1}, \ldots, \bm{W}_{\mathrm{p},
        N})}(\bm{w}_{\mathrm{p}, 1}, \ldots, \bm{w}_{\mathrm{p}, N}; \bm{a}).
    \label{eq:likelihood}\end{aligned}$$ We recover the parameter by maximizing this function for given data, namely the observed values of $\bm{w}_{\mathrm{p}, 1}, \ldots, \bm{w}_{\mathrm{p},
  N}$. We denote the maximum-likelihood estimate (MLE) by $\hat{\bm{a}}$. The MLE is itself the realization of a random variable, which we denote $\hat{\bm{A}}$. It is asymptotically normal (i.e. it is normally distributed in the infinite-data limit) with mean $\bm{a}_0$ (the true parameter) and variance $\bm{I}(\bm{a}_0)^{- 1}$, where the *expected Fisher information matrix* is $$\begin{aligned}
    \bm{I}(\bm{a}_0) = - {\operatorname{E}}\left( \dfrac{\partial^2 \ln L(\bm{a}_0;
      \bm{W}_{\mathrm{p}, 1}, \ldots, \bm{W}_{\mathrm{p}, N})}{\partial \bm{a}
      \partial \bm{a}^{\mathrm{t}}} \right)\end{aligned}$$ i.e. the negative of the expectation of the Hessian of the log-likelihood [see any text-book on the subject, for example @wasserman_all_2007]. The true parameter $\bm{a}_0$ may be approximated by $\hat{\bm{a}}$, and the expected Fisher information matrix by the *observed Fisher information matrix*, $$\begin{aligned}
    \bm{J}(\hat{\bm{a}}) = - \dfrac{\partial^2 \ln L(\hat{\bm{a}};
      \bm{w}_{\mathrm{p}, 1}, \ldots, \bm{w}_{\mathrm{p}, N})}{\partial \bm{a}
      \partial \bm{a}^{\mathrm{t}}}.
    \label{eq:fisher_information}\end{aligned}$$ In summary, the MLE is distributed as $$\begin{aligned}
    \hat{\bm{A}} \sim N(\hat{\bm{a}}, \bm{J}^{-1}(\hat{\bm{a}})).\end{aligned}$$ This allows us to determine the confidence region, $C$, defined by the boundary that is the solution to the equation $$(\bm{a} - \hat{\bm{a}})^\mathrm{t} \bm{J}^{-1}(\hat{\bm{a}}) (\bm{a} -
  \hat{\bm{a}}) = \chi_D^2(1 - 