When does an improper prior distribution lead to posterior impropriety?
Improper prior distributions can lead to posterior impropriety (improper posterior distribution) . To determine whether a posterior distribution is proper, you need to make sure that the normalizing constant is finite for all . If an improper prior distribution leads to an improper posterior distribution,…
Is the posterior proper if the prior is improper?
If the prior is improper and the likelihood is flat (because there are no meaningful observations), then the posterior equals the prior and is also improper. Usually you have some observations, and usually the likelihood is not flat, so the posterior is proper.
How to determine if a posterior distribution is proper?
To determine whether a posterior distribution is proper, you need to make sure that the normalizing constant is finite for all . If an improper prior distribution leads to an improper posterior distribution, inference based on the improper posterior distribution is invalid.
When is a prior distribution an informative prior?
An informative prior is a prior that is not dominated by the likelihood and that has an impact on the posterior distribution. If a prior distribution dominates the likelihood, it is clearly an informative prior. These types of distributions must be specified with care in actual practice.
How are improper priors used in Bayes theory?
See, e.g., Hartigan’s Bayes Theory, which formalises quite nicely the use of improper priors. Any measure d π with finite mass can be normalised into a probability measure with mass 1. See also this related Cross validated entry on improper priors.
What are the other names for the noninformative prior?
Other names for the noninformative prior are vague, diffuse, and flat prior. Many statisticians favor noninformative priors because they appear to be more objective. However, it is unrealistic to expect that noninformative priors represent total ignorance about the parameter of interest.