Bayesian repeated updates, likelihood functions with different nature Let's say we have a prior probability of some diseases 'D'. Then we have some data and likelihood function of symptoms (S) P(S|D) and we update priors. Then we have age (A) likelihood function P(A|D) an make one more update again. 
My questions is is it ok to make such updates with likelihood functions of different nature? And should I worry about something if I do it this way?
 A: As I understand it, you have a joint distribution that you can factor as follows:
\begin{equation}
P(A,D,S) = P(A|D)\,P(S|D)\,P(D) .
\end{equation}
I presume you are interested in the distribution of $D$ given that $A$ and $S$ are observed:
\begin{equation}
P(D|A,S) = \frac{P(A|D)\,P(S|D)\,P(D)}{P(A,S)} \propto P(A|D)\,P(S|D)\,P(D) . 
\end{equation}
At the purely formal level, there is no problem with this. 
Moreover, it seems natural to have a prior distribution for the disease, $P(D)$, and a distribution for symptoms $S$ that depends on the disease, $P(S|D)$. However, the idea that one has a distribution for age $A$ that depends on disease, $P(A|D)$, seems a bit unusual, but not impossible: If I tell you that the patient has a disease then you can tell me the probability that the patient is over any particular age. 
In what I have described above, the inference is conducted "all at once." But the inference could just as well have been conducted sequentially, for example via
\begin{equation}
P(D|A,S) \propto P(A|D)\,P(D|S) ,
\end{equation} 
where 
\begin{equation}
P(D|S) \propto P(S|D)\,P(D) . 
\end{equation}
One of the strengths of the Bayesian approach is that all relevant information can be brought to bear on a problem. 
