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Expectation Particle Belief Propagation (2015)
Thibaut Lienart, Yee Whye Teh, Arnaud Doucet

The paper proposes an update scheme for the proposal distribution in the message computation of particle belief propagation. The proposal distribution is in the exponential family and is iteratively updated in an expectation propgation (EP) framework. The computation at each iteration is quadratic in the number of particles.

Particle Belief Propagation

Let $\psi_{u}$ and $\psi_{uv}$ be node and edge potentials, respectively. Let $\Gamma_{u}$ be the set of neighbouring variables connected to node $u$. The loopy belief propagation (LBP) message updates are written as

\[\begin{aligned} m_{uv}^{t}(x_{v}) & =\int\psi_{uv}(x_{u},x_{v})\psi_{u}(x_{u})\prod_{w\in\Gamma_{u}\backslash v}m_{wu}^{t-1}(x_{u})\,\mathrm{d}x_{u},\\ B_{u}^{t}(x_{u}) & =\psi_{u}(x_{u})\prod_{w\in\Gamma_{u}}m_{wu}^{t}(x_{u}),\end{aligned}\]

where the superscript $\cdot^{t}$ denotes iteration number, $B_{u}$ denotes the belief at node $u$, and $m_{uv}$ is the message from node $u$ to node $v$. Particle belief propgation (PBP) uses importance sampling to compute the messages $m_{uv}$. Given a proposal distribution $q_{u}$ on node $u$, and a set of $N$ particles ${x_{u}^{(i)}}{i=1}^{N}\sim q{u}(x_{u})$, PBP messages $\hat{m}_{uv}^{\mathrm{PBP}}$ are written as:

\[\begin{aligned} & \int\psi_{uv}(x_{u},x_{v})\psi_{u}(x_{u})\prod_{w\in\Gamma_{u}\backslash v}m_{wu}^{t-1}(x_{u})\,\mathrm{d}x_{u}\\ & =\int\psi_{uv}(x_{u},x_{v})\psi_{u}(x_{u})\prod_{w\in\Gamma_{u}\backslash v}m_{wu}^{t-1}(x_{u})\frac{q_{u}(x_{u})}{q_{u}(x_{u})}\,\mathrm{d}x_{u}\\ & =\frac{1}{N}\sum_{i=1}^{N}\psi_{uv}(x_{u}^{(i)},x_{v})\frac{\psi_{u}(x_{u}^{(i)})}{q_{u}(x_{u}^{(i)})}\prod_{w\in\Gamma_{u}\backslash v}m_{wu}^{t-1}(x_{u}^{(i)})\\ & :=\sum_{i=1}^{N}\psi_{uv}(x_{u}^{(i)},x_{v})w_{uv}^{(i)}=:\hat{m}_{uv}^{\mathrm{PBP}}(x_{v}),\end{aligned}\]

where the importance weight $w_{uv}^{(i)}:=\frac{1}{N}\frac{\psi_{u}(x_{u}^{(i)})}{q_{u}(x_{u}^{(i)})}\prod_{w\in\Gamma_{u}\backslash v}m_{wu}^{t-1}(x_{u}^{(i)})$. The choice of $q_{u}$ determines the approximation quality.

Expectation Particle Belief Propagation

Let $\hat{m}_{uv}(x_{v})$ be the particle approximation of the exact message $m_{uv}$. The particle approximation to the belief at node $v$ is

\[\begin{aligned} \hat{B}_{u}(x_{u}) & \approx\frac{1}{N}\sum_{i=1}^{N}\frac{\psi_{u}(x_{u}^{(i)})\prod_{w\in\Gamma_{u}}\hat{m}_{wu}(x_{u}^{(i)})}{q_{u}(x_{u}^{(i)})}\delta_{x_{u}^{(i)}}(x_{u}),\end{aligned}\]

where $\delta_{x}$ is a delta measure at $x$. It can be seen that the best proposal $q_{u}$ is the belief itself, which is unknown. An idea to approximately achieve that is to use a tractable exponential family distribution for $q_{u}$ instead: \(q_{u}(x_{u})\propto\eta_{u}(x_{u})\prod_{w\in\Gamma_{U}}\eta_{wu}(x_{u}),\) where $\eta_{u}$ and $\eta_{wu}$ are exponential family approximations of $\psi_{u}$ and $\hat{m}_{wu}$ respectively. The exponential family distribution for $q_{u}$ is computed with EP. Specifically, each $\eta_{wu}$ is iteratively updated with

\[\eta_{wu}=\arg \min_{\eta\in\mathrm{ExpFam}}\mathrm{KL}\left[\hat{m}_{wu}(x_{u})q_{u}^{\backslash w}(x_{u})\,\big\|\,\eta(x_{u})q_{u}^{\backslash w}(x_{u})\right],\]

where the cavity distribution \(q_{u}^{\backslash w}\propto q_{u}/\eta_{wu}\). It is known that the solution $\eta_{wu}$ is the one such that the moments (expected sufficient statistics) of the tilted distribution \(\hat{m}_{wu}q_{u}^{\backslash w}\) match that of $\eta q_{u}^{\backslash w}$. The computation of moments can be performed crudely as $q_{u}$ is only used as a proposal distribution.

It is unclear if this scheme will work in the case that $q_{u}$ is high dimensional.