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STS551 Stephen Wu et al.
boarder range of engineering problems, such as reliability estimation due to
rare events (Au and Beck, 2001). In particular, hierarchical Bayesian model is a
powerful modeling tool that offers great flexibility for uncertainty
quantification, yet classical Markov Chain Monte Carlo approach is usually
impractical for even a simple ordinary differential equation model. This is
because of the inherently high dimensional problem setup as explained in this
study. Existing approaches for hierarchical Bayesian modeling usually attempt
to use simple stochastic models that can lead to analytical results, but usually
ends up with impractical assumptions, or approximating the stochastic models
with surrogate models that can result in analytical solutions. In my study, I
begin with a basic illustration of the power of hierarchical Bayesian model, and
then continue with a demonstration of its applications to engineering problems by
incorporating high performance computing and specifically designed sampling
methods. Our method focuses on the ability to recalculate the problem for many
times, especially when there are newly added data. The applications, including
pharmacokinetics and molecular dynamics, involve fairly complicated models that
classical models used for Bayesian inference often lead to misleading results.
Therefore, it is important to use a reliable, yet computationally efficient algorithm for
these problems.

2. Methodology
2.1 Problem setup for the general case
Consider the following probability model:

~(|(, ), ) ⟺ = (, ) + , ~( |0, ), (1)
⃗
⃗

where (|, ) denotes a normal distribution on a 1D variable with mean and
standard deviation , and denotes input and output variable of a model
(function) with model parameters . A hierarchical Bayesian model has
⃗
hyperparameters for to define a probability model for the parameter space.
⃗⃗
⃗
However, one can typically find two different types of such hierarchical Bayesian
models in the literature (Wu et al., 2018). Here, I illustrate using a simple linear
example.
2.2 Simple example: single parameter with Gaussian prior model
Consider a simple linear function with only one parameter, i.e., = , and a
⃗
Gaussian model for the parameter with mean and standard deviation as the
hyperparameters, i.e., = { , } :
⃗⃗

(, ) = (2)
with = + , ~ ( |0, ) ⟺ ~(| , )

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