NW:
Thank you for the excellent questions and for giving me the opportunity to clarify.
As you point out, the conclusion of the argument that is made by Bayesian parameter estimation (the posterior PDF) is consistent with one of the premises to this argument (the prior PDF). As Bayes theorem logically follows from the precepts of the probabilistic logic, if this premise (the prior PDF) is true, the conclusion (the posterior PDF) is true. However, under the law of non-contradiction the prior PDF over TECS cannot be true and thus the conclusion (the posterior PDF over TECS) is deductively unproved.
In the construction of a model, the resulting model is usually deductively unproved, as information needed for a deductive conclusion is missing. Thus, in this respect, there is nothing unusual about the deductive unprovability of the posterior PDF over TECS. However, when a model is “scientific,” it is by definition susceptible to falsification. In view of the unobservability of the equilibrium temperature, the posterior PDF over TECS is non-falsifiable and thus it would be improper to describe the posterior PDF over TECS as a “scientific” model. Climatologists err when they imply that it is one.