Abstract: In applications, statistical models are often restricted to what produces reasonable estimates based on the data at hand. In many cases, however, the principles that allow a model to be restricted can be derived theoretically, in the absence of any data and with minimal applied context. We illustrate this point with three well-known theoretical examples from spatial statistics and time series. First, we show that an autoregressive model for local averages violates a principle of invariance under scaling. Second, we show how the Bayesian estimate of a strictly-increasing time series, using a uniform prior distribution, depends on the scale of estimation. Third, we interpret local smoothing of spatial lattice data as Bayesian estimation and show why uniform local smoothing does not make sense. In various forms, the results presented here have been derived in previous work; our contribution is to draw out some principles that can be derived theoretically, even though in the past they may have been presented in detail in the context of specific examples.
Key words and phrases: ARMA, Bayesian statistics, conditional autoregression, image, scaling, sieve, spatial smoothing, spatial statistics, time series.