For a Gaussian likelihood, how to choose a prior that biases the MAP estimates away from zero (opposite of L2 regularization) Suppose that $x\in R^1$,  $y\in [-30,30]$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be
$$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or
$$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$
since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?
 e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If  $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.


Also, the MAP estimate should have an analytical expression.


Much thanks in advance!! Also, the more options the better!


edit:

*

*This should work for any value of $y$ and the prior needs to be chosen before seeing $y$.


*sorry made a mistake,instead of $y\in R^1$, $y\in [-30,30]$. As long as it's bounded by a negative number and a positive number it would work fine.


In general (i.e., for nonzero prior means), the posterior mean in the normal-normal location model is
$$
\frac{1/\tau^2}{1/\tau^2+n/\sigma^2}\mu_0+\frac{n/\sigma^2}{1/\tau^2+n/\sigma^2}\bar{x}
$$
for a prior mean $\mu_0$, a prior variance $\tau^2$, a population variance (assumed known here) $\sigma^2$ and a sample size $n$ (so notation is slightly different from yours, see https://www2.bcs.rochester.edu/sites/jacobslab/cheat_sheet/bayes_Normal_Normal.pdf.
Hence, the posterior mean is a weighted average of sample mean and prior mean. So if you want to the posterior mean be be further away from zero than the sample mean, you would need to pick $\mu_0>\bar x$ if $\bar x>0$ and $\mu_0<\bar x$ if $\bar x<0$.
But, since you should pick your prior before seeing the data, it is not clear how you would achieve that in general.
