How to obtain the distributions of linear model parameters? For a linear model
$$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + …… + \sigma^2$$
Assuming $\beta_0, \beta_1,……$ are random varibles,
how to obtain the distributions of these parameters?
It seems that Bayesian regression obtains the posterior distributions of $\beta_0, \beta_1,……$, but the posterior distributions are not the real distributions of $\beta_0, \beta_1,……$, because the variances of the posterior distributions will all converge to zeros when there are enough samples.
 A: This is a good question. The answer is that in the usual conceptual approach, the regression coefficients are not assumed to be random variables, but fixed. So you can think of an underlying process on the population level with constant parameters, and it is only your estimates of $\beta$ based on finite samples that have distributions due to, for example, sampling variability. Note that the underlying process $f(Y|X=x)$ described by the linear model is nevertheless not deterministic. This is due to the error term.
On the population level, you can think of the process for example as a machine that takes an input and has a fixed protocol to produce an output, but still produces outputs with some variability. This is not because the inputs or the protocol are stochastic, but a single realization of the protocol is stochastic.
So, for $n \rightarrow \infty$, you will have $\text{Var}(\hat{\beta}) \rightarrow 0$, $\text{Var}(\hat{Y}|X=x) \rightarrow 0$ ($\hat{Y}$ being the predicted conditional mean) and also $\text{Var}(\hat{\sigma}^2) \rightarrow 0$, but the width of your prediction intervals will still always be $>0$ (assuming $\sigma^2>0$).
Edit: My answer is more from a frequentist perspective dealing with sampling distributions. In a Bayesian approach, you indeed obtain probability distributions ober $\beta$. It is crucial to interpret this as your subjective belief about the location of $\beta$, however, not the true distribution of $\beta$ itself. Of course, having more data will make your beliefs converge stronger to a certain value.
