Algorithm \[alg:SMCfull\], described in Section \[sec:SMC\], to estimate the quantities needed for computing the proposal and the acceptance probability. Clearly, PMH0 and PMH1 are special cases obtained by using the corresponding proposal from in the algorithm. Note that, while the algorithm make explicit reference to the auxiliary variable $u$, it only depends on this variable through the estimates $\widehat{p}_{\theta'}(\mathbf{y})$, $\widehat{\mathsf{S}}_T(\theta')$ and $\widehat{\mathsf{I}}_T(\theta')$.

<span style="font-variant:small-caps;">Inputs:</span> Algorithm \[alg:SMCfull\]. $M>0$ (no. MCMC steps), $\theta_0$ (initial parameters), $\gamma$ (step length).\
<span style="font-variant:small-caps;">Output:</span> $\theta=\{\theta_1,\ldots,\theta_M\}$ (samples from the posterior).

Run Algorithm \[alg:SMCfull\] to obtain $\widehat{p}_{\theta_0}(\mathbf{y})$, $\widehat{\mathsf{S}}_T(\theta_0)$ and $\widehat{\mathsf{I}}_T(\theta_0)$. Sample $\theta' \sim q(\theta'|\theta_{k-1},u_{k-1})$ by , $\widehat{\mathsf{S}}_T(\theta_{k-1})$ and $\widehat{\mathsf{I}}_T(\theta_{k-1})$. Run Algorithm \[alg:SMCfull\] to obtain $\widehat{p}_{\theta'}(\mathbf{y})$, $\widehat{\mathsf{S}}_T(\theta')$ and $\widehat{\mathsf{I}}_T(\theta')$. Sample $\omega_k$ uniformly over $[0,1]$. $\theta_k \leftarrow \theta'$. $\{ \widehat{p}_{\theta_k}(\mathbf{y}), \widehat{\mathsf{S}}_T(\theta_{k}), \widehat{\mathsf{I}}_T(\theta_{k}) \} \leftarrow \{ \widehat{p}_{\theta'}(\mathbf{y}) , \widehat{\mathsf{S}}_T(\theta') , \widehat{\mathsf{I}}_T(\theta') \}$. $\theta_k \leftarrow \theta_{k-1}$. $\{ \widehat{p}_{\theta_k}(\mathbf{y}), \widehat{\mathsf{S}}_T(\theta_{k}), \widehat{\mathsf{I}}_T(\theta_{k}) \} \leftarrow \{ \widehat{p}_{\theta_{k-1}}(\mathbf{y}) , \widehat{\mathsf{S}}_T(\theta_{k-1}) , \widehat{\mathsf{I}}_T(\theta_{k-1}) \}$.

\[alg:PMH2order\]

Properties of the PMH1 and PMH2 proposals
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In the sequel, we use a single step size $\Gamma = \gamma^2 I_d$ for all the parameters in the (standard) proposal. This is done to illustrate the advantage of adding the Hessian information, which rescales the step lengths according to the local curvature. Hence, 