Thursday, September 16, 2010

Dirac integration trick

Suppose X is distributed as n-dimensional Gaussian with 0 mean and concentration matrix $A$ and you need conditional distribution of $P(\mathbf{x}|\mathbf{vx}=\mathbf{0})$ where $\mathbf{v}$ is some unit norm vector. To normalize this density you need to integrate $\exp(-\mathbf{x}'A\mathbf{x})$ over subspace of $\mathbb{R}^n$ orthogonal to $\mathbf{v}$, how do you do it?

Take the Dirac delta function. A nice property is that
$$\int dx \delta(x) f(x) = f(0)$$

Write it in terms of Fourier transform

$$\delta(x)=\int dk\exp(-2\pi i k x)$$

Now we can integrate our Gaussian over whole domain, but multiply by $\delta(\mathbf{v}' \mathbf{x})$ to set to 0 density in regions not orthogonal to $\mathbf{v}$. Changing integration order we get

$$Z_v=\int dk \int d\mathbf{x} \exp(-\mathbf{x}'A\mathbf{x}-2\pi i k \mathbf {v}' \mathbf{x})$$

Second integral is a standard Gaussian integral and can be solved by completing the square (Appendix B, Bishop's "Neural Networks"). The first then becomes another Gaussian integral, with final result

$$Z_v=\left(\frac{\pi^{d-1}} {|A| \mathbf{v'}A^{-1}\mathbf{v}}\right)^\frac{1}{2}$$


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