of the posterior distribution. We suggest to incorporate this information into the proposal construction. With $(\theta', u')$ being the current state of the Markov chain we simulate $\theta'' \sim q(\cdot | \theta', u')$ and $u'' \sim m_{\theta''}(\cdot)$, using some proposal $q$ (see Section \[sec:constructPMH2\]).

It follows that the (standard) MH acceptance probability for the extended target is given by $$\begin{aligned}
  \nonumber
  \alpha(\theta'',u'', \theta', u')
  &=  1  \wedge
  \frac{ \widehat{p}_{\theta''}(\mathbf{y} | u'') m_{\theta''}(u'') p(\theta'')}{ \widehat{p}_{\theta'}(\mathbf{y} | u') m_{\theta'}(u') p(\theta') }
  \frac{ m_{\theta'}(u') q(\theta' | \theta'', u'') }{ m_{\theta''}(u'') q(\theta'' | \theta', u' )} \\
  &=1  \wedge
  \frac{ \widehat{p}_{\theta''}(\mathbf{y} | u'') p(\theta'')}{ \widehat{p}_{\theta'}(\mathbf{y} | u') p(\theta') }
  \frac{ q(\theta' | \theta'', u'') }{ q(\theta'' | \theta', u' )}.
  % &=  1
  % \wedge
  % \frac{ \widehat{p}_{\theta''}(\mathbf{y} | u'') }{ \widehat{p}_{\theta'}(\mathbf{y} | u')  }
  % \frac{ p(\theta'') }{ p(\theta') }
  % \frac{ q(\theta' | \theta'', u'') }{ q(\theta'' | \theta', u' )}.
  \label{eq:PMHaprob}\end{aligned}$$ Note that $ q(\theta'' | \theta', u' )$ may depend on the auxiliary variable $u'$ in a (formally) arbitrary way. In particular, in Section \[sec:SMC\] we propose a construction making use of *biased* estimates of the gradient and Hessian of the log-posterior. Nevertheless, expression still defines a correct MH acceptance 