Can a neural network output represent a posterior probability? I seem to remember from years ago when I first read Bishop's ANN book that it is possible to construct a neural network such that the outputs should represent the posterior probability that I would have found if I used the likelihood approach to separate the same data. Is this true or have I made this up in my mind? If it is true what are the conditions, I seem to remember it has to have no loops and limits on the number of hidden layers. Although I also seem to remember a result from Komogorov showing that many hidden layers aren't helpful if the network is well trained. Unfortunately I can't find the book anymore and I can't seem to find much on this result which makes me worried it's me.
I think I probably need to flesh out my question a bit as pointed out by @bayerj
I have a few variables (I'll assume I only have one for now) which I am using to find the probability of one of three hypotheses being true given the three variables. I have some data which which a can construct PDFs. I am using
$$P(H_{k}|x) = \frac{P(x|H_{k})P(H_k)}{P({x})}$$
where 
P(x) is the unconditional probablity density
\begin{equation}
P(x) = \sum P(X|H_k)P(H_k)
\end{equation} 
I'm trying to verify whether it is possible to train a network, using the data with which I construct my PDFs, such that the value of the output node equals P(H_k|x). I'm sure this is possible I just can't remember the working. Since I posted the question I have found 
http://www-vis.lbl.gov/~romano/mlgroup/papers/neural-networks-survey.pdf
which I think is saying what I want to do is possible.
 A: The conditions under which the output of a neural net can be treated as estimates of posterior probabilities are fairly broad, I remember the following paper as being pretty interesting and informative (caveat: but I've not read it since 2002)
Marco Saerens, Patrice Latinne, Christine Decaestecker: Any reasonable cost function can be used for a posteriori probability approximation. IEEE Transactions on Neural Networks 13(5): 1204-1210 (2002)
I suspect it has references to the classic papers in the text as well.  Prof. Saerens has written several really nice papers; I would recommend anyone seriously interested in neural nets to look up his papers in Google scholar etc.
You could view the output layer of a neural net as being a logistic regression model (with outputs that represent probabilities), and the hidden layer as being a non-linear transformation of the inputs.  It is the output layer that matters regarding the interpretation of the outputs, so the MLP is essentially just a non-linear logistic regression model.
