+\noindent
+Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output. \\
+
+\noindent
+A binary regression model assumes that for some link function $g$, the probability that $Y=1$ conditionally to $X=x$ is given by $g(\langle \beta , x \rangle +b)$, where $\beta\in \R^{d}$ is the vector of regression coefficients and $b\in\R$ is the intercept. Popular examples of link functions are the logit link function where for any real $z$, $g(z)=e^z/(1+e^z)$ and the probit link function where $g(z)=\Phi(z),$ with $\Phi$ the cumulative distribution function of the standard normal ${\cal N}(0,1)$. \\
+If now we want to modelise heterogeneous populations, let $K$ be the number of populations and $\omega=(\omega_1,\cdots,\omega_K)$ their weights such that $\omega_{j}\geq 0$, $j=1,\ldots,K$ and $\sum_{j=1}^{K}\omega{j}=1$. Define, for $j=1,\ldots,K$, the regression coefficients in the $j$-th population by $\beta_{j}\in\R^{d}$ and the intercept in the $j$-th population by $b_{j}\in\R$. Let $\omega =(\omega_{1},\ldots,\omega_{K})$, $b=(b_1,\cdots,b_K)$, $\beta=[\beta_{1} \vert \cdots,\vert \beta_K]$ the $d\times K$ matrix of regression coefficients and denote $\theta=(\omega,\beta,b)$.
+The model of population mixture of binary regressions is given by:
+\begin{equation}
+\label{mixturemodel1}
+\PP_{\theta}(Y=1\vert X=x)=\sum^{K}_{k=1}\omega_k g(<\beta_k,x>+b_k).
+\end{equation}
+
+\noindent
+We assume that the random variable $X$ has a Gaussian distribution. We now focus on the situation where $X\sim \mathcal{N}(0,I_d)$, $I_d$ being the identity $d\times d$ matrix. All results may be easily extended to the situation where $X\sim \mathcal{N}(m,\Sigma)$, $m\in \R^{d}$, $\Sigma$ a positive and symetric $d\times d$ matrix. \\
+
+\noindent