Could you elaborate a bit? Do you already have these two models or are you asking about how to build them?
What are “those parameters”?
In cases where are no (easy) analytic answers, one tends to builds a model and use an MCMC-based sampling procedure to approximate a posterior which reflects credible values for all your parameters. That’s part of why I asked my first question above.
That link explains two different specifications of a single distribution. The \mu and \sigma are not “of the dataset” any more than the \alpha and \beta are “of the dataset”.