On the Meaning of Parameters in Approximation Models

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Models provide the foundation to statistics and are generally viewed as approximations to the underlying true state of nature. Data are used to estimate parameters in these approximating models. Assuming there exists a true underlying model, the parameters in the models used for data analysis are actually functions of the parameters of an underlying true model. Therefore, in order to fully understand what a proposed model actually represents, it is useful to examine how the parameters in an approximating model relate to the parameters in the true model. That is, given a statistic, this paper seeks to determine what the statistic is actually estimating in terms of an underlying true model. Examples and illustrations of the meaning of parameters in an approximating model to a true underlying model are provided. This is accomplished by fitting an approximating model to an assumed true model, similar to the way an approximating model is fit to a data set. The ideas are also illustrated with latent variable models, in particular, using mixture models.

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