Particle marginal Metropolis–Hastings

Fix observations $y_{1:T}$, a standard-Borel parameter space $(\mathrm{\Theta },\mathcal{T})$, a prior probability ${\pi }_0$, and the finite-horizon Feynman–Kac path model in Feynman–Kac path targets and normalizers. Write $L_{\theta }(y_{1:T})={\gamma }_{\theta ,T}(1),C={\int }_{\mathrm{\Theta }}{L}_{\theta }(y_{1:T}){\pi }_0(d\theta )\mathrm{.}$ Require every prefix normalizer $0<Z_{\theta ,t}<\mathrm{\infty }$ for $t=0,\dots ,T$ and ${\pi }_0$-almost every $\theta$, together with $0<C<\mathrm{\infty }$; then the parameter posterior is ${\pi }_y(d\theta )=C^{-1}{L}_{\theta }(y_{1:T}){\pi }_0(d\theta )\mathrm{.}$

For fixed particle count $N\ge 1$, let $U$ contain the full output of Sequential importance resampling for Feynman–Kac path targets at $\theta$ and a terminal index $K\in \{1,\dots ,N\}$ drawn conditionally with its final normalized weight. Let $m_{\theta }(du)$ be this measurable auxiliary-variable probability kernel and let ${\widehat{L}}_{\theta }(U)={\widehat{Z}}_T$ be jointly measurable and nonnegative in $(\theta ,U)$. The required interface is $\int {\widehat{L}}_{\theta }(u)m_{\theta }(du)=L_{\theta }(y_{1:T});$ The sequential particle estimator is unbiased for Feynman–Kac path integrals supplies this identity under its finite-horizon support and integrability hypotheses. Define the extended target probability by $\tilde{\pi }(d\theta ,du)=C^{-1}{\pi }_0(d\theta )m_{\theta }(du){\widehat{L}}_{\theta }(u)\mathrm{.}$

Given current $(\theta ,U)$, propose ${\theta }^{\mathrm{\prime }}\sim R(\theta ,\cdot )$ and then $U^{\mathrm{\prime }}\sim m_{{\theta }^{\mathrm{\prime }}}$. Apply the general Metropolis–Hastings construction Metropolis–Hastings kernel to this proposal kernel and target $\tilde{\pi }$. On rejection retain the entire current pair $(\theta ,U)$, including its stored likelihood estimate and selected path index. When ${\pi }_0$ and $R$ have densities $p_0$ and $r$, and the current extended target density is positive, the acceptance probability is $1\wedge \frac{p_0({\theta }^{\mathrm{\prime }}){\widehat{L}}_{{\theta }^{\mathrm{\prime }}}(U^{\mathrm{\prime }})r(\theta \mid {\theta }^{\mathrm{\prime }})}{p_0(\theta ){\widehat{L}}_{\theta }(U)r({\theta }^{\mathrm{\prime }}\mid \theta )}\mathrm{.}$ For the displayed density-ratio formula, initialize at a current state where $p_0(\theta ){\widehat{L}}_{\theta }(U)>0$. The density-free version uses the accepted-flow Radon–Nikodym construction in the cited general-state algorithm; it does not evaluate a pointwise prior mass or require this density condition. Proposed zero estimates are rejected. The input is the current extended state $(\theta ,U)$; output the proposed extended state $({\theta }^{\mathrm{\prime }},U^{\mathrm{\prime }})$ on acceptance and retain $(\theta ,U)$ on rejection.

The selected path stored in $U$ is part of the auxiliary state. Refreshing its estimate after rejection defines the different Monte Carlo within Metropolis procedure, not this algorithm.