Fig. 3. - Factor graph representation of the joint posterior PDF (19). Short notations are used. In particular, the time index n and the functional dependencies of the factors are neglected: x=x_{n}, M^{j}= M_{n}^{(j)}, y^{j}_{s}=y_{s,n}^{ (j)}, y^{j}_{m}=y_{m,n}^{(j)}, q^{j}_{0}=q_{P}(x _{n},a^{(j)}_{00,n};z^{(j)}_{n}), q^{j}_{s}=q_{S}(y_{s,n}^{(j)},a^{(j)}_{ss,n},x_{n}; z^{(j)}_{n}), q^{j}_{ss'}=q_{D}( y_{s,n}^{(j)},y_{s', n}^{(j)},a^{(j)}_{ss',n},x_{n};z ^{(j)}_{n}){ q_{D}(y_ {s',n}^{(j)},y_{s,n}^{(j)},a^{(j) }_{s's,n},x_{n};z^{(j)}_{n})}, q^{j}_{m}=q_{S}(y^{(j)}_{m,n},a^{(j)}_{m,n},x_{n};z^{(j) }_{m,n}), f= f(x_{n}|x_{n-1}), f^{1}_{s}= f(y_{s,n}|y_{s, n-1}), f^{j}_{s}= f^{(j)}(y^{(j)}_{s,n}|y^{(j-1)}_{s,n}), alpha_{n}=alpha(x_{n}), alpha_{s}^{j}=alpha_{s}(y^{(j)}_{s,n}), beta_{ss'}^{j}=beta_{ss'}(a_{ss',n }^{(j)}){ beta_{s's}(a_{s's,n}^{(j)} )}, gamma_{ss'}^{j}=gamma^{(j)}_{ss'}(x_{n}){ gamma^{(j)}_{s's}(x_{n})}, xi_{m}^{j}=xi(a^{(j)}_{m,n}), eta_{ss'}^{j}=eta(a_{ss',n}^{(j)}){ eta(a_{s's,n}^{(j)})}, nu_{ss'}^{j}=nu_{m->ss'}^{(p)}(a_{ss' ,n}^{(j)}){ nu_{m->s's}^{(p)}(a_{s' s,n}^{(j)})}, zeta_{m}^{j}=zeta(a_{m,n}^{(j)}), zeta_{m}^{j}=zeta_{ss'->m}^{(p)}(a_{m,n}^ {(j)}){ zeta_{s's->m}^{(p)}(a_{m,n} ^{(j)})}, tilde{rho}_{ss'}^{j}=rho_{ss'}(y^{(j)}_{s}){ rho_{s's}(y^{(j)}_{s})}, and phi_{m}^{j}=phi(y_{s}^{(j)}). For the numbers of MVAs, the short notations reads S^{1}= S_{n-1}, S^{j}= S_{n}^{(j)}, and (cdot)_{S^{jj}}=(cdot)_{S_{n}^{(j)}S_{n}^{(j)}}. The dashed lines with arrows indicate messages representing the agent and PMVAs beliefs of time n-1, n+1 or of anchors j-1, j+1. These messages are either only sent to the next time step (e.g., from n-1 to n) or only to the next anchor (e.g., from j-1 to j).
