Is there any software to estimate dependence between Bernoulli
trials?
I have N observations on k-vectors of 0s and 1s, and I need to
model the dependence. Consider the simple case with k = 2. There are 4
possible outcomes: (0, 0), (0, 1), (1, 0), and (1, 1). I assume 0 =
"failure" and 1 = "success". I assume further that there
are 3
independent failure mechanisms, one that makes the first outcome 0 with
probability (1-q1), another that makes the second outcome 0 with
probability (1-q2), and a third that make them both 0 with probability
(1-q12). I compute the probabilities of these 4 outcomes as follows:
P00 = Pr(0, 0) = q1*q2*q12 = probability of no failures.
P01 = Pr(0, 1) = q1*(1-q2)*q12 = probability of a failure from the
second mechanism and none from the other 2.
P10 = Pr(1, 0) = (1-q1)*q2*q12 = probability of a failure from the
first mechanism and none from the other 2.
P11 = Pr(1, 1) = 1-Pr(0,0)-Pr(0,1)-Pr(1,0).
With this setup, I can write a likelihood function, parameterize
it appropriately, and estimate what I need. However, I wonder if
something already exists.
Best Wishes,
Spencer Graves