I am reading the paper "Classical and Bayesian inference in Neuroimaging" by K. Friston. It starts with a hierarchical linear observation model

$y = X^{(1)}\Theta^{(1)} + \epsilon^{(1)}$

$\Theta^{(1)} = X^{(2)}\Theta^{(2)} + \epsilon^{(2)}$

$\Theta^{(n-1)} = X^{(n)}\Theta^{(n)} + \epsilon^{(n)}$

Where the vector $y=(y_1 , y_2, y_3,...)$stands for the response variable (=dependent variable), $\Theta^{(n)}$ is the parameter (vector) of the nth level, $X$ are matrices containing explanatory variables and $\epsilon$ is a normally distributed error term.

Rcursive substitution gives:

$y = \epsilon^{(1)} + X^{(1)}\epsilon^{(2)} + ...+X^{(1)}....X^{(n-1)}\epsilon^{(n)}+X^{(1)}...X^{(n)}\Theta^{(n)}$

Now, the next step is what I do not understand. They say:

The covariance partitioning implied is

$E\{ yy^T\}=C_\epsilon ^{(1)}+...+X^{(1)}...X^{(i-1)}C_\epsilon ^{(i)}X^{(i-1)T}+...+X^{(1)}...X^{(n)}\Theta ^{(n)}\Theta ^{(n)T}X^{(n)T}X^{(1)T}$

With $C_\epsilon ^{(1)}=Cov\{ \epsilon ^{(1)}\}$

Where does the equation for $E\{ yy^T\}$ come from? It looks like an expectation value to me, but I am confused by the curly brackets (have they a mathematical meaning I am not aware of?). Secondly what is "covariance partitioning"? I have never heard that term before.

  • $\begingroup$ The curly brackets do not have any special meaning, you can replace them by ordinary brackets if you want! $\endgroup$ Commented Oct 6, 2016 at 20:07


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