Leveraging cluster backbones for improving MAP inference in statistical relational models

Abstract

A wide range of important problems in machine learning, expert system, social network analysis, bioinformatics and information theory can be formulated as a maximum a-posteriori (MAP) inference problem on statistical relational models. While off-the-shelf inference algorithms that are based on local search and message-passing may provide adequate solutions in some situations, they frequently give poor results when faced with models that possess high-density networks. Unfortunately, these situations always occur in models of real-world applications. As such, accurate and scalable maximum a-posteriori (MAP) inference on such models often remains a key challenge. In this paper, we first introduce a novel family of extended factor graphs that are parameterized by a smoothing parameter χ ∈ [0,1]. Applying belief propagation (BP) message-passing to this family formulates a new family of W eighted S urvey P ropagation algorithms (WSP-χ) applicable to relational domains. Unlike off-the-shelf inference algorithms, WSP-χ detects the “backbone” ground atoms in a solution cluster that involve potentially optimal MAP solutions: the cluster backbone atoms are not only portions of the optimal solutions, but they also can be exploited for scaling MAP inference by iteratively fixing them to reduce the complex parts until the network is simplified into one that can be solved accurately using any conventional MAP inference method. We also propose a lazy variant of this WSP-χ family of algorithms. Our experiments on several real-world problems show the efficiency of WSP-χ and its lazy variants over existing prominent MAP inference solvers such as MaxWalkSAT, RockIt, IPP, SP-Y and WCSP.

Appendix A: Derivation of WSP-χ’s update equations

Here we derive the update equations for WSP-χ’s message passing. For simplicity, and without lose of generality, we consider the derivation of WSP-1 — a pure version of WSP-χ on $\hat {\mathcal {G}}$ when setting χ = 1 and γ = 0 in (8).

1.1 A1. Variable-to-factor

Let us start here by computing the update of the component $\mu ^{s}_{X_{j} \rightarrow \hat {f_{i}}}$. This component represents the probability that X_j is constrained by other extended factors to satisfy $\hat {f_{i}}$, and therefore, it is specified by the event that the variable X_j = s_i,j and its mega-node $P_{j} = Z^{j} \cup \{\hat {f_{i}}\}$. If we use $P_{j} = Z^{j} \cup \{\hat {f_{i}}\}$ as a notation representing the following event for a ground atom X_j

$$ \hat{f_{i}} \in P_{j} \text{and} Z^{j} = P_{j} \setminus \{\hat{f_{i}}\} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(j) $$

(18)

Then we can compute $\mu ^{s}_{X_{j} \rightarrow \hat {f_{i}}}$ as follows:

$$ \begin{array}{@{}rcl@{}} \mu^{s}_{X_{j} \rightarrow \hat{f_{i}}} & =& \sum\limits_{Z^{j} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=s_{i,j}, P_{j} = Z^{j} \cup\{\hat{f_{i}}\} \right\} \end{array} $$

(19a)

$$ \begin{array}{@{}rcl@{}} \mu^{s}_{X_{j} \rightarrow \hat{f_{i}}} & =& \sum\limits_{Z^{j} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \prod\limits_{\hat{f_{k}} \in Z^{j}} \eta^{s}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j) \setminus Z^{j}} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \end{array} $$

(19b)

$$ \begin{array}{@{}rcl@{}} & =& \left[ \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \left( \eta^{s}_{\hat{f_{k}} \rightarrow X_{j}} + \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \right) \right] \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \end{array} $$

(19c)

Similarly for $\mu ^{u}_{X_{j} \rightarrow \hat {f_{i}}}$. This component is specified by the event that X_j = u_i,j and its mega-node $P_{j} \subseteq \mathcal {F}^{u}_{\hat {f_{i}}}(j)$. Thus, we have:

$$ \begin{array}{@{}rcl@{}} \mu^{u}_{X_{j} \rightarrow \hat{f_{i}}} & =& \sum\limits_{Z^{j} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \left\{\eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=u_{i,j}, P_{j} = Z^{j} \right\} \end{array} $$

(20a)

$$ \begin{array}{@{}rcl@{}} & =& \sum\limits_{Z^{j} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(j), Z^{j} \neq \emptyset} \prod\limits_{\hat{f_{k}} \in Z^{j}} \eta^{s}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j) \setminus Z^{j}} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \end{array} $$

(20b)

$$ \begin{array}{@{}rcl@{}} && \ - \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \\ & =& \left[ \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \left( \eta^{s}_{\hat{f_{k}} \rightarrow X_{j}} + \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \right) - \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \right] \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \end{array} $$

(20c)

Finally, computing $\mu ^{*}_{X_{j} \rightarrow \hat {f_{i}}}$ is specified by the event that X_j = s_i,j with $P_{j} = \mathcal {F}^{s}_{\hat {f_{i}}}(j)$, and X_j = ∗ with P_j = ∅. Thus we have the following:

$$ \begin{array}{@{}rcl@{}} \mu^{*}_{X_{j} \rightarrow \hat{f_{i}}} & =& \sum\limits_{Z^{j} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \left\{\eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=s_{i,j}, P_{j} = Z^{j} \right\} + \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=*, P_{j} = \emptyset \right\} \end{array} $$

(21a)

$$ \begin{array}{@{}rcl@{}} & =& \sum\limits_{Z^{j} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(j), Z^{j} \neq \emptyset} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{s}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \\ && - \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} + \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \end{array} $$

(21b)

$$ \begin{array}{@{}rcl@{}} & =& \left[ \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j)} \left( \eta^{s}_{\hat{f_{k}} \rightarrow X_{j}} + \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \right) - \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \right] \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{u}_{\hat{f_{k}} \rightarrow X_{j}} \\ && + \prod\limits_{\hat{f_{k}} \in \mathcal{F}^{s}_{\hat{f_{i}}}(j) \cup \mathcal{F}^{u}_{\hat{f_{i}}}(j)} \eta^{*}_{\hat{f_{k}} \rightarrow X_{j}} \end{array} $$

(21c)

1.2 A2. Factor-to-Variables

Let us start here with the component $\eta ^{s}_{\hat {f_{i}} \rightarrow X_{j}}$. This component implies that X_j = s_i,j and $\hat {f_{i}} \in P_{j}$, and that the only possible assignment for the other ground atoms $X_{k} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}$ is u_i,k and their mega-nodes are $P_{k} \subseteq \mathcal {F}^{u}_{\hat {f_{i}}}(k)$. That is, it takes the form:

$$ \eta^{s}_{\hat{f_{i}} \rightarrow X_{j}} = \prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left( \overbrace{\sum\limits_{P_{k} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(k)} \left\{\mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=u_{i,k}, P_{k} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(k)\right\}}^{\text{From Eq.}~(20a) \text{this equals} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}}} \right) \times \overbrace{e^{\hat{w}_{i} \cdot y}}^{\text{a reward term, see (6)}} $$

(22)

Note that since the component $\eta ^{s}_{\hat {f_{i}} \rightarrow X_{j}}$ is constrained to satisfy $\hat {f_{i}}$, we multiply right hand side of (22) by the term $e^{\hat {w}_{i} \cdot y}$ which is the reward term of satisfying $\hat {f_{i}}$. Now, using the definition of $\mu ^{u}_{X_{k} \rightarrow \hat {f_{i}}}$ from (20a) into (22), we obtain the following:

$$ \eta^{s}_{\hat{f_{i}} \rightarrow X_{j}} = \left[\prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} \right] \times e^{\hat{w}_{i} \cdot y} $$

(23)

Now moving to the component $\eta ^{u}_{\hat {f_{i}} \rightarrow X_{j}}$. This component represents the probability that X_j can violate $\hat {f_{i}}$. That is to say, we have X_j = u_i,j and $P_{j} \subseteq \mathcal {F}^{u}_{\hat {f_{i}}}(j)$. This probability implies a combination of three possibilities (having weights labeled as W₁,W₂ and W₃) for the other ground atoms $X_{k} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}$ in a potential complete assignment:

• 1. There is one ground atom in $\mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}$ satisfying $\hat {f_{i}}$, and all the other ground atoms are violating it

  $$ \begin{array}{@{}rcl@{}} \text{W}_{1} & =& \sum\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \overbrace{\sum\limits_{Z^{k} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(k)} \left\{ \mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=s_{i,k}, P_{k} = Z^{k} \cup \{\hat{f_{i}}\} \right\} }^{\text{From Eq.}~(19a) \text{this equals} \mu^{s}_{X_{k} \rightarrow \hat{f_{i}}}} \\ && \times \prod\limits_{X_{i} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{k},X_{j}\}} \overbrace{\sum\limits_{Z^{i} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(i)} \left\{ \mu_{X_{i} \rightarrow \hat{f_{i}}} \bigg| X_{i}=u_{i,i}, P_{i} = Z^{i}\} \right\} }^{\text{From Eq.}~(20a) \text{this equals} \mu^{u}_{X_{i} \rightarrow \hat{f_{i}}}} \end{array} $$

  (24a)
  $$ \begin{array}{@{}rcl@{}} & = &\sum\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{s}_{X_{k} \rightarrow \hat{f_{i}}} \times \prod\limits_{X_{i} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{k},X_{j}\}} \mu^{u}_{X_{i} \rightarrow \hat{f_{i}}} \end{array} $$

  (24b)

• 2. There are two or more ground atoms in $\mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}$ satisfying $\hat {f_{i}}$ or equal joker ∗, and all other ground atoms are violating it

  $$ \begin{array}{@{}rcl@{}} \text{W}_{2} & =& \sum\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left[\sum\limits_{Z^{k} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(k)} \left\{\mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=s_{i,k}, P_{k} = Z^{k} \right\} + \left\{ \mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=*, P_{k} =\emptyset \right\} \right] \\ && \times \prod\limits_{X_{i} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{k},X_{j}\}} \sum\limits_{Z^{i} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(i)} \left\{\mu_{X_{i} \rightarrow \hat{f_{i}}} \bigg| X_{i}=u_{i,i}, P_{i} = Z^{i} \right\} \end{array} $$

  (25a)
  $$ \begin{array}{@{}rcl@{}} & =& \prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left[\mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} + \mu^{*}_{X_{k} \rightarrow \hat{f_{i}}} \right] - \sum\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{*}_{X_{k} \rightarrow \hat{f_{i}}} \times \prod\limits_{X_{i} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{k},X_{j}\}} \mu^{u}_{X_{i} \rightarrow \hat{f_{i}}} \\ && - \prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} \end{array} $$

  (25b)

  Note that the weight assigned to the event that each ground atom is either satisfying or ∗ is ${\prod }_{X_{k} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}} \left [\mu ^{u}_{X_{k} \rightarrow \hat {f_{i}}} + \mu ^{*}_{X_{k} \rightarrow \hat {f_{i}}} \right ]$, and the weight W₂ is given by subtracting from this quantity the weight assigned to the event that there are not at least two joker ground atoms ∗ or satisfying. This event is a combination of two disjoint events that either all other ground atoms in $X_{k} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}$ are violating (which weight ${\prod }_{X_{k} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}} \mu ^{u}_{X_{k} \rightarrow \hat {f_{i}}}$) or that only one ground atom is ∗ or satisfying (with weight ${\sum }_{X_{k} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}} \mu ^{*}_{X_{k} \rightarrow \hat {f_{i}}} \times {\prod }_{X_{i} \in \mathcal {X}_{\hat {f_{i}}} \setminus \{X_{k},X_{j}\}} \mu ^{u}_{X_{i} \rightarrow \hat {f_{i}}}$).

• 3. All other ground atoms in $\mathcal {X}_{\hat {f_{i}}} \setminus \{X_{j}\}$ are violating $\hat {f_{i}}$. So here, there is a penalty term $e^{-\hat {w}_{i} \cdot y}$ of violating $\hat {f_{i}}$ when updating the message:

  $$ \begin{array}{@{}rcl@{}} \text{W}_{3} & = &\left[\prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \overbrace{\sum\limits_{Z^{k} \subseteq \mathcal{F}^{u}_{\hat{f_{i}}}(k)} \left\{\mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=s_{i,k}, P_{k} = Z^{k} \right\}}^{\text{From Eq.}~(6) \text{this equals} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}}} \right] \times \overbrace{e^{-\hat{w}_{i} \cdot y}}^{\text{A penalty term, see (6)}} \end{array} $$

  (26a)
  $$ \begin{array}{@{}rcl@{}} & =& \left[ \prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} \right] \times e^{-\hat{w}_{i} \cdot y} \end{array} $$

  (26b)

Now, bringing together the weight forms of W₁, W₂, and W₃ from (24b), (25b) and (26b) results in:

$$ \begin{array}{@{}rcl@{}} &&\eta^{u}_{\hat{f_{i}} \rightarrow X_{j}} = \left[\prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left( \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} + \mu^{*}_{X_{k} \rightarrow \hat{f_{i}}} \right) + \prod\limits_{X_{i} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j},X_{k}\}} \mu^{u}_{X_{i} \rightarrow \hat{f_{i}}}\right. \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&\left.\sum\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left( \mu^{s}_{X_{k} \rightarrow \hat{f_{i}}} - \mu^{*}_{X_{k} \rightarrow \hat{f_{i}}} \right) \right] - \left[ \overbrace{(1-e^{-\hat{w}_{i} \cdot y})}^{\text{penalty}} \prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} \right] \end{array} $$

(27)

Finally, the component $\eta ^{*}_{\hat {f_{i}} \rightarrow X_{j}}$ represents the probability that X_j can be unconstrained by $\hat {f_{i}}$. This probability is a combination of two possibilities: either X_j is satisfying $\hat {f_{i}}$ and all other ground atoms are unconstrained, or X_j is unconstrained (i.e., X_j = ∗ with P_i = ∅). So we have:

$$ \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} = {\sum}_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left[ \sum\limits_{Z^{k} \subseteq \mathcal{F}^{s}_{\hat{f_{i}}}(k)} \left\{\mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=s_{i,k}, P_{k} = Z^{k} \right\} + \left\{ \mu_{X_{k} \rightarrow \hat{f_{i}}} \bigg| X_{k}=*, P_{k} =\emptyset \right\} \right] x $$

(28)

Note that the first part of (25a) and (25b) is identical to (28). Thus, we substitute the computation of this part from (25a) and (25b) into (28), and we have:

$$ \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} = \left[\prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \left( \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}}+\mu^{*}_{X_{k} \rightarrow \hat{f_{i}}}\right) \right] - \prod\limits_{X_{k} \in \mathcal{X}_{\hat{f_{i}}} \setminus \{X_{j}\}} \mu^{u}_{X_{k} \rightarrow \hat{f_{i}}} $$

(29)

1.3 A3. Estimating the Marginals

Now let us explain the derivation of ground atoms’ marginals over max-cores in $\hat {\mathcal {G}}$. Computing the unnormalized positive marginal of a ground atom X_j requires multiplying the satisfying income messages from the ground clauses in which X_j appears positively by the violating income messages from the ground clauses in which X_j appears negatively:

$$ \begin{array}{@{}rcl@{}} \tilde{\theta}^{+}_{j} & =& \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{s}(j)} {\sum}_{\mathcal{F}(i)} \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=s_{i,j}, P_{j} = \mathcal{F}^{s}(j) \right\} \times \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{u}(j)} {\sum}_{\mathcal{F}(i)} \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=u_{i,j}, P_{j} = \mathcal{F}^{u}(j) \right\} \end{array} $$

(30a)

$$ \begin{array}{@{}rcl@{}} & =& {\prod}_{\hat{f_{i}} \in \mathcal{F}^{+}(j)} \sum\limits_{\mathcal{F}(i)} \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=+, P_{j} = \mathcal{F}^{+}(j) \right\} \times \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{-}(j)} \sum\limits_{\mathcal{F}(i)} \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=-, P_{j} = \mathcal{F}^{-}(j) \right\} \end{array} $$

(30b)

$$ \begin{array}{@{}rcl@{}} & =& \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{-}(j)} \eta^{u}_{\hat{f_{i}} \rightarrow X_{j}} \times \left[ \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{+}(j)} \left( \eta^{s}_{\hat{f_{i}} \rightarrow X_{j}} + \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} \right) - \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{+}(j)} \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} \right] \end{array} $$

(30c)

Similarly, we can obtain the unnormalized negative marginal by multiplying the satisfying income messages from the factors in which X_i appears negatively by the violating income messages from the factors in which X_i appears positively:

$$ \tilde{\theta}^{-}_{j} = \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{+}(j)} \eta^{u}_{\hat{f_{i}} \rightarrow X_{j}} \times \left[ \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{-}(j)} \left( \eta^{s}_{\hat{f_{i}} \rightarrow X_{j}} + \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} \right) - \prod\limits_{\hat{f_{i}} \in \mathcal{F}^{-}(j)} \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} \right] $$

(31)

Finally, we can estimate the unnormalized joker marginal by multiplying all the unconstrained incoming messages from all factors in which X_j appears:

$$ \begin{array}{@{}rcl@{}} \tilde{\theta}^{*}_{j} & =& \prod\limits_{\hat{f_{i}} \in \mathcal{F}(j)} \left\{ \eta_{\hat{f_{i}} \rightarrow X_{j}} \bigg| X_{j}=*, P_{j} = \emptyset \right\} \\ & =& \prod\limits_{\hat{f_{i}} \in \mathcal{F}(j)} \eta^{*}_{\hat{f_{i}} \rightarrow X_{j}} \end{array} $$

(32a)

Now by normalizing the quantities in (30c), (31) and (32a), we obtain the marginal of X_j as follows:

$$ \begin{array}{@{}rcl@{}} \theta^{+}_{j} & =& \mathcal{Z}_{j}^{-1} \tilde{\theta}^{+}_{j} \end{array} $$

(33a)

$$ \begin{array}{@{}rcl@{}} \theta^{-}_{j} & =& \mathcal{Z}_{j}^{-1} \tilde{\theta}^{-}_{j} \end{array} $$

(33b)

$$ \begin{array}{@{}rcl@{}} \theta^{*}_{j} & =& \mathcal{Z}_{j}^{-1} \tilde{\theta}^{*}_{j} \end{array} $$

(33c)

and

$$ \mathcal{Z}_{j} = \tilde{\theta}^{+}_{j} + \tilde{\theta}^{-}_{j} + \tilde{\theta}^{*}_{j} $$

(34)

where $\mathcal {Z}_{i}$ is the normalizing constant, given the evidence E.