I am trying to compute and draw samples from a posterior distribution. Here is what I have:
My data, $\textbf{x}$, is a vectorized $N\times N$ image, i.e. it is of length $N^2$. An arbitrary shape (rectangle, square, circle, etc.) of constant size can be in one of $n$ different places, all with equal probability. Pixels without the shape are set equal to 0 and pixels with the shape are set equal to 1 (grayscale images). Thus, my prior is discrete and uniform, given by
$p(\textbf{x}) = \frac{1}{n}$
Now, I manipulate my image via an operator, $A$, and assume there is some noise $w \sim \mathcal{N}(\textbf{0}, I)$:
$\textbf{y} = A\textbf{x} + \textbf{w}$
Then it is clear that my likelihood is multi-variate Gaussian such that $\textbf{y} | \textbf{x} \sim \mathcal{N}(A\textbf{x}, I)$. Now, I know that my posterior is given by
$p(\textbf{x} | \textbf{y}) \propto p(\textbf{y}|\textbf{x})p(\textbf{x})$
and this is where I get lost. My attempt at the posterior is:
$p(\textbf{x} | \textbf{y}) \propto \frac{1}{n}\sum_{i=1}^{n}\mathcal{N}(A\textbf{x}, I)$
but I do not think that is correct. Thus, I am not sure how to:
- Compute this posterior. I think it should be a mixture, but I am not sure and do not have much experience with mixtures. I understand that I may not get a closed form expression: that is fine, I just want to see something written down to help my understanding.
- Draw sample images from $p(\textbf{x}|\textbf{y})$.
I posted a similar question here, but do not feel as though I did an adequate job expressing things.
