No we cannot justify (*). In fact, (*) will not even hold. However if (*) does not hold, I think that is just as bad as failing the Berford test. The tn sentences are themselves a sequence that is indistinguishable from coming a sequence coming from a weighted coin. Therefore failing to provide probability P to the sentences tn is a strong sign that the code will also give the wrong probability to ϕsn. The two are not qualitatively different.
A formal proof of why it fails is not written up, but if it is, the conclusion will be that either ϕsn OR tn will have incorrect limiting probabilities.
No we cannot justify (*). In fact, (*) will not even hold. However if (*) does not hold, I think that is just as bad as failing the Berford test. The tn sentences are themselves a sequence that is indistinguishable from coming a sequence coming from a weighted coin. Therefore failing to provide probability P to the sentences tn is a strong sign that the code will also give the wrong probability to ϕsn. The two are not qualitatively different.
A formal proof of why it fails is not written up, but if it is, the conclusion will be that either ϕsn OR tn will have incorrect limiting probabilities.