ABC-MCMC and ABC-sequential Monte Carlo

Use the prior and simulator setup of Approximate Bayesian computation and the tolerance-conditioned target: standard-Borel parameter and data spaces $(\mathrm{\Theta },\mathcal{T})$ and $(\mathsf{Y},\mathcal{Y})$, a prior probability $\mathrm{\Pi }$, a probability kernel $K(\theta ,\cdot )$, a measurable summary map $S:\mathsf{Y}\to {\mathbb{R}}^d$ with $d\ge 1$, an observed summary $s_0\in {\mathbb{R}}^d$, and tolerance $ϵ\ge 0$. Let $B_{ϵ}(s_0)=\{s\in {\mathbb{R}}^d:\mathrm{\parallel }s-s_0{\mathrm{\parallel }}_2\le ϵ\},A_{ϵ}=S^{-1}(B_{ϵ}(s_0))\mathrm{.}$ The prior-simulator joint law $H$ on the product sigma-algebra $(\mathrm{\Theta }\times \mathsf{Y},\mathcal{T}\otimes \mathcal{Y})$ is $H(C)={\int }_{\mathrm{\Theta }}K(\theta ,C_{\theta })\mathrm{\Pi }(d\theta ),C_{\theta }=\{y:(\theta ,y)\in C\},$ which is a probability measure by A prior and model kernel define a joint law. The finite product is standard Borel by Finite products of standard Borel spaces are standard Borel. Put $h_{ϵ}(\theta ,y)={\mathbf{1}}_{A_{ϵ}}(y),Z_{ϵ}={\int }_{\mathrm{\Theta }\times \mathsf{Y}}{h}_{ϵ}dH={\int }_{\mathrm{\Theta }}K(\theta ,A_{ϵ})\mathrm{\Pi }(d\theta )\mathrm{.}$ Thus $0\le Z_{ϵ}\le 1$, and $Z_{ϵ}$ is the prior-predictive probability $p_{ϵ}(s_0)$ in Approximate Bayesian computation and the tolerance-conditioned target. Only when $Z_{ϵ}>0$ define the augmented ABC tolerance target ${\tilde{\mathrm{\Pi }}}_{ϵ}(C)=\frac{1}{Z_{ϵ}}{\int }_C{h}_{ϵ}dH(C\in \mathcal{T}\otimes \mathcal{Y})\mathrm{.}$ It is the prior-simulator joint law conditioned on the summary-tolerance event. Its parameter marginal is identified in The augmented ABC target has the tolerance posterior as its parameter marginal. The definition does not assert that summary reduction or a positive tolerance preserves the full-data posterior.

Under the hypotheses of Augmented ABC tolerance target with $Z_{ϵ}>0$, the $\mathrm{\Theta }$-marginal of ${\tilde{\mathrm{\Pi }}}_{ϵ}$ is the tolerance-conditioned law ${\mathrm{\Pi }}_{ϵ}(\cdot \mid s_0)$ defined in Approximate Bayesian computation and the tolerance-conditioned target.

Fix $ϵ\ge 0$ with $Z_{ϵ}>0$ and let ${\tilde{\mathrm{\Pi }}}_{ϵ}$ be the target in Augmented ABC tolerance target. The state is $x=(\theta ,y)\in \mathrm{\Theta }\times \mathsf{Y}$, a standard-Borel product by Finite products of standard Borel spaces are standard Borel, and admissible initial states satisfy $y\in A_{ϵ}$. As proposal inputs take a parameter probability kernel $q(\theta ,d{\theta }^{\mathrm{\prime }})$ and the simulator kernel $K({\theta }^{\mathrm{\prime }},d{y}^{\mathrm{\prime }})$ from Augmented ABC tolerance target, such that $Q((\theta ,y),d{\theta }^{\mathrm{\prime }},d{y}^{\mathrm{\prime }})=q(\theta ,d{\theta }^{\mathrm{\prime }})K({\theta }^{\mathrm{\prime }},d{y}^{\mathrm{\prime }})$ is a probability kernel on the augmented state space. Construct the transition kernel $P_{ϵ}$ by applying Metropolis–Hastings kernel with target ${\tilde{\mathrm{\Pi }}}_{ϵ}$ and proposal $Q$. Each iteration proposes a new parameter from $q$, simulates new data from $K$, applies the general Metropolis–Hastings accepted-flow rule, and retains the current augmented state after rejection.

This definition uses one fixed tolerance and the augmented state $(\theta ,y)$. It does not claim irreducibility, uniqueness of a stationary law, convergence, a mixing rate, or recovery of the full-data posterior.

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.

Assume AC and the hypotheses of ABC Metropolis–Hastings on simulator-augmented states with $Z_{ϵ}>0$. Its output $P_{ϵ}$ is reversible with respect to and leaves invariant the augmented ABC tolerance target ${\tilde{\mathrm{\Pi }}}_{ϵ}$ on $\mathrm{\Theta }\times \mathsf{Y}$. This is invariance for the fixed tolerance target only; it gives no convergence or full-data posterior conclusion.

Assume the setup of Sequential ABC importance-resampling with tolerance populations and put ${\gamma }_t(d\theta ,dy)=\mathrm{\Pi }(d\theta )K(\theta ,dy){\mathbf{1}}_{A_{{ϵ}_t}}(y),{\pi }_t={\gamma }_t\mathrm{/}{Z}_{{ϵ}_t}={\tilde{\mathrm{\Pi }}}_{{ϵ}_t}\mathrm{.}$ For a measure $\mu$, write $\mu (f)=\int fd\mu$ for bounded nonnegative measurable $f$. For every bounded nonnegative product-measurable $f:\mathrm{\Theta }\times \mathsf{Y}\to [0,\mathrm{\infty })$, let $X_0^j=({\mathrm{\Theta }}_0^j,Y_0^j)$, $1\le j\le N$, be the initial $N$ independent prior-simulator draws with law $H$ and weights $a_0^j={\mathbf{1}}_{A_{{ϵ}_0}}(Y_0^j)$ satisfy $\mathbb{E}\text{\hspace{0.17em}⁣}[\frac{1}{N}{\sum }_{j=1}^N{a}_0^{j}f(X_0^j)]={\gamma }_0(f)\mathrm{.}$ For each $1\le t\le T$ reached with positive probability, let $Z_{t-1}=(({\mathrm{\Theta }}_{t-1}^i,Y_{t-1}^i{)}_{i=1}^N,(W_{t-1}^i{)}_{i=1}^N)$ be the previous population and let ${\lambda }_{t-1}$ be its law conditional on reaching stage $t$. Use the standard-Borel population space and probability kernel ${\varphi }_t^N(z,\cdot )$ from Sequential ABC importance-resampling with tolerance populations, and assume $\mathrm{\Pi }\ll {\varphi }_t^N(z,\cdot )$ for ${\lambda }_{t-1}$-almost every $z$. The kernel Radon–Nikodym lemma Jointly measurable Radon–Nikodym derivatives for probability kernels then supplies a product-measurable derivative; use the finite-valued version specified in Sequential ABC importance-resampling with tolerance populations after changing infinite values to zero. Write this version as $r_t^N(z,\theta )=d\mathrm{\Pi }\mathrm{/}d{\varphi }_t^N(z,\cdot )(\theta )$ for almost every $z$. Put $a_t^j={\mathbf{1}}_{A_{{ϵ}_t}}(Y_t^j)r_t^N(Z_{t-1},{\mathrm{\Theta }}_t^j)$ for the candidates produced by multinomial ancestor resampling, perturbation, and simulation. For every bounded nonnegative product-measurable $f:\mathrm{\Theta }\times \mathsf{Y}\to [0,\mathrm{\infty })$, $\mathbb{E}\text{\hspace{0.17em}⁣}[\frac{1}{N}{\sum }_{j=1}^N{a}_t^{j}f({\mathrm{\Theta }}_t^j,Y_t^j)|\text{previous population}]={\gamma }_t(f)\text{almost surely on reaching stage }t\mathrm{.}$ In particular, the average weight is conditionally unbiased for $Z_{{ϵ}_t}$. If candidates with positive total weight are resampled by the multinomial or residual scheme from normalized weights, then for every bounded measurable $f$ their conditional empirical mean equals the weighted candidate empirical mean in expectation. No unbiasedness is asserted for the self-normalized finite-$N$ estimate, and a finite sample may have zero total weight or fail to represent positive target mass.