I'm trying to understand message passing for compressed sensing. I came acrross this distribution:
As the title suggests, how does this distribution look like? I know the first products term in the right hand side is student distribution. But what happen when it's multiplied by the second term? Also, what is the "ds" in the left hand side of (2)?
Let's consider the example when $N = 3$ and $n = 2$.
In the case, we are in the 3-dimensional space and we are handling of constraint which is in a plane.
As the plane is two dimensional, Dirac delta has two components which is represented by product:
$$\delta_{\{y_1 = (As)_1\}} \cdot \delta_{\{y_2 = (As)_2\}}$$
where $(As)_1$ is the first component of multiplication of $As$.
If you calculate a probability for the whole space, you need to calculate multiple integrals.
For example,
\begin{aligned} \int_{s_1= -\infty}^{s_1 = \infty} & \int_{s_2 = -\infty}^{s_2 = \infty} \int_{s_3 = -\infty}^{s_3 =\infty}\mu(\mathrm{d}s) \\ & = \int_{s_1= -\infty}^{s_1 = \infty} \int_{s_2 = -\infty}^{s_2 = \infty} \int_{s_3 = -\infty}^{s_3 =\infty} P(s) \cdot \delta_{\{y_1 = (As)_1\}} \cdot\delta_{\{y_2 = (As)_2\}} \mathrm{d} s_1 \mathrm{d} s_2 \mathrm{d} s_3 \\ & = \int_{a= -\infty}^{a = \infty} \int_{b = -\infty}^{b = \infty} P(a, b) \mathrm{d} a \mathrm{d} b \end{aligned}
where
$$P(s) = \dfrac{1}{Z}\prod\limits_{i=1}^{N}\exp(-\beta|s_i|)$$
and $\mathrm{d}a$ and $\mathrm{d}b$ are differentials on the hyperplane defined by the Dirac delta.
The Dirac delta function is to take the value which is satisfied by the hyperplane constraint. The integral in the 3D space is converted into the integral on the 2D hyperplane.
