Say we start with a linear regression model of the form
$$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon, \quad \epsilon \sim N(0, \sigma^2)$$
with the conjugate prior
$$
\begin{align*}
&\sigma^2 \sim \text{Inv-}\chi^2(\nu_n, \sigma_n^2)\\
&\beta | \sigma^2 \sim N(\mu_0, \sigma^2\Omega_0^{-1})\\
&\mu_0 = (0,0,0)\\
&\Omega_0 = 0.01 \cdot I_3\\
&\nu_0=1\\
&\sigma^2 = 10^4
\end{align*}
$$
Question: Now say that we learn that earlier research shows strong evidence that $x_2$ has no effect on $y$. How could we can change the model to account for this information? (There may be many different ways but I'm mostly looking for a standard way if there is such a thing).
How I have tried: My first thought was to change the prior in some way, but the prior mean $\mu_0$ is already $0$ so there's not much to do there. My second thought was then that the prior variance around the mean $0$ should decrease for $\beta_2$ since the earlier research increases our prior belief that the mean is $0$, so $\Omega_0^{-1}$ would have to be decreased in the entry corresponding to $\beta_2$, i.e. increasing the entry corresponding to $\beta_2$ in $\Omega_0$. Not sure if this is a reasonable approach. I was also thinking that setting up a hierarchical model would be suitable but I just started learning about that so not sure how I would do it.
