**1. Introduction**

Suppose you have a complex question with numerous variables that are not well understood. The challenge that confronts us all is that such situations are not unusual—there are many examples of situations where there are numerous variables that can contribute to occurrences in our world, whether these occurrences involve medical issues, infrastructure issues, or science questions. Statistical methods have been developed to address limited information, but most do not permit the incorporation of new information except Bayesian methods.

The development of Bayesian methods that include priors that can be updated with new data or can respond as a result of added data overcomes initial limitations of most models. Bayesian methods developed as a result of information theory, assuming that the absolute or unconditional probability density function *p(x)* on *X* is the underlying distribution found through curve-fitting. Priors can be determined based on any combination of subjective or numeric information in the absence of real data, or as data are collected, including parameters such as the mean, variance, and range. Utilization of the observations from the prior data leads to the posterior probability function, which incorporates observations from *x* in the sample space *S*, although revealing additional information about the true content of the sample space *S* is subject to the influence of the proper prior distribution assumptions for *x* [1–3].

Many Bayesian practitioners stop with the posterior function, but the ability to develop true statistical inference requires further effort to create a predictive Bayesian solution. The Bayesian posterior methods were used in prior studies [2–12]. Predictive Bayesian methods are an extension of traditional Bayesian approaches, in which unconditional distributions for the quantity of interest are found by integrating over probabilities of parameters of the distribution for the quantity of interest, incorporating both uncertainty and variability in the quantity of interest. They have been termed "believed probabilities."

Press [13] noted that there are advantages to the predictive Bayesian approach. Practical experience and subjectivity can be accounted for explicitly by fitting known or subjective data to a probability function that can be updated as added information becomes available [13]. Predictive Bayesian methods continually improve the statistical inference based on increased amounts of data (hence more data should advance the understanding of statistical relationships and provide greater confidence in the prior and therefore the solution). The predicted distributions are also important for checking goodness of fit of the resulting predictive model to actual data. However, the analysis can become problematic when the information is so scarce that the analysis yields nothing useful [11].

The use of Monte Carlo methods makes the solutions easier. Through randomized sampling, the resulting predictions are simulated from the posterior predictive distribution, which is the distribution of the unobserved future results based on prior observed data. The more confidence that exists with the priors, the more likely results of likely outcomes can be derived.

However, many times Bayesian methods have been limited to situations where there are one or two variables that contribute to an outcome. While high-quality answers can be derived, the design of the algorithm often oversimplifies the real world where many variables may contribute to the outcome. The ability to incorporate many variables that are unknown or uncertain makes the calculations intractable in the traditional Bayesian processes.

Equally important is the ability to study the events, where multiple variables may impact the probability and impact of a given consequence. Regression models are often used, along with principal component analysis to address such situations. However, both rely on complete data sets, and for many situations, the lack of complete data may be extensive. Examples include much of the public or municipal infrastructure that we rely on so heavily for a functioning society, health risks or impacts, natural disaster risk, and extreme event prediction. This is where hierarchical application to Bayesian methods has value.

*Applications of Hierarchical Bayesian Methods to Answer Multilayer Questions with Limited… DOI: http://dx.doi.org/10.5772/intechopen.104784*
