Integrated Nested Laplace Approximation

Blangiardo, M., Cameletti, M., Baio, G. and Rue, H., 2013. Spatial and spatio-temporal models with R-INLA. Spatial and spatio-temporal epidemiology, 4, pp.33-49.

Advantages of the Bayesian Approach

• The specification of prior distributions allows the formal inclusion of information that can be obtained through previous studies or from expert opinion
   
• The posterior probability that a parameter does/does not exceed a certain threshold is easily obtained from the posterior distribution, providing a more intuitive and interpretable quantity than a frequentist p-value
   
• Easy to specify a hierarchical structure on the data and parameters

Bayesian Approach for Epidemiological Data

• Epidemiological data are often characterized by a spatial and/or temporal structure which needs to be taken into account in the inferential process
  -  if the data consists of aggregated counts of outcomes and covariates, typically disease mapping and ecological regression can be specified
  -  if the outcome or risk factors data are observed at point locations, then geostatistical models are considered as suitable representations of the problem
   
• Both models can be specified in a Bayesian framework by simply extending the concept of hierarchical structure
  -  allowing to account for similarities based on the neighborhood (for areal data) or on the distance (for point-reference data)

Bayesian Inference in Spatial Epidemiological Studies

• Bayesian inference has become very popular in spatial analyses in recent years
  -  due to the availability of computation methods to tackle fitting of spatial models
  -  e.g. Besag, York, and Mollie (1991) proposed a method to fit a spatial model using MCMC, which has been extensively used and extended to consider different types of fixed and random effects of spatial and spatio-temporal analysis
   
• Fitting these models has been possible because of the availability of different computational techniques, e.g. MCMC
  -  however, for large models or big data sets, MCMC can be tedious
  -  reaching the required number of samples (burn-in period) can take a long time, and autocorrelation may arise and an increased number of iterations may be required

Approximation of Posterior Distribution

• Posterior distributions may be approximated in some way
  -  however, most models are highly multivariate and approximating the full posterior distribution may not be possible in practice
   
• The integrated nested Laplace approximation (INLA) approach has been developed as a computationally efficient alternative to MCMC
   
• INLA focuses on the posterior marginals for latent Gaussian models
  -  although these models may seem rather restricted, they appear in a fair number of fields and are usually enough to handle most of the problems in spatial analyses
  -  ranging from generalized linear mixed to spatial and spatio-temporal models
   
• INLA can also be combined with the Stochastic Partial Differential Equation (SPDE) approach to implement spatial and spatio-temporal models for point-reference data

Spatial Data

• Spatial data are defined as realizations of a stochastic process indexed by space
  
   
• The actual data can be represented by a collection of observations
  
  
   
• The set {s1,...sn} indicates the spatial units at which the measurements are taken
   
• Depending on D being a spatial surface or a countable collection of 2-dimensional spatial units, the problem can be specified as a spatially continuous or discrete random process, respectively 

Examples

• Spatially continuous random process (for point-reference data):
  -  a collection of air pollutant measurements obtained by monitors located in the set (s1,...sn) of n points
  -  in this case, y is a realization of the air pollution process that changes continuously in space
   
• Spatially discrete random process (for areal data):
  -  spatial pattern of a certain health condition observed in a set (s1,...,sn) of n areas (such as census tracts or counties)
  -  in this case, y represent a suitable summary (e.g. the number of cases observed in each area)

Spatial Model within the Bayesian Framework

• The first step is to identify a probability distribution for the observed data (likelihood)
  -  usually we use a distribution from the Exponential family, indexed by a set of parameters θ accounting for the spatial correlation
   
• For point-referenced data:
  -  the parameters are defined as a latent stationary Gaussian field, a function of some hyper-parameters ψ associated with a suitable prior distribution p(ψ)
  -  this is equivalent to assuming that θ has a multivariate Normal distribution with mean µ=(µ1,...,µn) and spatially structured covariance matrix Σ
  
  
  
   
• For areal data:
  -  it is possible to reformulate the problem in terms of the neighborhood structure

INLA

• We are usually interested in estimating the effect of a set of relevant covariates on some function (typically the mean) of the observed data, while accounting for the spatial or spatio-temporal correlation implied in the model

the mean for the i-th unit

intercept

• We can vary the form of the functions fl() to accomodate a wide range of models, from standard and hierarchical regression, to spatial and spatio-temporal models

INLA (cont'd)

• The INLA approach exploits the assumptions of the model to produce a numerical approximation to the posteriors of interest, based on the Laplace approximation
   
• INLA proceeds by first exploring the marginal joint posterior for the hyper-parameters in order to locate the mode
   
• A grid search is then performed and produces a set of relevant points together with a corresponding set of weights to give the approximation to this distribution
   
• Each marginal posterior can be obtained using interpolation based on the computed values and correcting for skewness

Lab: R-INLA

R-INLA

• The R-INLA package provides an interface to INLA so that models can be fitted using standard R commands
   
• Assuming a vector of two covariates x1, x2, and a function f() indexed by a third covariate z1:
  
   
• The coefficients α, β1, β2 are by default given independent prior Normal distributions with zero mean and small precision
   
• The term f() is used to specify the structure of the function f(), using the following notation:
  
  -  the default choice is model="iid"
  -  the specification can be used to build standard hierarchical models
   
• Once the model has been specified, we can run the INLA by:

git pull

Spatial Areal Data: Suicides in London

• To investigate suicide mortality in n=32 London boroughs in the period 1989-1993
  
  
  
  
   
• Assuming a Besag-York-Mollie (BYM) specification, υi is the spatially structured residual, modeled using an intrinsic conditional autoregressive structure

the number of areas which share boundaries with the i-th one (i.e. its neighbors)