---
title: "Conditional Autoregressive Models"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(message = FALSE)
library(tidyverse)
library(mnormt)
library(spdep)
set.seed(03202021)
```


## Gaussian Model

Suppose the full conditionals are specified as

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Then using Brooks' Lemma, the joint distribution is
$$p(y_1, \dots, y_n) \propto \exp \left(-\frac{1}{2}\boldsymbol{y}^T D^{-1} (I - B) \boldsymbol{y} \right),$$
where $B$ is a matrix with entries $b_{ij}$ and D is a diagonal matrix with diagonal elements $D_{ii} = \tau_i^2$.

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The previous equation suggests a multivariate normal distribution,

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Now the full conditional distribution can be written as
$$Y_i|y_j, j\neq i \sim N \left(\sum_j w_{ij} y_j / w_{i+}, \tau^2 / w_{i+} \right)$$

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Similarly the joint distribution is now
$$p(y_1, \dots, y_n) \propto \exp \left(-\frac{1}{2 \tau^2}\boldsymbol{y}^T  (D_w - W) \boldsymbol{y} \right)$$
where $D_w$ is a diagonal matrix with diagonal entries $(D_w)_{ii} = w_{i+}$ 

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The joint distribution can also be re-written as
$$p(y_1, \dots, y_n) \propto \exp \left(-\frac{1}{2 \tau^2} \sum_{i \neq j} w_{ij} (y_i - y_j)^2\right)$$

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However, both these formulations results in an improper distribution. This could be solved with a constraint, such as $Y_i = 0$.

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The result is the joint distribution is improper, despite proper full conditional distributions. 

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## IAR

The IAR cannot be used to model data directly, rather this is used a prior specification 

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The impropriety can be remedied by defining a parameter $\rho$ such that


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With or without $\rho,$ $p(\boldsymbol{y})$ (or the Bayesian posterior when the CAR specification is placed on the spatial random effects) is proper.

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When using $\rho$, the full conditional becomes 

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## Simultaneous Autoregression Model

Rather than specifying the distribution on $\boldsymbol{Y}$, as in the CAR specification, the distribution can be specified for $\boldsymbol{\epsilon}$ which induces a distribution for $\boldsymbol{Y}$.

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## SAR Model

If the matrix $(I - B)$ is full rank, then 
$$\boldsymbol{Y} \sim N \left(\boldsymbol{0},(I - B)^{-1} \tilde{D} ((I - B)^{-1})^T \right)$$

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If $\tilde{D} = \sigma^2 I$, then $\boldsymbol{Y} \sim N \left(\boldsymbol{0},\sigma^2 \left[(I - B)  (I - B)^T \right]^{-1} \right)$

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#### Choosing B

There are two common approaches for choosing B

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SAR Models are often introduced in a regression context, where the residuals $(\boldsymbol{U})$ follow a SAR model.

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Let $\boldsymbol{U} = \boldsymbol{Y} - X \boldsymbol{\beta}$ and then $\boldsymbol{U} = B \boldsymbol{U} + \boldsymbol{\epsilon}$ which results in
$$\boldsymbol{Y} = B \boldsymbol{Y} + (I-B) X \boldsymbol{\beta} + \boldsymbol{\epsilon}$$

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Hence the model contains a spatial weighting of neighbors $(B \boldsymbol{Y})$ and a regression component $((I-B) X \boldsymbol{\beta} )$.

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