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 <- ysym(L1chr)
L1

## {{ 1,  0,  0,  0},
##  {-a,  1,  0,  0},
##  {-a,  0,  1,  0},
##  {-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 <- solve(L1)
K1 <- t(L1) %*% L1
V1 <- L1inv %*% t(L1inv)

cat(
  "\\begin{align} 
    K_1 &= ", tex(K1), " \\\\ 
   V_1 &= ", tex(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 <- ysym(L2chr)
L2

## {{  1,   0,   0,   0},
##  {-a1,   1,   0,   0},
##  {-a2,   0,   1,   0},
##  {-a3,   0,   0,   1}}

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

## {{w1,  0,  0,  0},
##  { 0, w2,  0,  0},
##  { 0,  0, w2,  0},
##  { 0,  0,  0, w2}}

L2inv <- solve(L2)
K2 <- t(L2) %*% solve(Ve) %*% L2
V2 <- L2inv %*% Ve %*% t(L2inv)

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

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