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Introduction and Datasets
Published in Andrew B. Lawson, Using R for Bayesian Spatial and Spatio-Temporal Health Modeling, 2021
Estimation in Bayesian HMs is based on a parameter posterior distribution, which is a function of a data likelihood and prior distributions for model parameters. Bayesian HMs are often relatively sophisticated and require approximation methods to address parameter estimation. Two major approaches to this approximation are commonly found: (1) posterior sampling where an iterative algorithm is used to provide a sample of parameter values that can be summarized to yield estimates, (2) numerical approximation to an integral, based on an approximate form of the posterior distribution. Posterior sampling is often carried out using Monte Carlo methods and a set of methods called Markov chain Monte Carlo (McMC) have been developed to facilitate this estimation approach. Essentially in this approach the posterior distribution is approximated by samples, and sampled values are used to summarize the posterior quantities of interest (such as mean, variance etc). Numerical approximation of the posterior distribution commonly involves a Laplace approximation which matches a Gaussian distribution to the form of the posterior distribution and provides estimates based on this approximation. Numerical approximation can be computationally advantageous especially in large-scale problems (big data) where sampling becomes inefficient. On the other hand McMC can provide a large amount of information concerning the form of the posterior distribution, which is not usually available immediately from numerical approximation.
Bayesian Background
Published in Emmanuel Lesaffre, Gianluca Baio, Bruno Boulanger, Bayesian Methods in Pharmaceutical Research, 2020
Emmanuel Lesaffre, Gianluca Baio
Integrated nested Laplace approximation (INLA) has been proposed as a computationally convenient alternative for MCMC methods. The classical Laplace approximation is a method to approximate Bayesian parameter estimation, based on a second-order Taylor approximation of the log posterior around the maximum a posteriori estimate. This results in a Gaussian approximation to the posterior, thereby avoiding sampling from the posterior, which can be too time consuming. Note that the Laplace approximation is a special case of adaptive Gaussian quadrature, but there is also non-adaptive Gaussian quadrature. Gaussian quadrature methods replace the integral by a weighted sum: . Q is the order of the approximation. The higher Q, the more accurate the approximation will be. With non-adaptive Gaussian quadrature, the nodes and weights are fixed, independent of f(z)ϕ(z). More specifically the design points are located around zero with dispersion independent of f(z)ϕ(z). With adaptive Gaussian quadrature, the nodes and weights are adapted to the ‘support’ of f(z)ϕ(z) with the design points located around the mean of f(z)ϕ(z) with appropriate dispersion. Note that the Laplace approximation corresponds to adaptive Gaussian quadrature with Q = 1.
Modeling dragonfly population data with a Bayesian bivariate geometric mixed-effects model
Published in Journal of Applied Statistics, 2023
Yulan B. van Oppen, Gabi Milder-Mulderij, Christophe Brochard, Rink Wiggers, Saskia de Vries, Wim P. Krijnen, Marco A. Grzegorczyk
When studying the related literature, we found that hardly any software for modeling bivariate count data is available. To fill this gap, we decided to make our R/JAGS software available on our GitHub repository [27]. A computational bottleneck of our algorithm is the expensive computation of the bivariate geometric (BGe) likelihood. Our future work might aim to reduce the computational inference costs. One potential idea is to switch to the marginal likelihood and to approximate it by a Laplace approximation. Another interesting route of research might be to try to extend the bivariate geometric distribution to multivariate geometric distributions. In principle, the probability generating function (PGF) from [18] can easily be extended to define multivariate geometric distributions, but the mathematical challenge would be to derive the corresponding probability mass function (PMF).
Satisfaction with access and quality of healthcare services for people with spinal cord injury living in the community
Published in The Journal of Spinal Cord Medicine, 2020
Elias Ronca, Anke Scheel-Sailer, Hans Georg Koch, Stefan Essig, Mirjam Brach, Nadja Münzel, Armin Gemperli
Spatially structured linear and logistic regression models were computed using the R-INLA library in the R programming language.24 This suite of functions allow for fast Bayesian inference using Laplace approximation. The two Rasch-constructed interval-scaled response variables were investigated using linear regression analyses, and the binary response variables were examined using logistic regression models. The regression models accounted for residual spatial autocorrelation by including region-specific random effects with conditional autoregressive prior distribution within the hierarchical Bayesian model.25 The effect of travel time was modelled as smoothed curve using a random walk of order 2.26 Missing covariate values were imputed using the missForest library with all variables used in the imputation process.27 Statistical analyses were performed using the R programming language version 3.3.2.28
Out-of-pocket spending for contraceptives in Latin America
Published in Sexual and Reproductive Health Matters, 2020
Lucas Godoy Garraza, Federico Tobar, Iván Rodríguez Bernate
To assess the relationship between macroeconomic variables and contraceptive retail sales, we used different regression approaches, including ordinary least squares (OLS), mixed-effect models fitted by restricted maximum likelihood (ME), and hierarchical models estimated using a fully Bayesian approach (BH). We regressed contraceptive sales on each candidate macroeconomic predictor separately, one at a time. In all cases, we used first differences, that is, the change in the value of each variable with respect to the previous year. For the outcome variable (i.e. contraceptive retail sales) as well as for inflation, we focused on the relative change (rather than the absolute change) to deal with more comparable figures across countries. However, we used the absolute difference in the case of unemployment and poverty rates. All predictors are lagged one period (i.e. they correspond to their value in the previous year) since we are interested in assessing the extent to which changes in the purchase of contraceptives can be anticipated. Using the first difference is a frequent approach to address residual serial correlation, which would otherwise invalidate the OLS estimation. Nevertheless, both the ME and BH models incorporated random intercepts by country and a first-order autoregressive process for the residuals. For Bayesian estimation, the choice of prior distributions for the model parameters favors simpler models over more complex ones when compatible with the data.28 Posterior distributions were estimated using nested Laplace approximation.29 All analysis was implemented with R,30 including the packages survey,31,32 nlme,33 and INLA.29