Monte Carlo within Metropolis

Use the parameter and likelihood-estimator inputs of Particle marginal Metropolis–Hastings, restricted to the common-density case where the prior has density $p_0$ and the proposal kernel has transition density $r({\theta }^{\mathrm{\prime }}\mid \theta )$ with respect to a common parameter reference measure. Initialize at a parameter with $p_0(\theta )>0$, and assume ${\widehat{L}}_{\vartheta }(U)>0$ for $m_{\vartheta }$-almost every $U$ at every parameter $\vartheta$ that the procedure can propose or visit. These conditions make the displayed ratio defined almost surely. The input is the current parameter $\theta$; the output is the next parameter.

At each iteration draw a fresh $U\sim m_{\theta }$, propose ${\theta }^{\mathrm{\prime }}\sim R(\theta ,\cdot )$, and independently draw $U^{\mathrm{\prime }}\sim m_{{\theta }^{\mathrm{\prime }}}$. Accept ${\theta }^{\mathrm{\prime }}$ with probability $1\wedge \frac{p_0({\theta }^{\mathrm{\prime }}){\widehat{L}}_{{\theta }^{\mathrm{\prime }}}(U^{\mathrm{\prime }})r(\theta \mid {\theta }^{\mathrm{\prime }})}{p_0(\theta ){\widehat{L}}_{\theta }(U)r({\theta }^{\mathrm{\prime }}\mid \theta )}\mathrm{.}$ Output ${\theta }^{\mathrm{\prime }}$ on acceptance and $\theta$ on rejection. Discard both auxiliary values before the next iteration and draw a new current estimate.

This is Monte Carlo within Metropolis (MCWM). It does not retain a single extended state on rejection, so the pseudo-marginal invariance proof does not apply. This definition makes no exactness or stationarity claim.