# prior for initial values of Kalman Filter

I'm studying Carter and Kohn's (1994) implementation of the Gibbs sampler for Bayesian analysis of state space models. In their paper, they assume the starting value, call it $\beta_0$, of the state vectors has a proper distribution. Applications of their algorithm (such as the time-varying vector autoregression of Primiceri (2005)) also list priors for the initial values of the state vector such as $\beta_0 \sim N(0, \Omega_0)$ for the state space model

$$y_t = \beta_t y_{t-1} + e_t$$ $$\beta_t = \beta_{t-1} + \epsilon_t,$$

where $e_t$ and $\epsilon_t$ are independent of each other and normally distributed. Another common prior is centered on the ordinary least squares estimate of $\beta$, $\hat{\beta}$.

I don't understand how a prior on $\beta_0$ is implemented here. Is the prior implemented through the initial value of the Kalman Filter? That is to say, a prior for the starting value $\beta_0$ where $\beta_0\sim~N(0, \Omega_0)$ begins the filter recursions with state vector $\beta_0 = 0$ and prediction covariance $\Omega_0$? Or is there something I'm missing?

Another post has referenced the use of priors in the time-varying parameter VAR, but it doesn't explain the derivation of the implementation/derivation of the conditional posterior from the prior.

Is the prior implemented through the initial value of the Kalman Filter? That is to say, a prior for the starting value $\beta_0$ where $\beta_0 \sim N(0,\Omega_0)$ begins the filter recursions with state vector $\beta_0=0$ and prediction covariance $\Omega_0$? Or is there something I'm missing?
If my memory is correct, the Carter and Kohn paper does not refer to these $\beta_0$ state vectors as parameters. The parameters are the values that govern the distributions of these observations and state vectors. The algorithm they propose alternates sampling between the conditional posterior of the states conditioning on the parameters (and the observations), and the conditonal posterior of the parameters given the states (and the observations).