Let's say that we have a model:
$y + x = \epsilon$, $\epsilon \sim N(0, 1)$.
After observing a value for $y$, we can write $x$ as:
$x = \epsilon - y$
Since $y$ is just a constant, and $\epsilon$ is standard normally distributed, it seems that we can claim:
$\epsilon - y \sim N(-y, 1)$
$p(x \mid y) = N(x; -y, 1)$ where $N$ is the normal pdf.
But this makes no sense as there is no mention of a prior for $x$ anywhere. So what went wrong with this "equational" reasoning argument?