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

Naive Bayes model

Consider this model: x_i = ax₀ + e_i,  i = 1, …, 4 and x₀ = e₀. All terms e₀, …, e₃ are independent and N(0, 1) distributed. Let e = (e₀, …, e₃) and x = (x₀, …x₃). Isolating error terms gives that e = L₁x where L₁ 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 Var(e) = LVar(x)L′ so the covariance matrix is V1 = 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 = "")

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 = "")