Joint Lattice of Reconstructability Analysis and Bayesian Network General Graphs

Abstract

This paper integrates the structures considered in Reconstructability Analysis (RA) and those considered in Bayesian Networks (BN) into a joint lattice of probabilistic graphical models. This integration and associated lattice visualizations are done in this paper for four variables, but the approach can easily be expanded to more variables. The work builds on the RA work of Klir [22], Krippendorff [26], and Zwick [44], and the BN work of Pearl [33,34,35,36], Verma and Pearl [39], Heckerman et al. [20], Chickering [9], Andersson et al. [1], and others. The RA four variable lattice and the BN four variable lattice partially overlap: there are ten unique RA general graphs, ten unique BN general graphs, and ten general graphs common to both RA and BN. For example, the specific graph having probability distribution p(A)p(C)p(B | AC) is unique to BN, the RA specific graph AB:AC:BC, which contains a loop, is unique to RA, and the specific graph ACD:BCD with probability distribution p(A | CD)p(B | CD)p(D | C)p(C) is common to both RA and BN. The joint RA-BN lattice of general graphs presented in this paper expands the set of general graphs with unique independence structures beyond what was previously available by either RA alone or BN alone, thus allowing for representations of complex systems which are (i) more accurate relative to data and/or (ii) simpler and thus more comprehensible and more generalizable than would be possible by modeling only with RA or only with BN.