next up previous
Next: Bibliography Up: Multi-variate Gaussian Integration: Previous: Special case: White noise

Special case: Scalar $ F$ and White noise

Consider $ F$ as a scalar, and $ B = 1^{\mathrm{\textsf{T}}}$ (a column vector of ones). This represents $ BF$ being a constant (or mean) vector. Therefore, $ B^{\mathrm{\textsf{T}}}B = N$ which gives the integral as
$\displaystyle \int \exp\left( \frac{-1}{2 \sigma^2} (Y - 1^{\mathrm{\textsf{T}}}F)^{\mathrm{\textsf{T}}}(Y - 1^{\mathrm{\textsf{T}}}F) \right) \; dF$ $\displaystyle =$ $\displaystyle (N)^{-1/2} (2 \pi \sigma^2)^{1/2} \exp\left( \frac{-N\ensuremath{\mathrm{Var}}(Y)}{2\sigma^2} \right)$ (21)

with $ R_c = \frac{1}{2 \sigma^2} (I - \frac{1}{N} B B^{\mathrm{\textsf{T}}})$. Note that $ Y^{\mathrm{\textsf{T}}}R_c Y = \frac{N\ensuremath{\mathrm{Var}}(Y)}{2\sigma^2}$ where $ \ensuremath{\mathrm{Var}}(Y) = \frac{1}{N} \sum_j (Y_j^2)
- (\frac{1}{N} \sum_...
...{\textsf{T}}}Y - \frac{1}{N} Y^{\mathrm{\textsf{T}}}B B^{\mathrm{\textsf{T}}}Y)$ represents the estimated variance of the data vector, $ Y$.