What is the difference between a prior distribution and a posterior distribution?

What is the difference between a prior distribution and a posterior distribution?

• A. Prior is the belief after observing data.
• B. Posterior is the belief before observing data.
• C. Prior is the belief before observing data.
• D. Posterior is independent of data.

Answer: (C)

In Bayesian thinking, the prior distribution represents what you believe about a parameter before seeing any data. After you observe data, you combine that information with how likely the observed data are under different parameter values (the likelihood) to obtain the posterior distribution, which is your updated belief about the parameter after seeing the data. The posterior is proportional to likelihood times prior and is then normalized. So the belief before observing data is the prior, and the belief after observing data is the posterior. The posterior depends on the data through the likelihood, so it is not independent of data. That’s why the correct statement is that the prior is the belief before observing data.

In Bayesian thinking, the prior distribution represents what you believe about a parameter before seeing any data. After you observe data, you combine that information with how likely the observed data are under different parameter values (the likelihood) to obtain the posterior distribution, which is your updated belief about the parameter after seeing the data. The posterior is proportional to likelihood times prior and is then normalized.

So the belief before observing data is the prior, and the belief after observing data is the posterior. The posterior depends on the data through the likelihood, so it is not independent of data.

That’s why the correct statement is that the prior is the belief before observing data.