What is the Bayesian posterior distribution?

What is the Bayesian posterior distribution?

• A. The distribution of data under the model.
• B. The distribution of unknown parameters after updating the prior with the likelihood.
• C. The prior distribution before observing data.
• D. The sampling distribution of an estimator.

Answer: (B)

In Bayesian analysis, after observing data you update your beliefs about the unknown parameters by combining your prior information with what the data say through the likelihood. The result is the posterior distribution: the distribution of the unknown parameters given the observed data. It captures both what you believed before and the evidence from the data, and it’s obtained by multiplying the prior by the likelihood and then normalizing so it sums to one. This is what you use to quantify uncertainty about the parameters after seeing the data. This differs from the distribution of data under the model (the predictive or sampling distribution of data given parameters), the prior distribution before observing data, and the sampling distribution of an estimator (which describes how an estimator would behave under repeated samples). An example: with a Beta prior for a probability parameter and observed coin flips, the posterior is another Beta distribution with updated parameters, reflecting both prior beliefs and data.

In Bayesian analysis, after observing data you update your beliefs about the unknown parameters by combining your prior information with what the data say through the likelihood. The result is the posterior distribution: the distribution of the unknown parameters given the observed data. It captures both what you believed before and the evidence from the data, and it’s obtained by multiplying the prior by the likelihood and then normalizing so it sums to one. This is what you use to quantify uncertainty about the parameters after seeing the data.

This differs from the distribution of data under the model (the predictive or sampling distribution of data given parameters), the prior distribution before observing data, and the sampling distribution of an estimator (which describes how an estimator would behave under repeated samples). An example: with a Beta prior for a probability parameter and observed coin flips, the posterior is another Beta distribution with updated parameters, reflecting both prior beliefs and data.