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STS551 Stephen Wu et al.
Several factors can contribute to such discrepancies, such as the choice of the
force field and its calibration, computational errors and experimental uncertainties.
Furthermore, the calibration of force fields in MD simulations often relies on
experimental data that exhibit a special structure. The experimental data, which is the
measurements of some physical quantities, often contains repeated data set using
different measurement techniques or under variable environmental conditions.
Therefore, this is a perfect example to demonstrate the use of hierarchical Bayesian
models (Wu et al., 2015a; Wu et al., 2015b). We can observe from Fig. 4 that non-
hierarchical model tends to under-estimate the uncertainty of the parameters, and
instead capturing some strong correlation between the two MD model
parameters.

Figure 4: Comparing results of (a) non-hierarhical Bayesian model inference
and (b) hierarhical Bayesian model inference.

The second example we give is on the calibration of pharmacokinetics
models based on actual clinical data. Once again, these data, which can be
coming from different patients or same patient but different period of time,
exhibit a similar data structure to our simple linear example. Therefore, this is
also a perfect example to demonstrate the use of hierarchical Bayesian models
(Wu et al., 2018).

4. Discussion and Conclusion
Hierarchical Bayesian model is an essential tool for many engineering
problems, because most of the experiments in practice are performed
repeatedly with some inevitable environmental changes. Ignoring such
uncertainties will often result in misleading conclusions. It is known that when
a likelihood model only consider additive noise, increase of data will lead to

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