Improper Prior Distribution
- A prior that fails to integrate to 1 or that has no finite mean or variance.
- Such priors are not always invalid but can produce unintuitive or undefined posterior quantities.
- Common examples include a uniform prior on the entire real line and Pareto priors with shape < 1.
Definition
Section titled “Definition”Improper prior distributions are those that do not integrate to 1, or do not have a finite mean or variance. This means that the probabilities assigned to certain values in the distribution do not sum to 1, or that the distribution does not have a well-defined center or spread.
Explanation
Section titled “Explanation”Improper priors may still be used in Bayesian analysis, but they can lead to unintuitive or unexpected results. Because they fail to provide a normalizable probability measure or finite moments, posterior distributions obtained using improper priors can have properties (such as infinite variance or undefined means) that make standard inference meaningless or misleading.
Examples
Section titled “Examples”Uniform prior over the entire real line
Section titled “Uniform prior over the entire real line”The uniform distribution over the entire real line assigns equal weight to all real numbers, but the probabilities do not sum to 1 because the real line is infinite. If a uniform prior is used to estimate the mean of a normally distributed data set, the posterior distribution will be a non-normal distribution with infinite variance. As a result, the posterior mean will be undefined and any inferences based on it will be meaningless.
Pareto prior with shape parameter less than 1
Section titled “Pareto prior with shape parameter less than 1”A Pareto distribution with a shape parameter less than 1 assigns increasing relative weight to larger values and does not sum to 1 over its support. For example, if a Pareto prior with a shape parameter of 0.5 is used to estimate the maximum value in a data set, the posterior distribution will be a Pareto distribution with a shape parameter of 0.5. In that case the posterior maximum will be undefined and any inferences based on it will be meaningless.
Notes or pitfalls
Section titled “Notes or pitfalls”- Improper priors do not necessarily provide meaningful or useful information about the data and can fail to capture the underlying distribution of the data.
- Proper priors, such as the normal or gamma distributions (as cited), provide more informative prior information and can lead to more reliable inferences in Bayesian analysis.
Related terms
Section titled “Related terms”- Proper prior distributions