Who Needs Data? Restricting Image Models by Pure Thought

In applications, statistical models for images are often restricted to what produces reasonable estimates for the data at hand. In many cases, 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 present three theoretical examples. We interpret local smoothing of spatial lattice data as Bayesian estimation and show why uniform local smoothing does not make sense. In time series, we show that an autoregressive model for local averages violates a principle of invariance under scaling. Finally, we show how the Bayesian estimate of a strictly-increasing time series, using a uniform prior distribution, depends on the scale of estimation. Concerns about the behavior of models and estimates under rescaling are especially important for image analysis. It is desirable for substantive inference about an image to not depend on the (often arbitrary) scale of "pixels," and it -is important to know which families of models can be dismissed on theoretical grounds alone.