And why is this a weakness?
> In this way, the case of total ignorance, for example, can not be represented
> properly.
Not so. One way of finding a probability distribution that represents total
ignorance is through the use of group invariance arguments, and this has been
known for at least 30 years. See the following papers by Jaynes:
E. T. Jaynes, "Prior Probabilities," _IEEE Transactions on Systems Science and
Cybernetics_, SSC-4, Sept. 1968, pp. 227--241.
E. T. Jaynes, "The Well-Posed Problem," _Foundations of Physics_ 3 (1973),
pp. 477--493.
E. T. Jaynes, "Marginalization and Prior Probabilities," _Bayesian Analysis in
Econometrics and Statistics_, A. Zellner (ed.), North-Holland Publishing Co.,
Amsterdam, 1980.
(The last introduces another method for finding noninformative priors, based on
marginalization.) All three of these are also found in
E. T. Jaynes, _Papers on Probability, Statistics and Statistical Physics_,
D. Reidel Publishing Co., 1983.
> Another important problem is the restriction to directed acyclic
> graphs.
This is a restriction of Bayesian networks, but not of Bayesian probability
theory in general. There are other tools for constructing large joint
probability distributions, e.g., maximum entropy techniques.