Updating belief Lets say I think that event 'E' has a probability 'p' of happening. Then I go around and ask a number of friends if they think event 'E' will happen, they can either answer Yes or No. Now I know that a number of my friends a more or (unfortunately) less knowledgeable about the subject, lets say I can divide them in a number of levels. 
By how much should I then update my belief (i.e. the probability) of event 'E' happening, given the opinions of my friends and their knowledge level? I have done a few calculations using Bayes Rule and I am able to generate weights per Level. But this approach seems to trivial. And I'm also looking for a way to generate confidence/credibility intervals.
What changes when I want to use an additive model instead of a multiplicative one?
Any help or pointers would be greatly appreciated!
 A: To make this easier to discuss, I'm going to rephrase the problem:


*

*A coin is flipped, which we encode as a random variable $z \in \{1, 0\}$. Your prior probability of it being heads is $p(z = 1) = \theta$.

*Your friends $j \in 1, \dotsc, D$ see the coin and tell you what it is, which we encode as a random variable $y_j \in \{1, 0\}$. For simplicity, we're going to assume these $y_j$ are independent when conditioned on $z$.

*Friend $j$ will give you honest answer $\phi_j \in [0,1]$ of the time. 


Since the $y_j$ are independent when conditioned on $z$, when the coin is heads the likelihood is
$$p(y|z=1) = \prod_j p(y_i|z=1) = \prod_j \phi_j^{y_j}(1-\phi_j)^{1-y_j} $$
So by Bayes' rule the posterior probability of heads is
$$p(z=1|y) \propto p(y|z=1)p(z=1) = \theta \prod_j \phi_j^{y_j}(1-\phi_j)^{1-y_j}$$
Having built the model this far, there's two things you could do next: 


*

*Change the $\phi_j$ from constants to random variables, allowing you to model your uncertainty in the trustworthiness of your friends. 

*Relax the indpedendence assumptions on the $y_j$, allowing you to model collusion amongst your friends.
In either case, if you want to take it further you'll probably want to do some reading on "graphical models".
