**Abstract**

There are many types of problems that include variables that are not well defined. Seeking answers to complex problems that involve many variables becomes mathematically challenging. Instead, many investigators use methods like principal component analysis to reduce the number of variables, or linear or logistic regression to rank the impact of the variables and eliminating those with the limited impact. However, eliminating variables can create a loss of integrity, especially for variables that might be associated with low likelihood but have high impact events. The use of hierarchical Bayesian methods resolves this issue by utilizing the benefits of information theory to help answer questions by incorporating a series of prior distributions for a number of variables used to solve an equation. The concept is to create distributions for the range and likelihood for each variable, and then create additional distributions to define the mean and shape values. At least three levels of analysis are required, but the hierarchical solution can include added levels beyond the initial variables (i.e., distributions related to the priors for the shape parameters). The results incorporate uncertainty, variability, and the ability to update the confidence in the values of the variables based on the receipt of new data.

**Keywords:** predictive Bayesian, hierarchical, Drake, infrastructure, dose–response, risk, extreme events
