My understanding is that when using a Bayesian approach to estimate parameter values:
- The posterior distribution is the combination of the prior distribution and the likelihood distribution.
- We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution).
- Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean.
But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it?