# Prior and posterior in Bayesian regression

I am referring to the Bayesian inference in Simple linear regression as explained in Probability & statistics - Schervish/Degroot( 2012).

For a simple one variable regression, the authors assume an improper prior = 1/Tau. The posterior joint distribution of both beta coefficients is then the bi-variate normal, the posterior distribution of Tau is Gamma(2, n-2) ,& the marginal distribution of each Beta coefficient is the T(n-2).

I have some questions regarding this model:

1. Once the posterior calculations are done , and the appropriate confidence intervals for Beta and Tau are constructed, what happens if we have new information in the form of some new data points? In this case whether the prior would be the normal ? What would be the posterior in this case ?

2. Which approach would be better and why? a) Add the new data to the existing data & estimate the posterior based on complete data. b) Assume the posterior calculated earlier as a prior & then estimate the posterior ( Is it analytically possible to estimate the posterior)

Thanks Kedar