be computed in closed form, but they can be estimated from the random variable $u'$ (see Section \[sec:SMC\]). This suggests three different versions of the PMH algorithm, each resulting from a specific choice of the proposal: $$\begin{aligned}
q(\theta''|\theta',u') =
\begin{dcases}
\textsf{N}\left( \theta', \Gamma \right), & \text{[PMH0]}  \\
\textsf{N}\left( \theta' + {\textstyle \frac{1}{2}} \Gamma \widehat{\mathsf{S}}_T(\theta'|u'), \Gamma \right), & \text{[PMH1]} \\
\textsf{N}\left( \theta' + \widehat{\textsf{G}}(\theta'|u'), \widehat{\textsf{H}}(\theta'|u') \right). & \text{[PMH2]}
\end{dcases}
\label{eq:2orderproposal}\end{aligned}$$ Here, we use the notation $\widehat{\textsf{G}}(\theta|u) = \frac{1}{2} \Gamma\, \widehat{\mathsf{I}}^{-1}_T(\theta|u) \, \widehat{\mathsf{S}}_T(\theta|u)$ and $\widehat{\textsf{H}}(\theta|u) = \Gamma\, \widehat{\mathsf{I}}^{-1}_T(\theta|u)$ for the natural gradient and scaled inverse Hessian, respectively. Furthermore, $\Gamma$ denotes a scaling matrix that controls the step lengths of the proposal. For PMH0 and PMH1, $\Gamma$ can be chosen as the inverse of an estimate of the posterior covariance matrix. However, computing this estimate typically requires costly and tedious trial runs. For PMH2, the curvature of the problem is captured by the Hessian matrix, i.e. a single step length can by used which can significantly simplify the tuning. It is also possible to choose different step lengths for the drift term and for the covariance matrix of the proposal.

The final PMH2 algorithm is presented in Algorithm \[alg:PMH2order\]. It makes use of 