**3.2 Example 2: dose response**

The use of predictive Bayesian methods for dose–response relationships has also been investigated by a number of authors [10–12, 54–56]. Beaudequin et al. [57] developed QMRA with the use of hierarchical Bayesian networks to address the data paucity, combine quantitative and qualitative information including expert opinion, and the ability to offer a systems approach to characterize complexity. They outlined how the Bayesian networks are the current method of choice for determining the risk to human health from exposure to pathogens because of their ability to separate risk and uncertainly, predict outcomes, and deal with poorer quality data [57]. Hence, as subjective data are incorporated, the prior distributions self-adjust [58]. Bloetscher et al. [9] used six sets of *Cryptosporidium* data to show how the dose–response function changes with new, additional data. As a result of new data, the dose–response is expected to improve, demonstrating that the process can be applied to other organisms. In addition, the paper creates a Predictive Bayesian MCMC solution for the Pareto II distribution with two uncertain parameters.

Given the unlikelihood of reinfection during a single incident (due to a short period of time), the likelihood of infection can be described by the binomial distribution. As such, a binomial function is used to represent the probability of exposure. **Figure 3** shows the conceptual model with three levels of probability distributions.

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

**Figure 3.** *Conceptual risk model for cryptosporidium.*

Because of the intrinsic difficulty in solving a predictive Bayesian equation with multiple embedded distributions through double integration, an analytical mathematical solution is not achievable. Instead, a probabilistic solution was developed using a Markov Chain Monte Carlo (MCMC) program developed in MATLAB18® with uncertain values for *a* and *k*. Six different models of 10,000 iterations were run, each model including an additional dataset and the prior for *α* increased to account for the additional data.

When compared with the beta-Poisson models developed by Haas et al. [59], the predictive Bayesian equation derived in this study is less conservative by a factor of over 10 than the beta-Poisson model used by Haas et al. [59] (see **Figure 4**). However, the beta-Poisson does not accept new data, and therefore cannot be updated, is the likely explanation for the difference.
