An older post defines a saturated model as one having as many parameters as observations. I understand how you calculate residual deviance (and its relation to scaled deviance) when the scale is known. What about the alternative?
Suppose, for example, you wish to assume that two independent observations are defined by a normal distributions with unknown means and equal but unknown variance. It seems like you have to choose some mapping from two to three parameters before the saturated model can be defined. Is there a correct choice?