Logistic regression

import MaximumLikelihoodProblems

import Distributions
import ForwardDiff
import LogDensityProblems
import StatsFuns
import TransformVariables

struct LogisticRegression{Ty, TX}
    y::Ty
    X::TX
end

function (problem::LogisticRegression)(θ)
    y = problem.y
    X = problem.X

    β = θ.β

    η = X * β
    μ = StatsFuns.logistic.(η)
    log_likelihood = sum(Distributions.logpdf.(Distributions.Bernoulli.(μ), y))
    return log_likelihood
end

N = 10_000

# the first column (the column of all ones) is the intercept
X = hcat(ones(N), randn(N))

size_β = (2,)
β_true = [1.0, 2.0]
η_true = X * β_true
μ_true = StatsFuns.logistic.(η_true)
y = rand.(Distributions.Bernoulli.(μ_true))

problem = LogisticRegression(y, X)
transformation = TransformVariables.as((β = TransformVariables.as(Array, size_β), ))
transformed_problem = LogDensityProblems.TransformedLogDensity(transformation,
                                                               problem)
transformed_gradient_problem = LogDensityProblems.ADgradient(:ForwardDiff,
                                                             transformed_problem)

β_hat_initial_guess = zeros(size_β)
θ_hat_initial = (; β = β_hat_initial_guess)

(β = [0.0, 0.0],)

θ_hat:

θ_hat = MaximumLikelihoodProblems.fit(transformed_gradient_problem,
                                      θ_hat_initial)

(β = [1.0399054172020537, 2.094422989526576],)

β_hat:

β_hat = θ_hat[:β]

2-element Array{Float64,1}:
 1.0399054172020537
 2.094422989526576

Value of the log likelihood function evaluated at θ_hat:

MaximumLikelihoodProblems.loglikelihood(transformed_gradient_problem, θ_hat)

-4216.010543058957

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