Special case: White noise

White noise with variance $\sigma^2$ gives $V^{-1} = \frac{1}{2\sigma^2} I$. In this case the above integral becomes:

$$\int \exp\left( \frac{-1}{2 \sigma^2} (Y - B F)^{\mathrm{\textsf{T}}}(Y - B F) \right) \; dF = \vert\det(B^{\mathrm{\textsf{T}}}B)\vert^{-1/2} (2 \pi \sigma^2)^{M/2} \exp\left( \frac{- Y^{\mathrm{\textsf{T}}}R_w Y}{2 \sigma^2} \right)$$ (20)

where $R_c = \frac{1}{2 \sigma^2} R_w$ and $R_w = I - \frac{1}{2 \sigma^2} B K K^{\mathrm{\textsf{T}}}B^{\mathrm{\textsf{T}}}= I - B (B^{\mathrm{\textsf{T}}}B)^{-1} B^{\mathrm{\textsf{T}}}$.