Find likelihood for a given posterior

Question

Let $p(z)=\mathcal{N}(z|0,1)$ be the standard normal density. Is there a conditional density $p(x|z)$ that could be expressed analytically and that would make the posterior $p(z|x)=\mathcal{N}(z|\phi(x), \bullet)$ a normal distribution whose mean is a non-linear function of $x$ (say, $\phi(x)=x^2$ for example) ? The variance could be arbitrary.

Answer 1

I take what you have written as the prior distribution for $z$ being $\mathcal{N}(0,1)$ a standard normal, i.e. with mean $0$ and variance $1$.

If the conditional density for $x$ given $z$ is for example $\frac 1 {x\sqrt{2\pi}}\ e^{-{\left(\log_e(x)-z\right)^2}/{2}}$ (log-normal with parameters $z$ and $1$) for $x \gt 0$, then the posterior distribution for $z$ having observed $x$ would be $\mathcal{N}\left(\frac12\log_e(x),\frac12\right)$