oversimplified. On the other hand, Bayesian analysis, which incorporates prior knowledge, directly estimates the probability of a hypothesis, with greater flexibility to examine clinically relevant effect sizes. Here, we studied the effects of Bayesian models on phase III trial interpretation. We further hypothesized that Bayesian approaches, using differential priors for efficacy and futility, would improve trial efficiency. Methods: Phase III superiority-design, two-arm oncology trials were screened from ClinicalTrials.gov for this meta-epidemiological study. Individual patient-level data were manually reconstructed from the Kaplan-Meir curves of the primary endpoint. First, Bayesian Cox regressions of reconstructed data applied skeptical N (0, 0.355), enthusiastic N (-0.41, 0.4), and neutral priors N (0, 10 6 ), where N (mean, standard deviation) denotes a normal distribution, to estimate posterior probabilities of the primary endpoint effects with Markov chain Monte Carlo sampling. Minimum clinically important differences (MCID) in the experimental arm were defined by HR < 0.8 per ASCO criteria (Ellis et al, J Clin Oncol 2014). Second, a single event or enrollment-driven interim analysis with Bayesian stopping rules was simulated 100 times for each trial using in silico models of randomly varying accrual kinetics with published patient outcomes. Interim efficacy and futility were defined per simulation by probabilities ≥ 85% for achieving effect sizes larger than the MCID/3 using a skeptical or enthusiastic prior, respectively. Early trial closure was recommended if ≥ 75% of simulations met efficacy or futility criteria. Results: After screening,194,129 patient outcomes from 230 trials were reconstructed. Overall survival was the primary endpoint in 90 trials (39%). All trials interpreted as positive had > 90% probabilities of marginal benefits (HR < 1). However, 38% of trials interpreted as positive had ≤ 90% probabilities of achieving the MCID (HR < 0.8), even under an enthusiastic prior. Conversely, 24% of trials interpreted as negative had > 90% probability of achieving marginal benefits, even under a skeptical prior. In the interim analysis simulations, early closure was recommended for 82 trials (36%). Bayesian interim analysis was associated with >99% probability of reducing enrollment sizes. The trial and its simulated interim analysis remained concordant (Bayesian Cohen’s κ, 0.95). Conclusions: Bayesian models add unique interpretative value for clinically relevant effects, and may improve trial efficiency without compromising trial interpretation. Bayesian models should be increasingly incorporated in phase III trials.

This chapter provides a brief introduction to the use of Bayesian graphical models in R. In these models, parameters are treated as random quantities on an equal footing with the random variables. This allows complex stochastic systems to modeled, often using Markov chain Monte Carlo (MCMC) sampling methods. We first consider a series of examples, ranging from simple repeated sampling, linear regression models, random coefficient regression models, to the chest clinic example of Chap. 3. We formulate these as Bayesian graphical models, and represent them graphically in a compact form using plates. We then describe a special case, in which the unknown parameters are all discrete, and explain how probability propagation methods described in Chap. 3 may be used to compute the posterior distributions. We then turn to the general case, when MCMC sampling is required, and explain the computations involved in Metropolis-Hastings sampling and Gibbs sampling. Finally we illustrate a Bayesian linear regression analysis using the R2WinBUGS package.

Variable selection is very important in many fields, and for its resolution many procedures have been proposed and investigated. Among them are Bayesian methods that use Markov chain Monte-Carlo (MCMC) sampling algorithms. A problem with MCMC sampling, however, is that it cannot guarantee that the samples are exactly from the target distributions. This drawback is overcome by related methods known as perfect sampling algorithms. In this paper, we propose the use of two perfect sampling algorithms to perform variable selection within the Bayesian framework. They are the sandwiched coupling from the past (CFTP) algorithm and the Gibbs coupler. We focus our attention to scenarios where the model coefficients and noise variance are known. We indicate the condition under which the sandwiched CFTP can be applied. Most importantly, we design a detailed scheme to adapt the Gibbs coupler algorithm to variable selection. In addition, we discuss the possibilities of applying perfect sampling when the model coefficients and noise variance are unknown. Test results that show the performance of the algorithms are provided.

ABSTRACTWe propose a new variational Bayes (VB) estimator for high-dimensional copulas with discrete, or a combination of discrete and continuous, margins. The method is based on a variational approximation to a tractable augmented posterior and is faster than previous likelihood-based approaches. We use it to estimate drawable vine copulas for univariate and multivariate Markov ordinal and mixed time series. These have dimension rT, where T is the number of observations and r is the number of series, and are difficult to estimate using previous methods. The vine pair-copulas are carefully selected to allow for heteroscedasticity, which is a feature of most ordinal time series data. When combined with flexible margins, the resulting time series models also allow for other common features of ordinal data, such as zero inflation, multiple modes, and under or overdispersion. Using six example series, we illustrate both the flexibility of the time series copula models and the efficacy of the VB estimator for copulas of up to 792 dimensions and 60 parameters. This far exceeds the size and complexity of copula models for discrete data that can be estimated using previous methods. An online appendix and MATLAB code implementing the method are available as supplementary materials.

In the context of Bayesian inversion for scientific and engineering modeling, Markov chain Monte Carlo (MCMC) sampling strategies have become the benchmark due to their flexibility and robustness in dealing with arbitrary posterior probability density functions (PDFs). However, these algorithms have been shown to be inefficient when sampling from high-dimensional posterior distributions or exhibit multimodality and/or strong parameter correlations. In such contexts, transitional MCMC (TMCMC) provides a more efficient alternative. Despite the recent applicability for Bayesian updating and model selection across a variety of disciplines, TMCMC may require a prohibitive number of tempering stages when the prior pdf is significantly different from the target posterior. Furthermore, the need to start with an initial set of samples from the prior distribution may present a challenge when dealing with implicit priors, e.g., based on feasible regions. Finally, TMCMC cannot be used for inverse problems with improper prior PDFs that represent a lack of prior knowledge on all or a subset of parameters. A generalization of TMCMC is proposed that alleviates such challenges and limitations toward providing a more robust and efficient tempering sampling strategy. We present convergence analysis, proving that the distance between the intermediate distributions and the target posterior distribution monotonically decreases as the algorithm proceeds. We also demonstrate the advantages of the proposed generalization through a series of test problems and an engineering application in the oil and gas industry.

Entity resolution is the process of merging and removing duplicate records from multiple data sources, often in the absence of unique identifiers. Bayesian models for entity resolution allow one to include a priori information, quantify uncertainty in important applications, and directly estimate a partition of the records. Markov chain Monte Carlo (MCMC) sampling is the primary computational method for approximate posterior inference in this setting, but due to the high dimensionality of the space of partitions, there are no agreed upon standards for diagnosing nonconvergence of MCMC sampling. In this article, we review Bayesian entity resolution, with a focus on the specific challenges that it poses for the convergence of a Markov chain. We review prior methods for convergence diagnostics, discussing their weaknesses. We provide recommendations for using MCMC sampling for Bayesian entity resolution, focusing on the use of modern diagnostics that are commonplace in applied Bayesian statistics. Using simulated data, we find that a commonly used Gibbs sampler performs poorly compared with two alternatives.

Small area statistics are required when the sample size is small to produce estimates with adequate precision. The assumptions underlying Battese Harter Fuller (BHF) unit-level models may often be unrealistic in some applications. Copula is an alternative approach when the assumptions are violated. This research discusses the performance of BHF and Copula in small area estimation (SAE) for estimating household per capita expenditure in sub-district levels. This study presents household per capita expenditure, which has a skewed distribution. Due to the fact that the data contains outliers, an appropriate method to handle outliers is also considered. In this research, the Gaussian and the Clayton Copulas are used. The results showed that the performance of BHF was better than Gaussian and Clayton Copulas, as indicated by small root mean square error (RMSE) with an average of 1.14, while the average RMSE of Gaussian copula was 2.71 and Clayton copula was 2.63. Furthermore, the coefficient of variation (CV) of BHF was also smaller compared to Gaussian and Clayton Copulas, and the resulting estimates can be categorized as reliable based on the CV of less than 25%.

Chapter 1 - Introduction to Bayesian Inference

Bayesian statistics have been widely used given the development of Markov chain Monte Carlo sampling techniques and the growth of computational power. A major challenge of Bayesian methods that has not yet been fully addressed is how we can appropriately evaluate the convergence of the random samples to the target posterior distributions. In this paper, we focus on Gelman and Rubin's diagnostic (PSRF), Brooks and Gleman's diagnostic (MPSRF), and Geweke's diagnostics, and compare the Type I error rate and Type II error rate of seven convergence criteria: MPSRF>1.1, any upper bound of PSRF is larger than 1.1, more than 5% of the upper bounds of PSRFs are larger than 1.1, any PSRF is larger than 1.1, more than 5% of PSRFs are larger than 1.1, any Geweke test statistic is larger than 1.96 or smaller than -1.96, and more than 5% of Geweke test statistics are larger than 1.96 or smaller than -1.96. Based on the simulation results, we recommend the upper bound of PSRF if we only can choose one diagnostic. When the number of estimated parameters is large, between the diagnostic per parameter (i.e., PSRF) or the multivariate diagnostic (i.e., MPSRF), we recommend the upper bound of PSRF over MPSRF. Additionally, we do not suggest claiming convergence at the analysis level while allowing a small proportion of the parameters to have significant convergence diagnosis results.

The Center for Drug Evaluation and Research (CDER) and the Center for Biologics Evaluation and Research (CBER) of the U.S. Food and Drug Administration (FDA) have been leaders in protecting and promoting the U.S. public health by helping to ensure that safe and effective drugs and biological products are available in the United States for those who need them. The null hypothesis significance testing approach, along with other considerations, is typically used to demonstrate the effectiveness of a drug or biological product. The Bayesian framework presents an alternative approach to demonstrate the effectiveness of a treatment. This article discusses the Bayesian framework for drug and biological product development, highlights key settings in which Bayesian approaches may be appropriate, and provides recent examples of the use of Bayesian approaches within CDER and CBER.

The estimation of finite fault earthquake source models is an inherently underdetermined problem: there is no unique solution to the inverse problem of determining the rupture history at depth as a function of time and space when our data are limited to observations at the Earth's surface. Bayesian methods allow us to determine the set of all plausible source model parameters that are consistent with the observations, our a priori assumptions about the physics of the earthquake source and wave propagation, and models for the observation errors and the errors due to the limitations in our forward model. Because our inversion approach does not require inverting any matrices other than covariance matrices, we can restrict our ensemble of solutions to only those models that are physically defensible while avoiding the need to restrict our class of models based on considerations of numerical invertibility. We only use prior information that is consistent with the physics of the problem rather than some artefice (such as smoothing) needed to produce a unique optimal model estimate. Bayesian inference can also be used to estimate model-dependent and internally consistent effective errors due to shortcomings in the forward model or data interpretation, such as poor Green's functions or extraneous signals recorded by our instruments. Until recently, Bayesian techniques have been of limited utility for earthquake source inversions because they are computationally intractable for problems with as many free parameters as typically used in kinematic finite fault models. Our algorithm, called cascading adaptive transitional metropolis in parallel (CATMIP), allows sampling of high-dimensional problems in a parallel computing framework. CATMIP combines the Metropolis algorithm with elements of simulated annealing and genetic algorithms to dynamically optimize the algorithm's efficiency as it runs. The algorithm is a generic Bayesian Markov Chain Monte Carlo sampler; it works independently of the model design, a priori constraints and data under consideration, and so can be used for a wide variety of scientific problems. We compare CATMIP's efficiency relative to several existing sampling algorithms and then present synthetic performance tests of finite fault earthquake rupture models computed using CATMIP.

PurposeThe purpose of this paper is to provide an analysis of the dependence structure between returns from real estate investment trusts (REITS) and a stock market index. Further, the aim is to illustrate how copula approaches can be applied to model the complex dependence structure between the assets and for risk measurement of a portfolio containing investments in REIT and equity indices.Design/methodology/approachThe usually suggested multivariate normal or variance‐ covariance approach is applied, as well as various copula models in order to investigate the dependence structure between returns of Australian REITS and the Australian stock market. Different models including the Gaussian, Studentt, Clayton and Gumbel copula are estimated and goodness‐of‐fit tests are conducted. For the return series, both the Gaussian and a non‐parametric estimate of the distribution is applied. A risk analysis is provided based on Monte Carlo simulations for the different models. The value‐at‐risk measure is also applied for quantification of the risks for a portfolio combining investments in real estate and stock markets.FindingsThe findings suggest that the multivariate normal model is not appropriate to measure the complex dependence structure between the returns of the two asset classes. Instead, a model using non‐parametric estimates for the return series in combination with a Studenttcopula is clearly more suitable. It further illustrates that the usually applied variance‐covariance approach leads to a significant underestimation of the actual risk for a portfolio consisting of investments in REITS and equity indices. The nature of risk is better captured by the suggested copula models.Originality/valueTo the authors', knowledge, this is one of the first studies to apply and test different copula models in real estate markets. Results help international investors and portfolio managers to deepen their understanding of the dependence structure between returns from real estate and equity markets. Additionally, the results should be helpful for implementation of a more adequate risk management for portfolios containing investments in both REITS and equity indices.

To describe the priors and decision thresholds in phase 2 and 3 randomized controlled trials (RCTs) evaluating drug efficacy using Bayesian methods. A systematic review of phase 2 and 3 RCTs evaluating drug efficacy through Bayesian inference was conducted across the MEDLINE, EMBASE, and Cochrane databases, with no date restrictions until September 2022. The type of prior used for the analysis of the primary endpoint and its characteristics (type and parameters of the distribution, justification, and sensitivity analysis), the use of a posterior probability decision threshold defined a priori, and its value, were extracted. From 1161 articles screened, 69 articles were ultimately included, encompassing a total of 91 comparisons, as some trials assessed multiple primary endpoints or treatments. The prior was assigned to treatment effect in 51% of the cases (n=46) to each arm in 37% (n=34) and was not explicitly defined in 12% (n=11). Prior distribution was described (with its parameters) in 59% of cases (n=54). A decision threshold was set a priori in 68% of the results (n=62), and its value ranged from 70% to 99% (median 95%). The inconsistent description of priors, along with the wide variation and occasional absence of decision thresholds, underscore the need for clear guidelines on the use and reporting of Bayesian methods. Bayesian methods are being used more frequently in clinical trials to assess drug's efficacy. These methods offer flexibility by incorporating prior knowledge into the analysis. However, the use of Bayesian approaches is still not widespread, and there are challenges with how results are interpreted, partly due to a lack of clear standards. We conducted a systematic review to describe how Bayesian methods were used and reported in phase 2 and 3 clinical trials assessing drug efficacy. We looked at the types of prior information used in the analyses and how decisions about efficacy were made based on the results. Out of 1161 studies reviewed, 69 were included in the analysis, covering 91 drug comparisons. The priors in Bayesian drug trials were assigned to treatment effects or each arm, often justified but variably describe. Similarly, decision thresholds for determining drug efficacy were preset in most studies, but with heterogeneity in the thresholds used to conclude. Our findings highlight the need for clearer guidelines on using Bayesian methods in clinical trials to improve transparency and consistency in how results are reported.