# Dependent Bernoulli trials

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?

• did you look at de Finetti theorem and exchangeable sequences ? stats.ox.ac.uk/~steffen/teaching/grad/definetti.pdf – user18129 Dec 27 '12 at 10:42
• 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

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