How to compute density of conditional probability from data

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A simple logit or probit model is a possibility, maximizing the sum or average of the log likelihood:

function logit(theta, s, x)
    p = 1.0./(1.0 .+ exp.(-x*theta))
    obj = s.*log.(p) .+ (log.(1.0 .- p)).*(1.0 .- s)
end

Another possibility is to use nonparametric kernel regression, rather than kernel density. The probability that s=1|X=x is also the conditional mean of s, given that X=x. Running a kernel regression might be a little easier.

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