arXiv subjects

Michael Stanley Smith

Publications and source records attributed to Michael Stanley Smith.

At least 19 recordsLinked to original sources

arXiv2026

Vector Vine Copula Models for Multivariate Longitudinal Data

Multivariate longitudinal data may exhibit non-Gaussian margins, nonlinear dynamics, and response vectors with composition that varies across waves. To account for these features, we introduce a vector drawable vine (VD-vine) copula that extends conventional drawable vine copulas from scalar to vector-valued nodes. Here, the response vector at each wave forms a multivariate marginal, and serial dependence is captured through a sequence of linking vector copulas. We establish that the VD-vine is itself a vector copula and reduces to a conventional drawable vine for scalar nodes. Recursive forward and backward conditional transports are derived that enable efficient likelihood evaluation and predictive simulation, with parsimonious reductions under finite-order Markov and stationary restrictions. Unconstrained parameterizations for Gaussian and FGM linking vector copulas, flexible multivariate marginals, and Bayesian variational inference provide a practical implementation. Simulations show improved predictive accuracy when the marginals are asymmetric and serial dependence is multivariate, with little loss under a correctly specified Gaussian panel vector autoregression. In an eight-wave Australian panel of 1,093 individuals with varying response vectors, the full VD-vine delivers the best cross-validated distributional forecasts among the models considered, establishing the benefit of capturing asymmetry and nonlinear dependence.

stat.ME↗

arXiv2026

A Tractable Unified Skew-t Distribution and Its Copula for Heterogeneous Asymmetries

Multivariate distributions that allow for asymmetry and heavy tails are important building blocks in many econometric and statistical models. The Unified Skew-t (UST) is a promising choice because it is both scalable and allows for a high level of flexibility in the asymmetry in the distribution. However, it suffers from parameter identification and computational hurdles that have to date inhibited its use for modeling data. In this paper we propose a new tractable variant of the unified skew-t (TrUST) distribution that addresses both challenges. Moreover, the copula of this distribution is shown to also be tractable, while allowing for greater heterogeneity in asymmetric dependence over variable pairs than the popular skew-t copula. We show how Bayesian posterior inference for both the distribution and its copula can be computed using an extended likelihood derived from a generative representation of the distribution. The efficacy of this Bayesian method, and the enhanced flexibility of both the TrUST distribution and its implicit copula, is first demonstrated using simulated data. Applications of the TrUST distribution to highly skewed regional Australian electricity prices, and the TrUST copula to intraday U.S. equity returns, demonstrate how our proposed distribution and its copula can provide substantial increases in accuracy over the popular skew-t and its copula in practice.

stat.ME↗

arXiv2026

Bayesian Modular Inference for Copula Models with Potentially Misspecified Marginals

Copula models of multivariate data are popular because they allow separate specification of marginal distributions and the copula function. These components can be treated as inter-related modules in a modified Bayesian inference approach called ''cutting feedback'' that is robust to their misspecification. Recent work uses a two module approach, where all $d$ marginals form a single module, to robustify inference for the marginals against copula function misspecification, or vice versa. However, marginals can exhibit differing levels of misspecification, making it attractive to assign each its own module with an individual influence parameter controlling its contribution to a joint semi-modular inference (SMI) posterior. This generalizes existing two module SMI methods, which interpolate between cut and conventional posteriors using a single influence parameter. We develop a novel copula SMI method and select the influence parameters using Bayesian optimization. It provides an efficient continuous relaxation of the discrete optimization problem over $2^d$ cut/uncut configurations. We establish theoretical properties of the resulting semi-modular posterior and demonstrate the approach on simulated and real data. The real data application uses a skew-normal copula model of asymmetric dependence between equity volatility and bond yields, where robustifying copula estimation against marginal misspecification is strongly motivated.

stat.ME↗

arXiv2026

Conjugating Variational Inference for Large Mixed Multinomial Logit Models and Consumer Choice

Heterogeneity in multinomial choice data is often accounted for using logit models with random coefficients. Such models are called "mixed", but they can be difficult to estimate for large datasets. We review current Bayesian variational inference (VI) methods that can do so, and propose a new VI method that scales more effectively. The key innovation is a step that updates efficiently a Gaussian approximation to the conditional posterior of the random coefficients, addressing a bottleneck within the variational optimization. The approach is used to estimate three types of mixed logit models: standard, nested and bundle variants. We first demonstrate the improvement of our new approach over existing VI methods using simulations. Our method is then applied to a large scanner panel dataset of pasta choice. We find consumer response to price and promotion variables exhibits substantial heterogeneity at the grocery store and product levels. Store size, premium and geography are found to be drivers of store level estimates of price elasticities. Extension to bundle choice with pasta sauce improves model accuracy further. Predictions from the mixed models are more accurate than those from fixed coefficients equivalents, and our VI method provides insights in circumstances which other methods find challenging.

stat.ME↗

arXiv2026

Bayesian Additive Regression Tree Copula Processes for Scalable Distributional Prediction

We show how to construct the implied copula process of response values from a Bayesian additive regression tree (BART) model with prior on the leaf node variances. This copula process, defined on the covariate space, can be paired with any marginal distribution for the dependent variable to construct a flexible distributional BART model. Bayesian inference is performed via Markov chain Monte Carlo on an augmented posterior, where we show that key sampling steps can be realized as those of Chipman et al. (2010), preserving scalability and computational efficiency even though the copula process is high dimensional. The posterior predictive distribution from the copula process model is derived in closed form as the push-forward of the posterior predictive distribution of the underlying BART model with an optimal transport map. Under suitable conditions, we establish posterior consistency for the regression function and posterior means and prove convergence in distribution of the predictive process and conditional expectation. Simulation studies demonstrate improved accuracy of distributional predictions compared to the original BART model and leading benchmarks. Applications to five real datasets with 506 to 515,345 observations and 8 to 90 covariates further highlight the efficacy and scalability of our proposed BART copula process model.

stat.ME↗

arXiv2025

Vector Copula Variational Inference and Dependent Block Posterior Approximations

The key to VI is the selection of a tractable density to approximate the Bayesian posterior. For large and complex models a common choice is to assume independence between multivariate blocks in a partition of the parameter space. While this simplifies the problem it can reduce accuracy. This paper proposes using vector copulas to capture dependence between the blocks parsimoniously. Tailored multivariate marginals are constructed using learnable transport maps. We call the resulting joint distribution a ``dependent block posterior'' approximation. Vector copula models are suggested that make tractable and flexible variational approximations. They allow for differing marginals, numbers of blocks, block sizes and forms of between block dependence. They also allow for solution of the variational optimization using efficient stochastic gradient methods. The approach is demonstrated using four different statistical models and 16 datasets which have posteriors that are challenging to approximate. This includes models that use global-local shrinkage priors for regularization, and hierarchical models for smoothing and heteroscedastic time series. In all cases, our method produces more accurate posterior approximations than benchmark VI methods that either assume block independence or factor-based dependence, at limited additional computational cost. A python package implementing the method is available on GitHub at https://github.com/YuFuOliver/VCVI_Rep_PyPackage.

stat.ML↗

arXiv2024

Natural Gradient Hybrid Variational Inference with Application to Deep Mixed Models

Stochastic models with global parameters and latent variables are common, and for which variational inference (VI) is popular. However, existing methods are often either slow or inaccurate in high dimensions. We suggest a fast and accurate VI method for this case that employs a well-defined natural gradient variational optimization that targets the joint posterior of the global parameters and latent variables. It is a hybrid method, where at each step the global parameters are updated using the natural gradient and the latent variables are generated from their conditional posterior. A fast to compute expression for the Tikhonov damped Fisher information matrix is used, along with the re-parameterization trick, to provide a stable natural gradient. We apply the approach to deep mixed models, which are an emerging class of Bayesian neural networks with random output layer coefficients to allow for heterogeneity. A range of simulations show that using the natural gradient is substantially more efficient than using the ordinary gradient, and that the approach is faster and more accurate than two cutting-edge natural gradient VI methods. In a financial application we show that accounting for industry level heterogeneity using the deep mixed model improves the accuracy of asset pricing models. MATLAB code to implement the method can be found at: https://github.com/WeibenZhang07/NG-HVI.

stat.ML↗

arXiv2024

Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns

Skew-t copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio (2003) allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A stochastic gradient ascent algorithm is used to solve the variational optimization. The methodology is used to estimate skew-t factor copula models with up to 15 factors for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. In a moving window study we show that the asymmetric dependencies also vary over time, and that intraday predictive densities from the skew-t copula are more accurate than those from benchmark copula models. Portfolio selection strategies based on the estimated pairwise asymmetric dependencies improve performance relative to the index.

econ.EM↗

arXiv2024

Cutting Feedback in Misspecified Copula Models

In copula models the marginal distributions and copula function are specified separately. We treat these as two modules in a modular Bayesian inference framework, and propose conducting modified Bayesian inference by "cutting feedback". Cutting feedback limits the influence of potentially misspecified modules in posterior inference. We consider two types of cuts. The first limits the influence of a misspecified copula on inference for the marginals, which is a Bayesian analogue of the popular Inference for Margins (IFM) estimator. The second limits the influence of misspecified marginals on inference for the copula parameters by using a pseudo likelihood of the ranks to define the cut model. We establish that if only one of the modules is misspecified, then the appropriate cut posterior gives accurate uncertainty quantification asymptotically for the parameters in the other module. Computation of the cut posteriors is difficult, and new variational inference methods to do so are proposed. The efficacy of the new methodology is demonstrated using both simulated data and a substantive multivariate time series application from macroeconomic forecasting. In the latter, cutting feedback from misspecified marginals to a 1096 dimension copula improves posterior inference and predictive accuracy greatly, compared to conventional Bayesian inference.

stat.ME↗

arXiv2024

Regression Copulas for Multivariate Responses

We propose a novel distributional regression model for a multivariate response vector based on a copula process over the covariate space. It uses the implicit copula of a Gaussian multivariate regression, which we call a ``regression copula''. To allow for large covariate vectors their coefficients are regularized using a novel multivariate extension of the horseshoe prior. Bayesian inference and distributional predictions are evaluated using efficient variational inference methods, allowing application to large datasets. An advantage of the approach is that the marginal distributions of the response vector can be estimated separately and accurately, resulting in predictive distributions that are marginally-calibrated. Two substantive applications of the methodology highlight its efficacy in multivariate modeling. The first is the econometric modeling and prediction of half-hourly regional Australian electricity prices. Here, our approach produces more accurate distributional forecasts than leading benchmark methods. The second is the evaluation of multivariate posteriors in likelihood-free inference (LFI) of a model for tree species abundance data, extending a previous univariate regression copula LFI method. In both applications, we demonstrate that our new approach exhibits a desirable marginal calibration property.

stat.ME↗

arXiv2022

Implicit copula variational inference

Key to effective generic, or "black-box", variational inference is the selection of an approximation to the target density that balances accuracy and speed. Copula models are promising options, but calibration of the approximation can be slow for some choices. Smith et al. (2020) suggest using tractable and scalable "implicit copula" models that are formed by element-wise transformation of the target parameters. We propose an adjustment to these transformations that make the approximation invariant to the scale and location of the target density. We also show how a sub-class of elliptical copulas have a generative representation that allows easy application of the re-parameterization trick and efficient first order optimization. We demonstrate the estimation methodology using two statistical models as examples. The first is a mixed effects logistic regression, and the second is a regularized correlation matrix. For the latter, standard Markov chain Monte Carlo estimation methods can be slow or difficult to implement, yet our proposed variational approach provides an effective and scalable estimator. We illustrate by estimating a regularized Gaussian copula model for income inequality in U.S. states between 1917 and 2018. An Online Appendix and MATLAB code to implement the method are available as Supplementary Materials.

stat.ME↗

arXiv2022

Variational inference for cutting feedback in misspecified models

Bayesian analyses combine information represented by different terms in a joint Bayesian model. When one or more of the terms is misspecified, it can be helpful to restrict the use of information from suspect model components to modify posterior inference. This is called "cutting feedback", and both the specification and computation of the posterior for such "cut models" is challenging. In this paper, we define cut posterior distributions as solutions to constrained optimization problems, and propose optimization-based variational methods for their computation. These methods are faster than existing Markov chain Monte Carlo (MCMC) approaches for computing cut posterior distributions by an order of magnitude. It is also shown that variational methods allow for the evaluation of computationally intensive conflict checks that can be used to decide whether or not feedback should be cut. Our methods are illustrated in a number of simulated and real examples, including an application where recent methodological advances that combine variational inference and MCMC within the variational optimization are used.

stat.ME↗

arXiv2021

Implicit Copulas: An Overview

Implicit copulas are the most common copula choice for modeling dependence in high dimensions. This broad class of copulas is introduced and surveyed, including elliptical copulas, skew $t$ copulas, factor copulas, time series copulas and regression copulas. The common auxiliary representation of implicit copulas is outlined, and how this makes them both scalable and tractable for statistical modeling. Issues such as parameter identification, extended likelihoods for discrete or mixed data, parsimony in high dimensions, and simulation from the copula model are considered. Bayesian approaches to estimate the copula parameters, and predict from an implicit copula model, are outlined. Particular attention is given to implicit copula processes constructed from time series and regression models, which is at the forefront of current research. Two econometric applications -- one from macroeconomic time series and the other from financial asset pricing -- illustrate the advantages of implicit copula models.

stat.ME↗

arXiv2021

Deep Distributional Time Series Models and the Probabilistic Forecasting of Intraday Electricity Prices

Recurrent neural networks (RNNs) with rich feature vectors of past values can provide accurate point forecasts for series that exhibit complex serial dependence. We propose two approaches to constructing deep time series probabilistic models based on a variant of RNN called an echo state network (ESN). The first is where the output layer of the ESN has stochastic disturbances and a shrinkage prior for additional regularization. The second approach employs the implicit copula of an ESN with Gaussian disturbances, which is a deep copula process on the feature space. Combining this copula with a non-parametrically estimated marginal distribution produces a deep distributional time series model. The resulting probabilistic forecasts are deep functions of the feature vector and also marginally calibrated. In both approaches, Bayesian Markov chain Monte Carlo methods are used to estimate the models and compute forecasts. The proposed models are suitable for the complex task of forecasting intraday electricity prices. Using data from the Australian National Electricity Market, we show that our deep time series models provide accurate short term probabilistic price forecasts, with the copula model dominating. Moreover, the models provide a flexible framework for incorporating probabilistic forecasts of electricity demand as additional features, which increases upper tail forecast accuracy from the copula model significantly.

stat.ME↗

arXiv2021

Fast and Accurate Variational Inference for Models with Many Latent Variables

Models with a large number of latent variables are often used to fully utilize the information in big or complex data. However, they can be difficult to estimate using standard approaches, and variational inference methods are a popular alternative. Key to the success of these is the selection of an approximation to the target density that is accurate, tractable and fast to calibrate using optimization methods. Most existing choices can be inaccurate or slow to calibrate when there are many latent variables. Here, we propose a family of tractable variational approximations that are more accurate and faster to calibrate for this case. It combines a parsimonious parametric approximation for the parameter posterior, with the exact conditional posterior of the latent variables. We derive a simplified expression for the re-parameterization gradient of the variational lower bound, which is the main ingredient of efficient optimization algorithms used to implement variational estimation. To do so only requires the ability to generate exactly or approximately from the conditional posterior of the latent variables, rather than to compute its density. We illustrate using two complex contemporary econometric examples. The first is a nonlinear multivariate state space model for U.S. macroeconomic variables. The second is a random coefficients tobit model applied to two million sales by 20,000 individuals in a large consumer panel from a marketing study. In both cases, we show that our approximating family is considerably more accurate than mean field or structured Gaussian approximations, and faster than Markov chain Monte Carlo. Last, we show how to implement data sub-sampling in variational inference for our approximation, which can lead to a further reduction in computation time. MATLAB code implementing the method for our examples is included in supplementary material.

stat.ME↗

arXiv2020

Marginally-calibrated deep distributional regression

Deep neural network (DNN) regression models are widely used in applications requiring state-of-the-art predictive accuracy. However, until recently there has been little work on accurate uncertainty quantification for predictions from such models. We add to this literature by outlining an approach to constructing predictive distributions that are `marginally calibrated'. This is where the long run average of the predictive distributions of the response variable matches the observed empirical margin. Our approach considers a DNN regression with a conditionally Gaussian prior for the final layer weights, from which an implicit copula process on the feature space is extracted. This copula process is combined with a non-parametrically estimated marginal distribution for the response. The end result is a scalable distributional DNN regression method with marginally calibrated predictions, and our work complements existing methods for probability calibration. The approach is first illustrated using two applications of dense layer feed-forward neural networks. However, our main motivating applications are in likelihood-free inference, where distributional deep regression is used to estimate marginal posterior distributions. In two complex ecological time series examples we employ the implicit copulas of convolutional networks, and show that marginal calibration results in improved uncertainty quantification. Our approach also avoids the need for manual specification of summary statistics, a requirement that is burdensome for users and typical of competing likelihood-free inference methods.

stat.ME↗

arXiv2020

Bayesian Variable Selection for Non-Gaussian Responses: A Marginally Calibrated Copula Approach

We propose a new highly flexible and tractable Bayesian approach to undertake variable selection in non-Gaussian regression models. It uses a copula decomposition for the joint distribution of observations on the dependent variable. This allows the marginal distribution of the dependent variable to be calibrated accurately using a nonparametric or other estimator. The family of copulas employed are `implicit copulas' that are constructed from existing hierarchical Bayesian models widely used for variable selection, and we establish some of their properties. Even though the copulas are high-dimensional, they can be estimated efficiently and quickly using Markov chain Monte Carlo (MCMC). A simulation study shows that when the responses are non-Gaussian the approach selects variables more accurately than contemporary benchmarks. A real data example in the Web Appendix illustrates that accounting for even mild deviations from normality can lead to a substantial increase in accuracy. To illustrate the full potential of our approach we extend it to spatial variable selection for fMRI. Using real data, we show our method allows for voxel-specific marginal calibration of the magnetic resonance signal at over 6,000 voxels, leading to an increase in the quality of the activation maps.

stat.ME↗

arXiv2020

Bayesian Inference for Regression Copulas

We propose a new semi-parametric distributional regression smoother that is based on a copula decomposition of the joint distribution of the vector of response values. The copula is high-dimensional and constructed by inversion of a pseudo regression, where the conditional mean and variance are semi-parametric functions of covariates modeled using regularized basis functions. By integrating out the basis coefficients, an implicit copula process on the covariate space is obtained, which we call a `regression copula'. We combine this with a non-parametric margin to define a copula model, where the entire distribution - including the mean and variance - of the response is a smooth semi-parametric function of the covariates. The copula is estimated using both Hamiltonian Monte Carlo and variational Bayes; the latter of which is scalable to high dimensions. Using real data examples and a simulation study we illustrate the efficacy of these estimators and the copula model. In a substantive example, we estimate the distribution of half-hourly electricity spot prices as a function of demand and two time covariates using radial bases and horseshoe regularization. The copula model produces distributional estimates that are locally adaptive with respect to the covariates, and predictions that are more accurate than those from benchmark models.

stat.ME↗