Dependent Bernoulli trials

Question

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?

Answer 1

The most flexible structure is the one that assings to all possible $n$ binary vectors $\left( x_1, \ldots, x_n \right)$ a probabilty $$P \left[ x_1 = i_1, \ldots, x_n = i_n \right] = p_{i_1, \ldots, i_n}$$ such that $$ \sum_{i_1 = 0}^1 \cdots \sum_{i_n = 0}^1 p_{i_1, \ldots, i_n} = 1$$ Thus, you have to specify $2^n - 1$ parameters (This is much more complicated than your i.i.d. case where you specify $n$ parameters $p_1, \ldots, p_n$.)

There are many ways to simplify the problem.