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The probability of a sequence of n independent Bernoulli trials can be easily expressed as $$p(x_1,...,x_n|p_1,...,p_n)=\prod_{i=1}^np_i^{x_i}(1-p_i)^{1-x_i}$$ but what if the trials are not independent?

How would one express the probability to capture the dependence?

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What is the dependence? E.g. Summing over the N trials must equal K? There must be an even number of 'true' results, etc. Once you define the kind of dependence it will be possible to write down the actual likelihood more concretely. – Nick Dec 27 '12 at 17:25

There are expressions you can write down, but I hope you realize how uninformative they are. Saying that the variables are not known to be indpendent, without saying anything else, gives no usable information. It's like saying that you have a friend whose name is not known to be Bob, then asking what you can say about your friend's height and age. So, here is a nearly meaningless restatement:

$$p(x_1,...,x_n) = \prod_i p(X_i=x_i|X_1=x_1,...,X_{i-1}=x_{i-1}).$$

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did you look at de Finetti theorem and exchangeable sequences ?

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Can you make this reply more self-contained, e.g. by providing a brief overview of the slides/references you linked to? – chl Dec 27 '12 at 11:24

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