confusing notion in Bayesian inference My confusion is on the observed data $D$, I think for $D$, (1) it could be only one observation, and it could be multiple observation, correct? (2) when there are multiple observations, do we need to assume the multiple observation are independent? I have the confusion here since I referred some text book and wiki, but do not see it is always mentioned the multiple observations are independent. Want to confirm with the expert here, (3) when there are multiple observations and they are independent, then $P(D_1,D_2,...,D_n|w)$ = $P(D_1|w)*P(D_2|w)*...*P(D_n|w)$, correct?
The following material is reproduced from Christopher Bishop's Pattern Recognition and Machine Learning.

 A: Exchangeability is often interpreted as the Bayesian version of i.i.d assumption . In i.i.d assumption we assume that the parameter we are interested in exists in some meaningful sense and the data at hand (1 or $n$ observations does not matter ) were generated as independent trials conditional on the parameter .
Exchangeability  is the condition that the joint density of the data remains the same under re-ordering or re-labeling of the indices of the data , so if we assume for example a binomial model then  under exchangeability  two sequence of random variables , each with the same length $n$ and the same proportion of ones would be assigned the same probability only the number of ones matters not the location of the ones .
So that  the assumption of exchangeability  in Bayesian inference means we believe that the data are exchangeable then it is as if there is a parameter (say $\theta$) that derives a stochastic model generating the data and a density over $\theta$ that does not depend on the data , this density interpretable as a prior density  thus  the existence of a prior density over a parameter is a  result of the exchangeability rather than an assumption
Edit based on Xi'an's comment : 
Some models allow for exchangeability and others do not .In the  hierarchical models also  we can not consider the entire sequence of data as being exchangeable instead the data within any given group might be considered exchangeable that means we condition on the group and that is so called conditional exchangeability 
A: Just on a technicality, the proper decomposition is:
$$P (D_1,...,D_n|w)=P(D_1|w)\times P (D_2|D_1 w)\times\dots P (D_n|D_1 D_2\dots D_{n-1}w) $$
This is always correct - but when assuming conditional independence, it simplifies to the expression you have in the OP.  Time series models are examples where you do not have conditional independence (eg auto-regressive or AR models).  Standard linear regression models are examples where you do have conditional independence.
A: While the answer of probabilityislogic is making the right point and is thus the answer so far, let me add that any dependence structure within a sample $\mathfrak{D}$ is achievable with the adequate probability model. Statistical inference is possible for such models if there is some degree of repeatability. For instance, assuming a Markov dependence as in the OP means that the conditional distribution of each datapoint given the previous one is the same for all indices. If one does not wish to make any assumption about the dependence structure of the sample $\mathfrak{D}$, this amounts to observing a single realisation of the vector $\mathfrak{D}$, with an unknown distribution for which inference based on $\mathfrak{D}$ only is impossible (or unreliable).
