What do you mean by sound? Mathematically sound means "there exists a model validating it", and of course it exists, but so what? If you mean you can bound the probability of error, then in your formulation you actually can't.
Bayesian inference is mathematically sound as it is based on a very generic postulates and allows to compare probabilities based in the current information and made a decision accordingly. With the proper approach the errors are automatically accounted for. I.e. if the errors are large, then one will see that probabilities are too close each other to make a sound decision. Still if one must made a decision, then one can just use the answer based on Bayesian reasoning.
The problem in practice is that accounting for the existing information is hard with guessing of priors etc. But that is the problem of applicability of Bayesian inference, not the problem with the principle itself.
I.e. Bayesian inference is a good answer to the philosophical problem of induction. It is sad that the article has not even touched on that subject.
