I wondered if a well-known formula exists or if there is any reference I can look into for the following Bayesian inference problem.
An observed data point $y$ is a sum of true underlying value $x$ and some noise $\epsilon$, which follows a normal distribution $N(0,\sigma_{\epsilon})$
$$y = x + \epsilon$$
The true value $x$ can be drawn either from $N(\mu_{H},\sigma)$ or $N(\mu_{L},\sigma)$, and I have a prior that with probability $p_{0}$, the true value is drawn from $N(\mu_{H},\sigma_{H})$.
In this case, the prior for the actual value $x$ will be a mixture of the two normal distributions with weight $p$, which results in a non-conjugate prior. Would there be a tractable formula for the posterior for $x$? (The posterior for the distribution part, I could derived myself.)