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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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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 ? http://www.stats.ox.ac.uk/~steffen/teaching/grad/definetti.pdf |
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