On the distinction between the conditional probability and the joint probability approaches in the specification of nearest-neighbour systems

Brook, D.

doi:10.1093/biomet/51.3-4.481

Cited by 168 publications

(61 citation statements)

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“…Hence, following the work of Cressie (1991), the CAR model is given by

(T_{italicijk} | T_{ijk'}, k' \neq k, τ_{italicjk}^{2}) ~ 𝒩 (ρ_{j} \sum_{k' \neq k} b_{kk'} T_{ijk'}, τ_{italicjk}^{2})

, k = 1, …, K , where K is the number of spatial nodes. Following Brook’s (1964) lemma, the resulting joint density function of T ijk , k = 1, …, K , takes the form

f (T_{italicij} | τ_{j}^{2}) \propto exp {- \frac{1}{2} T_{italicij}^{'} D_{τ_{j}^{2}}^{- 1} (I - ρ_{j} B) T_{italicij}},

where

τ_{j}^{2} = (τ_{j 1}^{2}, \dots, τ_{italicjK}^{2})'

, B is K × K matrix with B = ( b kk ′ ) and b kk = 0 and

D_{τ_{j}^{2}} = Diag (τ_{j}^{2})

is a K × K diagonal matrix with non-zero entries

{τ_{italicjk}^{2}, k = 1, \dots, K}

. From the classical CAR theory, it is clear that if the precision matrix

D_{τ_{j}^{2}}^{- 1} (I - ρ_{j} B)

is positive definite, then (2.1) is a proper distribution.…”

Section: A Generalized Latent Variable Modelmentioning

confidence: 99%

Bayesian latent variable models for spatially correlated tooth-level binary data in caries research

Zhang

Todem

Kim

et al. 2011

Statistical Modelling

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Analysis of dental caries is traditionally based on aggregated scores, which are summaries of caries experience for each individual. A well-known example of such scores is the decayed, missing and filled teeth or tooth surfaces index introduced in the 1930s. Although these scores have improved our understanding of the pattern of dental caries, there are still some fundamental questions that remain unanswered. As an example, it is well believed among dentists that there are spatial symmetries in the mouth with respect to caries, but this has never been evaluated in a statistical sense. An answer to this question requires the analysis to be performed at subunits within the mouth, which necessitates the use of methods for correlated data. We propose a Bayesian generalized latent variable model coupled with an undirected graphical model to investigate the unique spatial distribution of tooth-level caries outcomes in the mouth. Data from the Signal Tandmobiel® study in Flanders, a dental longitudinal survey, are used to illustrate the methodology.

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“…Hence, following the work of Cressie (1991), the CAR model is given by

(T_{italicijk} | T_{ijk'}, k' \neq k, τ_{italicjk}^{2}) ~ 𝒩 (ρ_{j} \sum_{k' \neq k} b_{kk'} T_{ijk'}, τ_{italicjk}^{2})

, k = 1, …, K , where K is the number of spatial nodes. Following Brook’s (1964) lemma, the resulting joint density function of T ijk , k = 1, …, K , takes the form

f (T_{italicij} | τ_{j}^{2}) \propto exp {- \frac{1}{2} T_{italicij}^{'} D_{τ_{j}^{2}}^{- 1} (I - ρ_{j} B) T_{italicij}},

where

τ_{j}^{2} = (τ_{j 1}^{2}, \dots, τ_{italicjK}^{2})'

, B is K × K matrix with B = ( b kk ′ ) and b kk = 0 and

D_{τ_{j}^{2}} = Diag (τ_{j}^{2})

is a K × K diagonal matrix with non-zero entries

{τ_{italicjk}^{2}, k = 1, \dots, K}

. From the classical CAR theory, it is clear that if the precision matrix

D_{τ_{j}^{2}}^{- 1} (I - ρ_{j} B)

is positive definite, then (2.1) is a proper distribution.…”

Section: A Generalized Latent Variable Modelmentioning

confidence: 99%

Bayesian latent variable models for spatially correlated tooth-level binary data in caries research

Zhang

Todem

Kim

et al. 2011

Statistical Modelling

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“…By Brook’s Lemma (Brook 1964), the full conditionals in Equation (2) imply the following joint prior on μ up to a normalizing constant:

π (μ ∣ τ^{2}) \propto \exp true{- 0.5 τ^{- 2} \sum_{i \sim j} {(μ_{i} - μ_{j})}^{2} true},

which is also called pairwise difference prior since it only depends on the differences of all neighbor pairs (Besag 1993). …”

Section: Modelmentioning

confidence: 99%

“…By Brook’s Lemma (Brook 1964), the prior of μ is guaranteed to exist. However, with our specification of the conditional priors on the μ i s in Equation (5), the prior of μ does not have a tractable density.…”

Section: Posterior Estimationmentioning

confidence: 99%

Pre-surgical fMRI Data Analysis Using a Spatially Adaptive Conditionally Autoregressive Model

Liu¹,

Berrocal²,

Bartsch³

et al. 2016

Bayesian Anal.

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Spatial smoothing is an essential step in the analysis of functional magnetic resonance imaging (fMRI) data. One standard smoothing method is to convolve the image data with a three-dimensional Gaussian kernel that applies a fixed amount of smoothing to the entire image. In pre-surgical brain image analysis where spatial accuracy is paramount, this method, however, is not reasonable as it can blur the boundaries between activated and deactivated regions of the brain. Moreover, while in a standard fMRI analysis strict false positive control is desired, for pre-surgical planning false negatives are of greater concern. To this end, we propose a novel spatially adaptive conditionally autoregressive model with variances in the full conditional of the means that are proportional to error variances, allowing the degree of smoothing to vary across the brain. Additionally, we present a new loss function that allows for the asymmetric treatment of false positives and false negatives. We compare our proposed model with two existing spatially adaptive conditionally autoregressive models. Simulation studies show that our model outperforms these other models; as a real model application, we apply the proposed model to the pre-surgical fMRI data of two patients to assess peri- and intra-tumoral brain activity.

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“…According to Brook’s (1964) theorem, this set of full-conditional distributions implies that the joint distribution for X satisfies…”

Section: Bayesian Hierarchical Models For Multivariate Lattice Datamentioning

confidence: 99%

Bayesian Inference for General Gaussian Graphical Models With Application to Multivariate Lattice Data

Dobra

Lenkoski

Rodríguez

2011

Journal of the American Statistical Association

104

138

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We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcomes and the spatial correlation structure, thereby allowing for adaptive smoothing and spatial autocorrelation parameters. Our methods are illustrated using a simulated example and a real-world application which concerns cancer mortality surveillance. Supplementary materials with computer code and the datasets needed to replicate our numerical results together with additional tables of results are available online.

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On the distinction between the conditional probability and the joint probability approaches in the specification of nearest-neighbour systems

Cited by 168 publications

References 0 publications

Bayesian latent variable models for spatially correlated tooth-level binary data in caries research

Bayesian latent variable models for spatially correlated tooth-level binary data in caries research

Pre-surgical fMRI Data Analysis Using a Spatially Adaptive Conditionally Autoregressive Model

Bayesian Inference for General Gaussian Graphical Models With Application to Multivariate Lattice Data

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