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. Improper prior distributions are not necessarily incorrect or problematic, but they can lead to unintuitive or unexpected results in Bayesian analysis.

One example of an improper prior distribution is the uniform distribution over the entire real line. This distribution assigns equal probabilities to all real numbers, but the probabilities do not sum to 1 because the real line is infinite. This can lead to counterintuitive results when used as a prior in Bayesian analysis. For instance, if we use a uniform prior to estimate the mean of a normally distributed data set, the posterior distribution will be a non-normal distribution with infinite variance. This means that the posterior mean will be undefined, and any inferences based on it will be meaningless.

Another example of an improper prior is the Pareto distribution with a shape parameter less than 1. This distribution assigns higher probabilities to larger values, but the probabilities do not sum to 1 because the Pareto distribution has infinite support. This can lead to unrealistic or implausible inferences in Bayesian analysis. For instance, if we use a Pareto prior with a shape parameter of 0.5 to estimate the maximum value in a data set, the posterior distribution will be a Pareto distribution with a shape parameter of 0.5. This means that the posterior maximum will be undefined, and any inferences based on it will be meaningless.

In both of these examples, the improper prior distributions do not provide meaningful or useful information about the data. They do not adequately capture the underlying distribution of the data, and they do not allow us to make meaningful inferences about the parameters of interest. In contrast, proper prior distributions, such as the normal or gamma distributions, provide more informative and useful prior information that can lead to more accurate and reliable inferences in Bayesian analysis.