the estimate of the negative Hessian with the inverse sample covariance matrix calculated using the trace of Markov chain when the estimate is not PD. This can be seen as a hybrid between the PMH2 algorithm and a *pre-conditioned PMH1 algorithm*. This resembles some other adaptive MH algorithms [@AndrieuThoms2008] in which the same procedure is used to adapt the covariance matrix of a random walk proposal. For this, we can make use of the last $L$ iterations of the MH algorithm after that the Markov chain has reached stationarity. During the burn-in phase, non–PD estimates can be handled using a regularization approach or by rejecting the proposed parameter. In this article, we refer to this method for handling non–PD estimates of the negative Hessian as the *hybrid PMH2 algorithm*, where we use the latter alternative during the burn-in phase. Note that this pre-conditioning can also be applied to the PMH0 and PMH1 algorithm, we return to this in Section \[sec:results:earth\].

<span style="font-variant:small-caps;">Inputs:</span> $\mathbf{y}$ (data), $R(\cdot)$ (propagation kernel), $\nu(\cdot)$ (weight function), $N > 0$ (no. particles), $0 < \Delta \leq T$ (lag).\
<span style="font-variant:small-caps;">Outputs:</span> $\widehat{p}_{\theta}(\mathbf {y})$ (est. of the likelihood), $\widehat{\mathsf{S}}_T(\theta)$ (est. of the gradient), $\widehat{\mathsf{I}}_T(\theta)$ (est. of the negative Hessian).

