Bayes theorem with errors

#1
I have a situation in which I want to calculate, for a given $y$ (which I measure experimentally), the probability distribution of $x$ i.e. $p(x|y)$ (actually what I need is the value of x for which this is maximized). Using Bayes theorem I have $p(x|y) = \frac{p(y|x)p(x)}{p(y)}$. I know both $p(x)$ and $p(y|x)$ which are both Gaussians. I don't know $p(y)$, but given that $y$ is constant for a given measurement, and all that I need is the maximum value over x, that shouldn't matter. However, in practice, I have an error associated to $y$, call it $dy$ ($y$ is Gaussian distributed). How can I account for this uncertainty on $y$ when trying to find the best $x$ (and the uncertainty on $x$)? Thank you!