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Assume I have a large number of data points (admitted, score) representing applicants who have either been admitted or rejected and who have some score. Assume we know there is a positive correlation between score and admission.
I want to predict the probability that a new applicant will be admitted based on his or her score. Note that I know that the correlation between score and admission is strong but don't know how strong, so I'll also need to somehow measure my confidence in my prediction.
You can use a non-parametric estimation. Let $p_*=(a_*, s_*)$ be a datapoint and $a_i=1$ if ith datapoint was admitted, $a_i=0$ otherwise. Then:
$P(a_* = 1|s_*<s) = \frac{\sum\limits_{ \left\{s_i<s\right\} } a_i}{\sum\limits_{ \left\{s_i<s\right\}}1}$
This is, you just divide how many with that score or lower have been admitted by how many are there with that score or lower. You must use intervals if your score is a continuous variable (points in a continous variable have probability almost zero).
In other to do estimate some confidence interval, you can refer to confidence interval in non-parametric methods.
If you want to use a parametric method, you must have to make some assumptions about the model.
This method is not affected by non observed cofounders or non-independence of observations, since it makes no assumptions on the underlying model.
You can use kernel density estimation (a fancier version of histogram) to estimate the density of score $P(s \, | \, \hbox{accept})$ assuming the applicant is accepted (from your sample of accepted scores), and similarly estimate the density of score $P(s \, | \, \hbox{reject})$ assuming rejected (from your sample of rejected scores). You also have prior probabilities $P(\hbox{accept})$ and $P(\hbox{reject}) = 1 - P(\hbox{accept})$ based on the empirical sample. Then, for a given new score $s$, you can calculate the acceptance probability $P(\hbox{accept} \, | \, s)$ using Bayes rule.
