A prior distribution is conjugate with respect to the likelihood when the resulting posterior is of the same type of distribution as the prior, except for different parameters. When both the prior and the likelihood are normally distributed, then the posterior is also normally distributed.

The conjugacy of the prior and likelihood implies a closed-form solution for the posterior that facilitates the update process and avoids the need to use numerical methods to approximate the posterior. Moreover, the resulting posterior can be used as prior for the next update step.

Let's illustrate this process using a binary classification example for stock price movements.