How do I construct the likelihood function for a series of observed Bernoulli-distributed datapoints?
I have observed N (a_i, p_i) pairs each drawn from a different Bernoulli distribution. Here a_i are observed amplitudes and p_i are observed probabilities of success for the i^{th} draw. I would like to model the full likelihood distribution (and not just via the MLE) with a view to identifying which draws belong to the successful class (and hence the statistics of just these draws, especially any irregular uncertainty distributions etc.). For example, I have a_i p_i p_true 10 0.2 0 100 0.9 1 11 0.1 0 99 0.93 1 12 0.25 0 I know the PDFs of the model and of the data (both are obviously Bernoulli distributions), but how do I combine them to obtain residuals that I can use to explore the joint distribution? What distribution does the joint distribution follow and how? I have tried to unpack the cross-entropy and data and model but don't have a clear solution. Is the continuous Bernoulli distribution a complete red herring? Note that ordering doesn't matter in my example. Some other points in response to comments: The p_i values come from an oracle - these represent the probabilities that the datapoints belong to a common class. The amplitudes a_i are just weights for the corresponding Bernoulli distributions. The higher the amplitude, the higher the scaling of the Bernoulli distribution in its contribution to the overall process. See also: "Weighted" Poisson binomial distribution Weighted sum of Bernoulli distributions https://math.stackexchange.com/questions/3481907/sum-of-wei…
