Sequential ABC importance-resampling with tolerance populations

Fix a finite horizon $T\ge 0$, $N\ge 1$, a positive nonincreasing tolerance schedule ${ϵ}_0\ge {ϵ}_1\ge \cdots \ge {ϵ}_T>0$, and assume $Z_{{ϵ}_t}>0$ for every $t$. Let ${\tilde{\mathrm{\Pi }}}_{{ϵ}_t}$ be the normalized augmented target from Augmented ABC tolerance target. The parameter proposal at stage $t$ is a probability kernel $Q_t(\theta ,d{\theta }^{\mathrm{\prime }})$ on $\mathrm{\Theta }$.

For each $t\ge 1$, write the previous weighted population as $Z_{t-1}=(({\theta }_{t-1}^i,y_{t-1}^i{)}_{i=1}^N,(W_{t-1}^i{)}_{i=1}^N)$ in the standard-Borel space $E_N=(\mathrm{\Theta }\times \mathsf{Y}{)}^N\times [0,1{]}^N,$ where $[0,1]$ has its Borel sigma-algebra and is standard Borel because the closed interval with the Euclidean metric is Polish. Thus $E_N$ is standard Borel by Finite products of standard Borel spaces are standard Borel. Let ${\lambda }_{t-1}$ be the law of $Z_{t-1}$ conditional on the algorithm reaching stage $t$, when that event has positive probability. For $z=(({\theta }_i,y_i{)}_{i=1}^N,(w_i{)}_{i=1}^N)\in E_N$, define the probability kernel ${\varphi }_t^N$ by ${\varphi }_t^N(z,B)=\{\begin{array}{ll}{\sum }_{i=1}^N{w}_i{Q}_t({\theta }_i,B),& {\sum }_i{w}_i=1,\\ \frac{1}{N}{\sum }_{i=1}^N{Q}_t({\theta }_i,B),& {\sum }_i{w}_i\mathrm{\ne }1.\end{array}$ For each fixed measurable $B\subseteq \mathrm{\Theta }$, $z\mapsto Q_t({\theta }_i,B)$ and $z\mapsto w_i$ are measurable. For nonnegative measurable $u,v$ on $E_N$, the superlevel sets satisfy $\{uv>a\}=E_N(a<0),\{uv>a\}={\bigcup }_{q\in \mathbb{Q},q>0}(\{u>q\}\cap \{v>a\mathrm{/}q\})(a\ge 0),$ so each product $w_i{Q}_t({\theta }_i,B)$ is measurable. Finite sums and scalar multiples are measurable by Closure properties of measurable functions used by the integral, making the branch condition and both branch evaluations measurable. Each branch is a convex combination of probability measures, so Measure kernel and probability kernel gives a probability kernel. The second branch only extends the kernel to ambient states outside the algorithm; every actual previous population has normalized weights and uses the proposal mixture in the first branch. Require $\mathrm{\Pi }\ll {\varphi }_t^N(z,\cdot )$ for ${\lambda }_{t-1}$-almost every $z$ at every stage that is reached with positive probability. Apply Jointly measurable Radon–Nikodym derivatives for probability kernels to the constant kernel $z\mapsto \mathrm{\Pi }$ and the kernel ${\varphi }_t^N$ to obtain a product-measurable derivative on $E_N\times \mathrm{\Theta }$. Set its value to zero where it is infinite; since its integral is one for ${\lambda }_{t-1}$-almost every $z$, this gives a finite-valued product-measurable version $r_t^N(z,\theta )=d\mathrm{\Pi }\mathrm{/}d{\varphi }_t^N(z,\cdot )(\theta )$ for almost every previous population.

At stage $0$, draw $({\theta }_i,y_i)$ independently from the prior-simulator law $H$, and set $a_0^i={\mathbf{1}}_{A_{{ϵ}_0}}(y_i)$. If ${\sum }_i{a}_0^i=0$, stop and report finite-particle degeneracy; otherwise normalize these weights to obtain $W_0^i$ and report the weighted empirical measure ${\widehat{\mathrm{\Pi }}}_0^N={\sum }_{i=1}^N{W}_0^i{\delta }_{({\theta }_i,y_i)}\mathrm{.}$

For each $t=1,\dots ,T$, form the proposal mixture ${\varphi }_t^N(\cdot )={\sum }_{i=1}^N{W}_{t-1}^i{Q}_t({\theta }_{t-1}^i,\cdot )\mathrm{.}$ Draw $N$ ancestor indices from the previous weights using the multinomial branch of Multinomial and residual offspring resampling. Conditional on those ancestors, independently draw one perturbed parameter from its $Q_t$ kernel and one datum from $K$ for each child. For a candidate $({\theta }_t^j,y_t^j)$ set $a_t^j={\mathbf{1}}_{A_{{ϵ}_t}}(y_t^j)r_t^N(Z_{t-1},{\theta }_t^j)\mathrm{.}$ If ${\sum }_j{a}_t^j=0$, stop and report finite-particle degeneracy; do not report a normalized target estimate. If the sum is positive, set $W_t^j=a_t^j\mathrm{/}{\sum }_k{a}_t^k$ and report the weighted empirical measure ${\widehat{\mathrm{\Pi }}}_t^N={\sum }_j{W}_t^j{\delta }_{({\theta }_t^j,y_t^j)}$.

If a stage degenerates, return that status and no normalized estimate for the stage or any later target. If every stage through $T$ succeeds, return ${\widehat{\mathrm{\Pi }}}_T^N$; in particular, when $T=0$ the returned estimate is ${\widehat{\mathrm{\Pi }}}_0^N$.

The mixture support condition is stage-specific and must hold almost surely under the law of the actual previous population. The algorithm makes no finite-$N$ accuracy, mixing, or posterior-recovery claim.