Bayesian update with the shifted and scaled data

Question

Suppose I have data $y$ (N observations) which follows a normal distribution:

$y \sim N(\alpha+\beta*\mu,\sigma^2)$

while $\alpha$ and $\beta$ are known parameters. I want to update $\mu$ and the prior is $\mu \sim N(\mu_0,\sigma^2_0)$.

My question is what is the posterior distribution for $\mu$?

I know the posterior when $X \sim N(\mu,\sigma^2)$ (here). But in my case, $y$ is a shifted and scaled version of $X$.

For my own derivation, I treat the above as a regression problem, and I get

$\mu \sim N(\frac{\beta^T(y-\alpha)+\mu_0/\sigma^2_0}{\beta^T\beta+1/\sigma^2_0},\frac{\sigma^2}{\beta^t\beta+1/\sigma^2})$

However, the above formula does seem correct. For instance, suppose I have two data, $y_1$ and $y_2$. Updating with $y_1$ then $y_2$ vs updating with $y_2$ then $y_1$ will result in different posteriors.

Any help is appreciated!

Answer 1

You can do a standard update for distribution parametrized by $\mu’ = \alpha + \beta\mu$. Next, given that it's a location-scale transformation you can just transform it. But your likelihood is characterized by $\mu$ not $\mu’$, so you cannot use the distribution for $\mu$ in the next update, but the one for $\mu’$.