NW (May 11, 2012 at 2:05 am):
Thank you for the stimulating comments. A prior PDF that is uninformative about the numerical value of the associated parameter is logically required in the circumstance that information about this value is not available. Generally and in the specific case of TECS, uninformative priors are of infinite number. When one of these priors (e.g., the ever popular uniform prior) is selected for use, a consequence is for Aristotle’s law of non-contradiction to be violated. In the argument that is made for the existence of a specified posterior PDF over TECS, the negated law plays the role of a false premise. As this premise is false, the existence of the specified posterior PDF is necessarily unproved. As the equilibrium temperature is unobservable, the notion of TECS is scientifically and logically nonsensical.
If, in addition to being uninformative, the prior PDF is unique, the existence of the posterior PDF is proved by Bayes’s theorem. There is a circumstance in which the prior is unique. It is manifested in a sequence of trials of an experiment. In a single trial, the relative frequency with which the experiment has a specified outcome is 0 or 1. In 2 trials, the relative frequency is 0 or 1/2 or 1. In N trials, the relative frequency is 0 or 1/N or 2/N or… or 1. Note that the relative frequency possibilities are equally spaced on the number line in the interval between 0 and 1.
Let N increase without limit. The relative frequency becomes known as the “limiting relative frequency.” The limiting relative frequency possibilities are evenly spaced on the interval between 0 and 1. Maximization of the missing information about the limiting relative frequency yields the conclusion that equal numerical values are assigned to the probabilities of the various limiting relative frequency possibilities. The probability density is the ratio of the probability value, namely 1/(N+1), to the distance between adjacent limiting relative frequency possibilities, namely 1/N. Thus, the probability density is uniform in the interval between 0 and 1 and equal to 1; otherwise, the probability density is nil. This phenomenon provides us with an exception to the rule that prior PDFs are of infinite number. This exception provides climatologists with a loophole that they could jump through enroute to logically sound models of the climate.