A Case Study of the Robustness of Bayesian Methods of Inference: Estimating the Total in a Finite Population Using Transformations to Normality

摘要

Bayesian methods of inference are the appropriate statistical tools for providing interval estimates in practice. The example presented here illustrates the relative ease with which Bayesian models can be implemented using simulation techniques to approximate posterior distributions but also shows that these techniques cannot be automatically applied to arrive at sound inferences. In particular, the example dramatizes three important messages. The first two messages are concrete and easily stated: Although the log normal model is often used to estimate the total on the raw scale (e.g., estimate total oil reserves assuming the logarithm of the values are normally distributed), the log normal model may not provide realistic inferences even when it appears to fit fairly well as judged from probability plots. Extending the log normal family to a larger family, such as the Box-Cox family of power transformations, and selecting a better fitting model by likelihood criteria or probability plots, may lead to less realistic inferences for the population total, even when probability plots indicate an adequate fit. In general, inferences are sensitive to features of the underlying distribution of values in the population that cannot be addressed by the data.