Convex Optimization · Problem 7.3

Geometric problems — Boyd & Vandenberghe

Mathematics — Optimization · probit-model-ml-estimation

Problem

Probit model. Suppose

y \in \{0, 1\}

is random variable given by

y = \begin{cases} 1 & a^T u + b + v \leq 0 \\ 0 & a^T u + b + v > 0, \end{cases}

where the vector

u \in \mathbf{R}^n

is a vector of explanatory variables (as in the logistic model described on page 354), and

v

is a zero mean unit variance Gaussian variable. Formulate the ML estimation problem of estimating

a

and

b

, given data consisting of pairs

(u_i, y_i)

i = 1, \ldots, N

, as a convex optimization problem.