Organizations: School of Computing & Information Sciences, Florida International University, Miami, USA · Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, USA
A neuro-symbolic system should combine only relevant answers from expert agents or other knowledge sources. Using only relevant experts for a given input query becomes ever more important for accuracy and scalability as the knowledge network grows and connects ever more knowledge sources. We extend the standard definition of relevance from the legal theory of evidence: An answer is relevant if and only if it is both probative and material for the matter at hand given the knowledge available. We apply this relevancy scheme to mixing the feedback fuzzy cognitive maps (FCMs) of real or artificial experts. The expert FCM directed causal graphs mix or combine into a common knowledge network. We compare the equilibrium limit-cycle predictions or answers of the experts to the answers of the mixture FCM and then estimate the probative status of answers by their difference or variation. A new Bayes-like theorem de-mixes the knowledge mixture into corresponding posterior-like FCMs that also allow relevancy-based comparisons. The result is a final set of relevancy mixing weights that produces a relevancy-mixed FCM knowledge network for the given query. So different queries produce different relevancy-mixed FCMs. We demonstrate this relevancy approach using generated synthetic FCMs from an actual 13-node expert FCM that models the causes and feedbacks of ischemic-stroke risk.
Figures & tables
Fig. 1 : Fuzzy Cognitive Map (FCM) mixture of relevant experts. The expert likelihood FCMs Fk on the left mix through convex combination with prior mixing weights wk to give the likelihood FCM mixture F . The likelihood FCM mixture gives Bayesian posterior-like FCMs Pk that produce limit cycle equilibria Lk . These posterior FCMs also mix with the same prior mixing weights wk to give the posterior FCM mixture P with its limit cycle L . The convex relevancy weights rk(x) for query x measure the relevance of the likelihood FCMs by comparing the posterior FCMs’ limit cycles Lk to the limit cycle L of the posterior FCM mixture through the norm of their difference. The likelihood FCMs mix with the final relevance weights rk(x) to produce the final relevant FCM given input x .
Fig. 2 : Ischemic stroke FCM [ 5 ] : The 13-variable FCM represents the causal relationships and weights among the factors that affect the risk of ischemic stroke. (a) The FCM connects 13 demographic, clinical, and lifestyle causal variables using signed weighted edges. Green edges denote positive causal influence and red edges denote negative influence on stroke risk C13 . (b) Causal edge matrix E : directed partial causal edge eij denotes the degree that Ci affects Cj . (c) Plot of FCM equilibrium from an initial state. The trajectory settles into a limit cycle where blood pressure C2 oscillates with diabetes C5 .
Fig. 3 : Synthetic likelihood FCMs from the perturbation of the ischemic stroke FCM [ 5 ] : Edge matrices corresponding to the 7 likelihood FCMs. Brighter color corresponds to higher edge weight. White corresponds to zero edge weight.
Fig. 4 : Nontrivial attractors for the synthetic likelihood FCMs from the perturbation of the ischemic stroke FCM [ 5 ] : (a) F1 has two fixed points. (b) F2 has one limit cycle with length 5 and two fixed points. (c) F3 has one fixed point, one limit cycle with length 4, and one limit cycle with length 2. (d) F4 has just a fixed point. (e) F5 has a limit cycle with length 3. (f) F6 has a limit cycle with length 2 and a fixed point. (g) F7 has a limit cycle with length 2 and one fixed point. The nodes are along the y -axis and the time steps are along the x -axis. Yellow corresponds to active nodes and black corresponds to inactive nodes.
Fig. 5 : Mixing the likelihood FCMs with the set of non-uniform weights ( w1=0.5,w2=0.15,w3=0.1,w4=0.1,w5=0.05,w6=0.05,w7=0.05 ): (a) The edge matrix for the non-uniformly mixed likelihood FCM. (b) One of the nontrivial attractors in which blood pressure, diabetes, smoking, and BMI remain active.
Posterior Pj
rj(x(1))
rj(x(2))
rj(x(3))
rj(x(4))
P1
0.083
0.127
0.097
0.083
P2
0.132
0.142
0.149
0.156
P3
0.177
0.156
0.149
0.177
P4
0.144
0.142
0.149
0.118
P5
0.177
0.180
0.159
0.177
P6
0.132
0.127
0.138
0.156
TABLE I : Relevancy mixing weights for the posterior FCMs with the non-uniform weights ( w1=0.5,w2=0.15,w3=0.1,w4=0.1,w5=0.05,w6=0.05,w7=0.05 ). We picked four initial states x(i) ’s from the four largest basins.
Fig. 6 : Mixing the likelihood FCMs with uniform weights ( w1=w2=w3=w4=w5=w6=w7=71 ): (a) The mixture Favg averages the 7 likelihood FCM weight matrices and produces a matrix whose entries blend the causal influences of the individual maps. (b) These are all the nontrivial equilibria for all the basins: two fixed points and two limit cycles.
Fig. 7 : Relevance FCM mixture from the weighted likelihood mixture with the uniform weights ( w1=w2=...=w7=71 ): We used initial state x′=[1,0,1,0,1,0,0,0,0,1,1,0,0] . The edge matrix of the relevant FCM mixture FR is on the left. The corresponding equilibrium is in the middle. The plot on the right shows the corresponding equilibrium from the likelihood FCM mixture Favg for the same input.
Fig. 8 : Relevance FCM mixture from the weighted likelihood mixture with the non-uniform weights ( w1=0.5,w2=0.15,w3=0.1,w4=0.1,w5=0.05,w6=0.05,w7=0.05 ): We tested the initial state x′′=[1,0,0,0,0,0,0,0,0,0,0,0,0] . The edge matrix of the relevant FCM mixture FR is on the left. The corresponding equilibrium is in the middle. The plot on the right shows the corresponding equilibrium from the likelihood FCM mixture Favg for the same initial state.
Posterior Pj
rj(x(1))
rj(x(2))
rj(x(3))
rj(x(4))
P1
0.133
0.175
0.132
0.101
P2
0.159
0.165
0.155
0.159
P3
0.084
0.130
0.103
0.084
P4
0.133
0.130
0.103
0.133
P5
0.168
0.154
0.161
0.139
P6
0.178
0.130
0.178
0.197
TABLE II : Relevancy mixing weights for the posterior FCMs with the uniform weights ( w1=w2=w3=w4=w5=w6=w7=71 ). We picked four initial states x(i) ’s from the four largest basins.