it allows for taking larger steps when the curvature is small and vice verse.

This property of PMH2 makes the algorithm scale-free in the same manner as a Newton algorithm in optimisation [@NocedalWright2006 Chapter 3]. That is, the proposal is invariant to affine transformations of the parameters. Note that, since the local information is used, this is different from scaling the proposal in PMH0 with the posterior covariance matrix estimated from a pilot run, as this only takes the geometry at the mode of the posterior into account.

Some analyses of the statistical properties are available for PMH0 [@SherlockThieryRobetsRosenthal2013], MH using a random walk [@RobertsGelmanGilks1997] and MALA [@RobertsRosenthal1998]. It is known from these analyses that adding the gradient into the proposal can increase the mixing of the Markov chain. Note that these results are obtained under somewhat strict assumptions. Also, we know from numerical experiments [@GirolamiCalderhead2011] that there are further benefits of also taking the local curvature into account.

Estimation of the likelihood, gradient, and Hessian {#sec:SMC}
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In this section, we show how to estimate the likelihood together with the gradient and Hessian using SMC methods.

Auxiliary particle filter
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An auxiliary particle filter (APF) [@PittShephard1999] can be used 