Disease surveillance is an important public health practice, as it provides information which can be used to make successful interventions. Innovative surveillance systems are being developed to improve early detection and investigation of outbreaks, with the Bayesian models attracting a lot of interest recently. Outbreak detection requires a system that will be able to flag areas that are differentially expressed. Both test-based and model-based techniques exist in the literature. Within the Bayesian framework, spatio-temporal hierarchical models are able to give robust results due to their flexibility. Through the specification of spatially and/or temporally structured random effects information is shared between areas and/or time points, increasing the strength of the parameter estimates. These models are designed to provide estimates and describe risk patterns, however very limited research exists in models that are able to provide a detection mechanism. In addition, these do not correct for multiple testing which is common in these studies, where a large set of comparisons is conducted. BaySTDetect is a recently developed method by Li (2012) able to detect outbreaks, and also to control for multiple testing through the specification of the False Discovery Rate (FDR). The objective of this work is to analyse mortality data on asthma disease by using the BaySTDetect method. Mortality data are obtained from the Small Area Health Statistics Unit (SASHU) at Imperial College. Important aspect of this work is to examine the relationship between mortality data on asthma and GP drug prescriptions on asthma, by including this as an explanatory variable in the model. GP drug prescription data are released monthly by the English National Health Service (NHS) for all general practices in England and all drugs. The dataset that is currently available to be used in this project includes 8004 practices along with the number of prescribed drugs for asthma each month. The BaySTDetect model fitted to the data includes a spatial random effect component at super output area (SOA) level and a temporal effect component at month level. The temporal coverage is from August 2010 to November 2013. OpenBUGS software is used for the implementation of the models, and GIS for the mapping. Next step of this work is to use Integrated Nested Laplace Approximation (INLA) for the implementation of the models. -----
In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This work proposes stochastic gradient-based inference [Welling and Teh, ICML, 2011] to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the calculation of stochastic gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structures on the covariance or reducing the number of input vectors. -----
Latent Gaussian models (LGMs) form a flexible subclass of Bayesian hierarchical models and have become popular in many areas of statistics and various fields of applications, as LGMs are both practical and readily interpretable. Although LGMs are well suited from a statistical modeling point of view their posterior inference becomes computationally challenging when latent models are desired for more than just the mean structure of the data density function; the number of parameters associated with the latent model increase; or when the data density function is non-Gaussian. We propose a novel computationally efficient Markov chain Monte Carlo (MCMC) scheme which serves to address these computational issue, we refer to as the MCMC split sampler. The sampling scheme is designed to handle LGMs where latent models are imposed on more than just the mean structure of the likelihood; to scale well in terms of computational efficiency when the dimensions of the latent models increase; and to be applicable for any choice of a parametric data density function. The main novelty of the MCMC split sampler lies in how the model parameters of a LGM are split into two blocks, such that one of the blocks exploits the latent Gaussian structure in a natural way and becomes invariant of the data density function. -----
The Integrated Nested Laplace Approximation and its associate R-INLA package provide a suitable framework for approximate Bayesian inference. In particular, R-INLA will fit complex Bayesian hierarchical models in a fraction of the time required by other computational intensive methods such as Markov Chain Monte Carlo. However, a limitation of INLA is that in order to fit a model it needs to be implemented within R-INLA. Also, INLA only provides marginal inference of the model parameters and other related quantities. In this poster we will show how to extend INLA and R-INLA by combining it with MCMC. In this way, we will provide an easy way to extend the number of models that R-INLA can fit, as well as other other important topics such as (low dimension) multivariate inference and handling missing values in the covariates of regression models. Joint work with Håvard Rue. -----
In this poster we will describe the newly implemented 'slm' latent effect in R-INLA to fit different spatial econometrics models. This new latent effect implements a random effect with a spatial autoregressive specification (SAR), so that new spatial models can be fitted with R-INLA. We will describe the different types of spatial econometrics models that can be fitted using the 'slm' latent effect. Furthermore, we will illustrate the use of some of these spatial econometrics models using examples on house prices in Boston and the probability of reopening a business in New Orleans in the aftermath of hurricane Katrina. Joint work with R. S. Bivand and H. Rue. -----
The Expected Value of Perfect Partial Information (EVPPI) is a decision-theoretic measure of the "cost" of uncertainty in decision making used principally in health economic decision making. Despite having optimal properties in terms of quantifying the value of decision uncertainty, the EVPPI is rarely used in practise. This is due to the prohibitive computational time required to estimate the EVPPI via Monte Carlo simulations. However, a recent development has demonstrated that the EVPPI can be estimated by non-parametric regression methods, which have significantly decreased the computation time required to approximate the EVPPI. Under certain circumstances, high-dimensional Gaussian Process regression is suggested, but this can still be prohibitively expensive. Exploiting the link between Gaussian Fields and Gaussian Markov Random Fields demonstrated in Lindgren et al. (2011) and using INLA for fast computation allows us to decrease the computation time for fitting 2-dimensional Gaussian Processes. To extend this method to higher dimensional inputs, we project from this high dimensional space into 2 dimensions. Results show that this new method for Gaussian Process regression is in line with the standard Gaussian Process regression method for EVPPI calculation. -----
Working in analysis, fitting the right model to the data is of key importance for several reasons: prediction, inference, etc. The problem we often run into is when the data at best closely resembles that of the assumed model, but clearly some outliers or other descrepancies do not fit the underlying model assumptions. For the sake of Gaussian analysis, we would like to start at a broader framework and establish a method of estimation that is able to reciprocate the base Gaussian model, if the data fits the model. At the same time we also aim to maintain some flexibility to accomodate possible descrepancies in the data. In this case, we introduce a measure of skewness to the Gaussian distribution and concentrate on the skew Gaussian model. -----
State space models may have more stochastic parameters than observations. The model fit can then be perfect so that the likelihood of the data becomes infinite. With Gaussian models the problem is associated with a zero variance of measurement errors making the normal density infinite. The problem is generally avoided by integration over a major part of the parameters --- analytically or numerically --- to obtain a predictive likelihood of observations conditional on a relatively small number of parameters --- typically comprising all variances. The space of such likelihoods is then analyzed to determine the posterior distribution of the non-integrated parameters. Relatively informative priors (RIP) allows numerical integration to take place over variances as well, leaving only a single parameter for the final analysis. In the context of Gaussian models RIP involves informative priors of variances introduced without any preceding estimation. The idea stems from the term "Boundary avoiding priors"" introduced by Gelman (2014). The distribution class is changed from Gamma to Inverse Gamma, though. Whether using RIP or not, the non-integrated parameters represent an index (generally multi-dimensional) to members of a class of predictive models of the observations. Watanabe's information criterion based on a log-likelihood and a penalty, guides decisions on which indexed model (or which ensemble of indexed models), to lean on. Models with higher effective dimension in terms of higher variance latent variables will be more penalized. I propose to apply the WAIC-penalized likelihood in the determination of a penalized posterior of non-integrated parameters --- although this is controversial and deviates from standard Bayesian posteriors. This is an efficient strategy in finding predictive models with low WAIC-score. Otherwise, substantial amounts of trial and error would be needed to find these models. The combination with RIP means that the dimension of the model index can be reduced to 1, with additional efficiency obtained. -----
A Bayesian latent Gaussian model is proposed for flood frequency analysis. The model uses monthly maxima as opposed to the most commonly used annual maxima to make better use of data. A linear mixed model is incorporated at the latent level to account for seasonal dependence of parameters and to allow the model to be extrapolated to river catchments where little or no data are available. The observed data come from twelve river catchments around Iceland. The generalized extreme value distribution is selected as a data distribution with a separate triplet of likelihood parameters for each combination of month and river. The result is a high dimensional model that comes with substantial computational costs. The Markov chain Monte Carlo inference method makes use of a newly developed sampling scheme called the MCMC split-sampler. The MCMC split-sampler utilizes the latent Gaussian structure such that efficient posterior sampling is obtained. The specification of prior distributions makes use of penalizing complexity priors to introduce a robust method to infer the latent parameters and the hyperparameters. The results indicate that using monthly maxima is a viable option in flood frequency analysis and that the latent linear mixed model improves the estimation of the likelihood parameters. -----
Broad scale tornado climatology is well described and physically understood. However local scale variations to this climatology are less documented and poorly understood. Factors that contribute to these differences remain largely unknown. Tornadoes are discrete events, clustered in space and time, and locally quite rare making it risky to interpret results from simple or routine statistical applications. The authors develop and implement a statistical model to examine variations in tornado climate caused by environmental factors. The model uses historical tornado tracks aggregated to the county level. The aggregation makes it easy to control for differences in county size and population density. The model is fit using the method of integrated nested Laplacian approximation (INLA) to solve the Bayesian integrals. The Bayesian setup makes it straightforward to include a correlated random effects term. A choropleth map of the random effects shows where tornado activity is high relative to the state-wide average after statistically controlling for some important non-meteorological factors. The approach is described using Kansas tornadoes. Similar maps are shown for a few other states for comparison. -----
A number of malaria surveys in Africa collect information about the disease and about coverage indicators of different interventions. It is of interest to assess the effects of interventions on the dynamics of the disease. The Roll Back Malaria Partnership has defined intervention coverage indicators related to ownership and usage of insecticide-treated mosquito nets. Some indicators may capture the effect on malaria risk, others not. Studies have shown that the effect of malaria interventions may not be constant but vary in space. Bayesian variable selection of spatiotemporally varying coefficients could address both questions: (i) identification of appropriate malaria intervention coverage indicators; and (ii) estimation of their effects over space and time. For such models, inference is computationally expensive due to intensive covariance matrix calculations. Gaussian predictive process approximations to spatial processes offer a flexible solution to address matrix calculations, but knot selection may influence variable selection. We employ approximate Bayesian inference with iteratively integrated nested Laplace approximations to calculate the marginal likelihood of all possible models. We use predictive processes and assess the sensitivity of Bayesian variable selection to knot selection. Results show that variable selection is sensitive to the predictive process approximation. The methodology can be extended to include correlation between the effects of interventions and to perform Bayesian model averaging. -----
The selection of appropriate hyperpriors for variance parameters is a sensible topic in all kinds of Bayesian regression models involving the specification of (conditionally) Gaussian prior structures. The variance parameters determine a data-driven, adaptive amount of prior variability or precision. We consider the special case of structured additive distributional regression where Gaussian priors are used to enforce specific properties such as smoothness on various effect types combined in predictors for multiple parameters related to the distribution of the response. Relying on a recently proposed class of penalised complexity priors, we derive scale-dependent hyperpriors with prior elicitation being facilitated by assumptions on the scaling of the different effect types. The posterior distribution is assessed with an adaptive Markov chain Monte Carlo scheme and conditions for its propriety are studied theoretically. We investigate the new type of scale-dependent priors in simulations with alternative priors and demonstrate the appropriateness compared to the standard inverse gamma priors along two real data illustrations. -----
Various characteristics in typical wind farms make it a challenging problem to generate competitive wind power forecasting that allows also for the quantification of uncertainties. Since wind is intermittent , there is a probability mass at zero wind power, which gives a discontinuity in the distribution function. Furthermore, the distribution of wind power is clearly not Gaussian. It is propose a hierarchical spatial model to describe marginal densities for several locations and to mimic the spatial dependence structure for the yearly average wind power. We model it using a right-skewed continuous distribution with a stochastic mean that includes both, covariates and spatial structure, resulting in the latent Gaussian random field with Matérn covariance function. It is also proposed a method to estimate the wind power instantaneously for a set of locations given that we do not observe them, which helps utility managers to plan for transmission, purchase and distribution of electricity. This type of setting is modelled with a mixture of a degenerated distribution at zero and a right-skewed continuous distribution for the non-zero values, both distributions sharing a Gaussian random field with with Matérn covariance function. Note that, for those models, the posterior marginals are not available in closed form owing to the non-Gaussian response variables. To evaluate probabilistic forecast, we use integrated Nested Laplace approximation and directly compute very accurate approximations to the posterior marginals, with computational benefits. The resulting predictions are evaluated on 344 wind farms in the western Denmark and compared with some benchmark methods. -----
Geostatistical hierarchical additive models often entail a linear component, also known as trend, and a latent field. The latter is used to describe complicated spatial dependencies and usually its variance or precision matrix depends on several parameters that are to be estimated. As the size of the datasets available to the scientific community has been steadily increasing over the years, it appears crucial to develop computational techniques capable of efficiently tackling them. In this work, we focus on a bivariate latent field which is a Gaussian Markov Random Field (GMRF) over a “big” regular grid. It depends on five parameters, which quantify the cross-dependencies in the data. From both a frequentist and Bayesian point of view, it is important to provide a mathematical description of the parameter space where the associated sparse precision is positive definite. We show how a convenient asymptotic approximation not only makes it possible to solve the above mentioned task, bus also leads to efficient algorithms, with respect to the grid size, to compute the log-determinant of the precision matrix and quadratic forms where the latter matrix is present. We benchmark our approach with the sparse linear algebra functions provided by the “spam” R package and show thereof a consistent gain in both computational speed and memory usage. Lastly, we outline how our thorough geometrical understanding of the positive-definiteness domain can be used to introduce a non-stationary generalization of the above described GMRF formulation, for instance by introducing a further layer in the precision matrix of the considered GMRF. -----
Inference for latent Gaussian models is computationally demanding, because we must deal with marginal distributions expressed as high-dimensional integrals. The posterior distribution may often be viewed as a sparse graphical model, with many pairs of conditionally independent variables. We aim to exploit this sparse structure to reduce the computational burden of inference. To do this, we combine methods for approximate storage of continuous distributions with new approaches for inference in graphical models with sparse, but not tree-like, structure. The result is an approximate method, with two tuning parameters which allow us to trade-off the error in the approximation with the cost of computation. -----
Usually in Ecology the availability of the data is not as good as we want. Most of the environmental studies are based on presence/absence data depending of the species. Working with animal species and particularly with small species (as our case amphibians and reptiles) the possibility to have more absences than presences is more than common. The main of this study is to define the best model working with data with a low level of presences using Watanabe-Akaike information criteria (WAIC) and finally compare with a model of species joined, using the presences and absences of related species together. -----
Analysing the collected evidence of a systematic review in form of a meta-analysis enjoys increasing popularity and provides a valuable instrument for decision making. The rationale for a network meta-analysis (NMA) is the inclusion of the whole body of evidence as opposed to direct pairwise treatment comparisons. A statistical framework for NMA was laid out by a broad and growing body of literature. Bayesian estimation of NMA models is propagated, especially if correlated random effects for multi-arm trials are included. The standard method for Bayesian estimation is Markov chain Monte Carlo (MCMC) sampling, which is a computational burden if the number of edges in the network is large and if cross-validation is needed to assess sources of inconsistency. An alternative to MCMC sampling is the recently suggested approximate Bayesian method of integrated nested Laplace approximations (INLA). INLA was demonstrated to deliver accurate approximations in various hierarchical models. We discuss the estimation of NMA with INLA and demonstrate its application for two well established NMA models. -----
Penalized regression on B-splines, proposed by Eilers and Marx (1996) and denoted as P-spline, is a flexible and stable approach for fitting smooth functional effects in generalized additive models. The Bayesian approach to P-splines assumes an intrinsic Gaussian Markov random field (IGMRF) prior on the spline coefficients, conditional on a precision hyper-parameter $\tau$. Prior elicitation of $\tau$ is a critical issue; one difficulty is that the assumptions made by the user about the number of knots can have a substantially impact on the scale of $\tau$. To overcome this scaling issue one may consider building priors on an interpretable property of the model, indicating the complexity of the smooth function to be estimated. Following this idea, we propose Penalized Complexity (PC) priors for the number of effective degrees of freedom in P-splines. In this poster, we present the general ideas behind the construction of these new PC priors, describe their properties and show how to implement them in P-splines for Gaussian data. Joint work with Håvard Rue. |