When sample size is small, informative priors can be valuable in increasing the precision of estimates. Pooling historical data and current data with equal weights under the assumption that both of them are from the same population may be misleading when heterogeneity exists between historical data and current data. This is particularly true when the sample size of historical data is much larger than that of the current data. One way of constructing an informative prior in the presence of the historical data is the power prior, which is realized by raising the likelihood of the historical data to a fractional power.

In this dissertation, we extend the power prior by considering the existence of nuisance parameters. When historical information is used as priors, we assume that the parameters of interest have not changed, while the nuisance parameter may change. The properties of power prior methods with nuisance parameters and its posterior distributions are examined for normal populations. The power prior approaches, with or without nuisance parameters, are compared empirically in terms of the mean squared error (MSE) of the estimated parameter of interest as well as the behavior of the power parameter.

To illustrate the implementation of the power prior with nuisance parameter approach, we apply it to lognormal models for operational risk data and the Rasch model for item response theory (IRT). In the application to the Rasch model, we extend the power prior with nuisance parameter approach further by incorporating it with the hierarchical Bayes model.