probability for the extended target, ensuring the validity of our approach. Note also that the aforementioned proposal construction opens up for a wide range of adapted proposals, possibly different from the ones considered in this work.

Constructing PMH1 and PMH2 {#sec:constructPMH2}
--------------------------

We now turn to the construction of a proposal that makes use of the gradient and Hessian of the log-posterior. Following @RobertCasella2004, we do this by a Laplace approximation of the parameter posterior around the current state $\theta'$. Hence, consider a second order Taylor expansion of $\log p(\theta''|\mathbf{y})$ at $\theta'$: $$\begin{aligned}
    \log p(\theta''|\mathbf{y})
    &\approx    
    \log p(\theta'|\mathbf{y}) 
    + (\theta'' - \theta')^{\top} \Big[ \Dtheta \log p(\theta|\mathbf{y}) \Big]_{\theta=\theta'} \\
    &+ \frac{1}{2} (\theta'' - \theta')^{\top} \Big[ \DDtheta \log p(\theta|\mathbf{y}) \Big]_{\theta=\theta'} (\theta'' - \theta').\end{aligned}$$ Taking the exponential of both sides and completing the square, we obtain $$\begin{aligned}
    p(\theta''| \mathbf{y})
    &\approx
    \textsf{N} \Big( \theta'';\theta' + \mathsf{I}^{-1}_T(\theta') \mathsf{S}_T(\theta'), \mathsf{I}^{-1}_T(\theta') \Big),\end{aligned}$$ where we have introduced $\mathsf{S}_T(\theta') = \Dtheta \log p(\theta|\mathbf{y})|_{\theta=\theta'}$ and $\mathsf{I}_T(\theta') = - \DDtheta \log p(\theta|\mathbf{y})|_{\theta=\theta'}$, for the gradient and the negative Hessian of the log-posterior, respectively. Here, we assume for now that the negative Hessian is positive definite; see Section \[sec:regularisation\] for further discussion on this matter.

As pointed out above, these quantities cannot 