What is the purpose of a posterior distribution?

• A. Use to ascertain a posterior probability value or a credible interval.
• B. Use to compute population means.
• C. Use to determine sample size.
• D. Use to test the null hypothesis via p-values.

Answer: (A)

In Bayesian analysis, the posterior distribution represents updated beliefs about a parameter after observing the data. It combines the prior information with the data through the likelihood, giving a full distribution for the parameter rather than a single point. From this distribution you can read the probability that the parameter lies in a particular range and you can construct a credible interval, which states the parameter lies within that interval with a certain probability given the data and prior. This is the main use of the posterior: to quantify uncertainty about the parameter after seeing the data.

It's not typically used to directly compute population means, nor to determine sample size, nor to test null hypotheses with p-values. Those latter concepts come from frequentist methods; Bayesian analysis focuses on posterior probabilities and credible intervals to summarize what we believe about the parameter after observing the data.