Feynman–Kac targets and sequential particle filters

Apply [F3] to $\lambda$ and each one-step kernel to obtain joint laws $P(dx,dy)=\lambda (dx)M(x,dy)$ and $Q(dx,dy)=\lambda (dx)R(x,dy)$. For any product-measurable $A$, the section integrals are measurable by [F4]. If $Q(A)=0$, then $R(x,A_x)=0$ for $\lambda$-almost every $x$ by [F5]; intersecting with the conull kernel-domination set gives $M(x,A_x)=0$ there, hence $P(A)=0$. Thus $P\ll Q$.

By [F6], choose a product-measurable density $h=dP\mathrm{/}dQ$. On $N=\{h<0\}$, positivity of $P$ gives $0\le P(N)={\int }_{N}hdQ=-{\int }_N{h}^{-}dQ$, so [F5] makes $h^{-}=0$ $Q$-almost everywhere on $N$. The Borel positive-part map preserves measurability by [F9], so replacing $h$ by $h^{+}=\max (h,0)$ leaves the density unchanged $Q$-almost everywhere and makes it nonnegative.

Let $\mathcal{C}$ be a countable measure-determining algebra on $F$ from [F7], containing $F$. For each $B\in \mathcal{C}$ and $D\in \mathcal{E}$, the rectangle laws and density identity [F8] give ${\int }_{D}M(x,B)\lambda (dx)=P(D\times B)={\int }_D({\int }_{B}h(x,y)R(x,dy))\lambda (dx)$. Both integrands are measurable by [F2] and [F4] and integrable because their integrals over $E$ are at most one. Uniqueness of Radon–Nikodym densities in [F6] makes them equal $\lambda$-almost everywhere for each fixed $B$.

Intersect the conull sets from step 3.1 over the countable algebra $\mathcal{C}$. For each $x$ in the intersection, the section $h(x,\cdot )$ is measurable by [F10], so [F11] makes $B\mapsto {\int }_{B}h(x,y)R(x,dy)$ a measure. The kernel section $M(x,\cdot )$ is a probability measure by [F2]. These two measures agree on $\mathcal{C}$ and have total mass one because $F\in \mathcal{C}$ and step 3.1 includes $B=F$. The measure-determining property [F7] makes them equal on every $B\in \mathcal{F}$, with the same exceptional null set for all $B$. Taking $r=h$ proves the assertion.

In the multinomial branch, $C_j={\sum }_{\mathrm{\ell }=1}^N{\mathbf{1}}_{\{J_{\mathrm{\ell }}=j\}}$ and each indicator has conditional expectation $p_j$ by the specified product mass. Linearity of a finite sum gives $\mathbb{E}[C_j\mid a]=N{p}_j$, including the uniform fallback.

In the residual branch, write $q_j=\lfloor N{p}_j\rfloor$ and $r_j=N{p}_j-q_j$. The remainder $R={\sum }_j{r}_j=N-{\sum }_j{q}_j$ is an integer. If $R=0$, every $r_j=0$ and $C_j=q_j=N{p}_j$; if $R>0$, then $C_j=q_j+{\sum }_{\mathrm{\ell }=1}^R{\mathbf{1}}_{\{K_{\mathrm{\ell }}=j\}}$ with each indicator having conditional expectation $r_j\mathrm{/}R$, so $\mathbb{E}[C_j\mid a]=q_j+R(r_j\mathrm{/}R)=N{p}_j$.

Steps 1.1 and 1.2 establish the same conditional expected-count identity for both schemes in the procedure, completing the assertion.

The coordinate projection $x_{0:k}\mapsto x_k$ is measurable by [F3], so $G_k(x_k)$ is measurable by [F4]. Its product with the nonnegative product-measurable $\phi$ is measurable by [F5]; hence $H(x_{0:k-1})=\int G_k(x_k)\phi (x_{0:k})M_k(x_{k-1},d{x}_k)$ is measurable by [F2].

By [F1], the finite path law at time $k$ is obtained by appending $M_k$ to the time-$(k-1)$ path law, and ${\gamma }_{k-1}=V_{k-1}d{P}_{k-1}^M$. Applying [F6] to $H$ and then expanding the ordered kernel integral gives $\int Hd{\gamma }_{k-1}=\int H{V}_{k-1}d{P}_{k-1}^M={\gamma }_k(\phi )$. These are nonnegative integrals, so the identity holds in $[0,\mathrm{\infty }]$ and is the stated recursion.

For each $0\le s\le t$, let $P_s^M$ and $Q_s$ be the target and proposal finite-prefix path laws supplied by [F3]. Initially choose a real-valued Radon–Nikodym representative ${\tilde{\rho }}_0$ from [F4]. Since ${\mu }_0$ is positive, for $N_{-}:=\{{\tilde{\rho }}_0<0\}$ its setwise identity gives $0\le {\mu }_0(N_{-})={\int }_{N_{-}}{\tilde{\rho }}_{0}d{\nu }_0\le 0$. Thus ${\int }_{N_{-}}{\tilde{\rho }}_0^{-}d{\nu }_0=0$, and [F9] implies ${\tilde{\rho }}_0^{-}=0$ ${\nu }_0$-almost everywhere. Replace ${\tilde{\rho }}_0$ by the measurable nonnegative version ${\rho }_0=\max ({\tilde{\rho }}_0,0)$; [F10] gives measurability and the modification preserves the setwise density identity. Hence $P_0^M\ll Q_0$ with base density ${\rho }_0$; after weighting by $G_0$ in [F1], ${\gamma }_0$ has density $W_0=G_0{\rho }_0$.

First record the nonnegative extension used below. If finite measures $P,Q$ satisfy $P(A)={\int }_{A}udQ$ for every measurable $A$, with $u\ge 0$, choose by [F7] simple functions $g_n\uparrow g$ for any nonnegative measurable $g$. The simple-function identity in [F5] gives $\int g_{n}dP=\int g_{n}udQ$ for each $n$; [F8] and monotone convergence of $g_{n}u$ give $\int gdP=\int gudQ$, with infinity allowed. All measures to which this fact is applied below are probabilities, so the finite-level-set condition in [F5] holds.

Fix $1\le k\le t$ and assume $P_{k-1}^M\ll Q_{k-1}$ with density $D_{k-1}(x_{0:k-1})={\rho }_0(x_0){\prod }_{j=1}^{k-1}{r}_j(x_{j-1},x_j)$. For nonnegative product-measurable $h$, define $\psi (x_{0:k-1})=\int h(x_{0:k-1},x_k)M_k(x_{k-1},d{x}_k)$; [F6] makes $\psi$ measurable. For $Q_{k-1}$-almost every history, the kernel Radon–Nikodym identity and step 1.2 give $\psi (x_{0:k-1})=\int h(x_{0:k-1},x_k)r_k(x_{k-1},x_k)R_k(x_{k-1},d{x}_k)$. Step 1.2 also extends the induction density identity from sets to $\psi$, so ordered integration from [F3] gives $\begin{array}{rl}{\displaystyle \int hd{P}_k^M}& {\displaystyle =\int \psi d{P}_{k-1}^M}\\ {\displaystyle }& {\displaystyle =\int D_{k-1}(x_{0:k-1})\psi (x_{0:k-1})Q_{k-1}(d{x}_{0:k-1})}\\ {\displaystyle }& {\displaystyle =\int h(x_{0:k})D_{k-1}(x_{0:k-1})r_k(x_{k-1},x_k)Q_k(d{x}_{0:k})\mathrm{.}}\end{array}$ Thus $P_k^M\ll Q_k$ with density $D_k=D_{k-1}{r}_k$. The kernel identity is needed only on a $Q_{k-1}$-conull set; the same is true for $P_{k-1}^M$ by the induction density, and changes on the exceptional set do not affect these nonnegative integrals.

Induction in step 2.1 gives the proposal density of every finite-horizon state path. The target measure ${\gamma }_s$ is obtained by integrating the nonnegative path potential $V_s={\prod }_{j=0}^s{G}_j$ against $P_s^M$. For each measurable $A$, apply step 1.2 to $g={\mathbf{1}}_A{V}_s$ and the density $D_s={\rho }_0{\prod }_{j=1}^s{r}_j$ to obtain ${\gamma }_s(A)={\int }_A{V}_s{D}_{s}d{Q}_s={\int }_A{W}_{s}d{Q}_s\mathrm{.}$ Dividing by $Z_s$ is valid exactly when $0<Z_s<\mathrm{\infty }$.

First, extend each setwise density identity needed here to nonnegative measurable integrands. If $P(A)={\int }_{A}udQ$ for every measurable $A$, with $u\ge 0$, choose simple $g_n\uparrow g$ by [F15]. The simple-function identity [F14] gives $\int g_{n}dP=\int g_{n}udQ$ for each $n$, and [F16] on both sides yields $\int gdP=\int gudQ$, allowing infinity. This applies to the initial law pair and, for almost every preceding proposal state, to each transition-kernel pair in [F5]. At time zero, the algorithm gives ${\widehat{\gamma }}_0^N(\phi )=N^{-1}{\sum }_{i=1}^N{G}_0(X_0^i){\rho }_0(X_0^i)\phi (X_0^i)$, with zero-weight terms interpreted by the nonnegative-integral convention $0\cdot \mathrm{\infty }=0$. Each $X_0^i$ has law ${\nu }_0$, so [F6], finite-sum linearity [F7], and the just-proved extension give $\mathbb{E}[{\widehat{\gamma }}_0^N(\phi )]=\int G_0{\rho }_0\phi d{\nu }_0={\gamma }_0(\phi )$. The finite target integral makes this random variable integrable.

The finite procedure defines measurable transition kernels. The initial IID draws from ${\nu }_0$ form a finite product kernel by [F21]. At each later step, the previous normalized weights are measurable functions of the previous paths by [F17, F18, F19, F20]. The multinomial and residual laws are probability kernels on the finite ancestor-index set: normalization uses division on the Borel branch where the sum is positive, each floor map is Borel since $\{x:\lfloor x\rfloor =m\}=[m,m+1)$, and the $R=0$ or $R>0$ branches are Borel. Since $R$ takes only the values $0,\dots ,N-1$, each residual output-list probability is a finite sum over possible draw vectors of Borel product masses. For a fixed ancestor vector, the proposal law of the next particles is the finite product of the pullbacks of the $R_k$ kernels along measurable coordinate projections; [F17, F18, F19] make each pullback a probability kernel, and [F21] makes their product a probability kernel. The finite mixture over ancestor vectors is again a probability kernel by finite sums and products [F20]. Thus [F12] applies to the particle-system transitions.

Every time-zero particle path has law $Q_0$. Inductively, suppose each previous particle path avoids every fixed $Q_{k-1}$-null history set almost surely. If $Q_k(A)=0$, the pullback of $R_k$ to the history space is a probability kernel by step 1.2, and [F11] makes $x_{0:k-1}\mapsto R_k(x_{k-1},A_{x_{0:k-1}})$ measurable. The proposal path recursion [F12] and zero-integral criterion [F13] show it vanishes for $Q_{k-1}$-almost every history. The kernel-derivative exceptional parent-state set has a null preimage under the last-coordinate marginal of $Q_{k-1}$, so every stored path avoids that preimage; resampling copies only existing paths, and propagation therefore avoids $A$. Each initial or kernel derivative has reference integral one by [F5]; it is therefore finite almost everywhere for its reference law. The finite-valued potentials and the finite particle count then make all particle weights finite almost surely.

Write ${\mathcal{F}}_{k-1}$ for the previous particle histories and weights. Let $H_k$ be the integral of ${\widehat{\gamma }}_k^N(\phi )$ against the algorithm's conditional finite-step transition law. By [F3], the expected number of children from parent $j$ is $N{\bar{w}}_{k-1}^j$. The algorithm gives ${\widehat{Z}}_k{\bar{w}}_k^i={\widehat{Z}}_{k-1}{a}_k^i\mathrm{/}N$ when the new weight sum is positive; when it is zero, both sides are zero. For each parent outside the exceptional proposal-marginal null set, [F18, F19, F20] make $x_k\mapsto G_k(x_k)\phi (x_{0:k})$ nonnegative measurable; apply the simple-approximation extension in step 1.1 to it. The setwise identity in [F5] then gives $\int G_k\phi r_{k}d{R}_k=\int G_k\phi d{M}_k$, including infinite values. Integrating each child's incremental weight and summing the expected counts therefore gives $H_k={\widehat{\gamma }}_{k-1}^N(g_k)$, where $g_k(x_{0:k-1})=\int G_k(x_k)\phi (x_{0:k})M_k(x_{k-1},d{x}_k)$. If the previous normalizer estimate is zero, both sides vanish by the algorithm's zero-weight convention. Measurability follows from [F11]. This is initially a nonnegative conditional-kernel integral, not yet an $L^1$ conditional expectation.

By [F4], the target integral of $g_k$ is ${\gamma }_{k-1}(g_k)={\gamma }_k(\phi )<\mathrm{\infty }$. The induction hypothesis therefore gives $\mathbb{E}[H_k]={\gamma }_k(\phi )$. For every prior event $B\in {\mathcal{F}}_{k-1}$, the ordered joint law of the prior histories and the algorithm's conditional finite-step transition law [F12], integrated with the nonnegative integrand ${\mathbf{1}}_B{\widehat{\gamma }}_k^N(\phi )$, gives $\mathbb{E}[{\mathbf{1}}_B{\widehat{\gamma }}_k^N(\phi )]=\mathbb{E}[{\mathbf{1}}_B{H}_k]\mathrm{.}$ This identity is valid before any integrability assumption, by nonnegative iterated integration. Taking $B=\mathrm{\Omega }$ shows that ${\widehat{\gamma }}_k^N(\phi )$ has finite expectation ${\gamma }_k(\phi )$, so it is in $L^1$; $H_k$ is also in $L^1$ by its finite expectation. Thus the event identities and measurability make $H_k$ a version of $\mathbb{E}[{\widehat{\gamma }}_k^N(\phi )\mid {\mathcal{F}}_{k-1}]$ by [F2] and [F10]. The tower property [F9] now yields $\mathbb{E}[{\widehat{\gamma }}_k^N(\phi )]={\gamma }_k(\phi )$, completing the induction step.

Apply the established identity at $k=t$ to the constant function $\phi =1$. Then ${\widehat{\gamma }}_t^N(1)={\widehat{Z}}_t{\widehat{\pi }}_t^N(1)={\widehat{Z}}_t$, while ${\gamma }_t(1)=Z_t$, proving the stated normalizer unbiasedness.