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You can only get a posterior distribution by writing out a full model with priors and a likelihood. A Bayesian model consists of a likelihood and some number of layers of prior distributions. The product of those probability densities is proportional to the posterior probability density. And what you care about is the entire posterior distribution. Once you have that, all you have to do is choose how to summarize it. The difficulties in Bayesian inference come in the forms of laying out the model (often the likelihood is obvious while the priors are less-so), and getting samples from the posterior. Incalculable blood, sweat, and tears have gone into ways to get samples from this distribution. But we have things like stan for that these days, so Bayesian inference gets to proceed top down from a model-first perspective.