# Bayesian linear regression - Defining hyperparameters

I am currently working on a linear regression task from a textbook and I have some questions. In the following you will find the task as well as some thoughts of mine to possible results.

Short description: A quadratic regression analysis shall be made for temperature data from France. The explained variable is given by: $temperature = \beta_0 + \beta_1 * time + \beta_2 * time^2 + \epsilon, \epsilon \sim N(0,\sigma^2)$

The conjugate prior is:

$\\ p(\beta, \sigma^2) = N-Inv-\chi^2(\beta_0, \sigma^2 \Omega^{-1}; v_0, s_0^2) \\ p(\beta | \sigma^2) = N(\beta_0, \sigma^2 \Omega^{-1}) \\ p(\sigma^2) = Inv-\chi^2(v_0, s_0^2)$

a) In the first task I shall determine the prior hyperparameters to suitable values:

$\mu_0 = Prior \ mean \\ \Omega_0 = Prior \ precision \\ v_0 = Degrees \ of \ freedom \\ \sigma^2 = Prior \ variance$

I can think about the prior mean as a guess of the average temperature in France (lets assume 20°C). But I can not make a sense out of the other parameters. How do I define the prior precision as well as the degrees of freedom and what are they telling me? I know that $\sigma^2$ can be derived from the Inv-$\chi^2$ distribution but how do I estimate $s_0^2$?