The structure of the concentration and covariance matrix in a naive Bayes model

Naive Bayes model

Consider this model: \[ x_i = a x_0 + e_i, \quad i=1, \dots, 4 \] and \(x_0=e_0\). All terms \(e_0, \dots, e_3\) are independent and \(N(0,1)\) distributed. Let \(e=(e_0, \dots, e_3)\) and \(x=(x_0, \dots x_3)\). Isolating error terms gives that \[ e = L_1 x \] where \(L_1\) has the form

L1chr <- diag(4)
L1chr[2:4, 1] <- "-a"
L1 <- as.Sym(L1chr)
L1

## Yacas matrix:
##      [,1] [,2] [,3] [,4]
## [1,] 1    0    0    0   
## [2,] -a   1    0    0   
## [3,] -a   0    1    0   
## [4,] -a   0    0    1

If error terms have variance \(1\) then \(\mathbf{Var}(e)=L \mathbf{Var}(x) L'\) so the covariance matrix is \(V1=\mathbf{Var}(x) = L^- (L^-)'\) while the concentration matrix (the inverse covariances matrix) is \(K=L' L\).

L1inv <- Simplify(Inverse(L1))
K1 <- Simplify(Transpose(L1) * L1)
V1 <- Simplify(L1inv * Transpose(L1inv))

cat(
  "\\begin{align} 
    K_1 &= ", TeXForm(K1), " \\\\ 
   V_1 &= ", TeXForm(V1), " 
  \\end{align}", sep = "")

\[\begin{align} K_1 &= \left( \begin{array}{cccc} 3 a ^{2} + 1 & - a & - a & - a \\ - a & 1 & 0 & 0 \\ - a & 0 & 1 & 0 \\ - a & 0 & 0 & 1 \end{array} \right) \\ V_1 &= \left( \begin{array}{cccc} 1 & a & a & a \\ a & a ^{2} + 1 & a ^{2} & a ^{2} \\ a & a ^{2} & a ^{2} + 1 & a ^{2} \\ a & a ^{2} & a ^{2} & a ^{2} + 1 \end{array} \right) \end{align}\]

Slightly more elaborate:

L2chr <- diag(4)
L2chr[2:4, 1] <- c("-a1", "-a2", "-a3")
L2 <- as.Sym(L2chr)
L2

## Yacas matrix:
##      [,1] [,2] [,3] [,4]
## [1,] 1    0    0    0   
## [2,] -a1  1    0    0   
## [3,] -a2  0    1    0   
## [4,] -a3  0    0    1

Vechr <- diag(4)
Vechr[cbind(1:4, 1:4)] <- c("w1", "w2", "w2", "w2")
Ve <- as.Sym(Vechr)
Ve

## Yacas matrix:
##      [,1] [,2] [,3] [,4]
## [1,] w1   0    0    0   
## [2,] 0    w2   0    0   
## [3,] 0    0    w2   0   
## [4,] 0    0    0    w2

L2inv <- Simplify(Inverse(L2))
K2 <- Simplify(Transpose(L2) * Inverse(Ve) * L2)
V2 <- Simplify(L2inv * Ve * Transpose(L2inv))

cat(
  "\\begin{align} 
    K_2 &= ", TeXForm(K2), " \\\\ 
   V_2 &= ", TeXForm(V2), " 
  \\end{align}", sep = "")

\[\begin{align} K_2 &= \left( \begin{array}{cccc} \frac{w_{1} a_{1} ^{2} w_{2} ^{2} + w_{1} w_{2} ^{2} a_{2} ^{2} + w_{1} w_{2} ^{2} a_{3} ^{2} + w_{2} ^{3}}{w_{1} w_{2} ^{3}} & \frac{ - a_{1}}{w_{2}} & \frac{ - a_{2}}{w_{2}} & \frac{ - a_{3}}{w_{2}} \\ \frac{ - a_{1}}{w_{2}} & \frac{1}{w_{2}} & 0 & 0 \\ \frac{ - a_{2}}{w_{2}} & 0 & \frac{1}{w_{2}} & 0 \\ \frac{ - a_{3}}{w_{2}} & 0 & 0 & \frac{1}{w_{2}} \end{array} \right) \\ V_2 &= \left( \begin{array}{cccc} w_{1} & w_{1} a_{1} & w_{1} a_{2} & w_{1} a_{3} \\ a_{1} w_{1} & w_{1} a_{1} ^{2} + w_{2} & a_{1} w_{1} a_{2} & a_{1} w_{1} a_{3} \\ a_{2} w_{1} & a_{2} w_{1} a_{1} & w_{1} a_{2} ^{2} + w_{2} & a_{2} w_{1} a_{3} \\ a_{3} w_{1} & a_{3} w_{1} a_{1} & a_{3} w_{1} a_{2} & w_{1} a_{3} ^{2} + w_{2} \end{array} \right) \end{align}\]