estimate of the Hessian {#sec:regularisation}
---------------------------------------------

The PMH2 proposal relies on the assumption that the observed information matrix is positive definite (PD). The estimator given in does not always satisfy this, especially when the Markov chain is located far from the posterior mode. Typically, the amount of information is limited in such regions and this results in that the curvature is difficult to estimate. To cope with this issue, one alternative is to regularize the Hessian by adding a diagonal matrix to shift the eigenvalues to be positive. The diagonal matrix can e.g. be selected such that $$\begin{aligned}
    \Delta \widehat{I}_T = \max \Big\{0, - 2 \lambda_{\min} \big( \widehat{I}_T \big) \Big\} I_d,
    \label{eq:Hessianregularization}\end{aligned}$$ where $\lambda_{\min}(\widehat{I}_T)$ denotes the smallest eigenvalue of $\widehat{I}_T(\theta|u)$. In this article, we make use of this method for handling non–PD estimates of the negative Hessian for the PMH2 algorithm. This heuristic is common for Newton-type optimisation algorithms [@NocedalWright2006 Chapter 3.4].

Note, that there are other solutions available for ensuring positive definiteness that only shifts the negative eigenvalues, see [@NocedalWright2006 Chapter 3]. We emphasise that this type of regularization keeps the Markov chain invariant, i.e. still targets the correct posterior distribution (recall Section \[sec:PMH\]).

Another alternative is to replace 