On a Class of Tensor Markov Fields.

Here, we introduce a class of Tensor Markov Fields intended as probabilistic graphical models from random variables spanned over multiplexed contexts. These fields are an extension of Markov Random Fields for tensor-valued random variables. By extending the results of Dobruschin, Hammersley and Clifford to such tensor valued fields, we proved that tensor Markov fields are indeed Gibbs fields, whenever strictly positive probability measures are considered. Hence, there is a direct relationship with many results from theoretical statistical mechanics. We showed how this class of Markov fields it can be built based on a statistical dependency structures inferred on information theoretical grounds over empirical data. Thus, aside from purely theoretical interest, the Tensor Markov Fields described here may be useful for mathematical modeling and data analysis due to their intrinsic simplicity and generality.

Network analysis of frailty and aging: Empirical data from the Mexican Health and Aging Study.

BACKGROUND: Frailty remains a challenge in the aging research area with a number of gaps in knowledge still to be filled. Frailty seems to behave as a network, and in silico evidence is available on this matter. Having in vivo evidence that frailty behaves as a complex network was the main purpose of our study. METHODS: Data from the Mexican Health and Aging Study (main data 2012, mortality 2015) was used. Frailty was operationalized with a 35-deficit frailty index (FI). Analyzed nodes were the deficits plus death. The edges, linking those nodes were obtained through structural learning, and an undirected graph associated with a discrete probabilistic graphical model (Markov network) was derived. Two algorithms, hill-climbing (hc) and Peter and Clark (PC), were used to derive the graph structure. Analyses were performed for the whole population and tertiles of the total FI score. RESULTS: From the total sample of 10,983 adults aged 50 or older, 43.8% were women, and the mean age was 64.6 years (SD = 9.3). The number of connections increased according to the tertile level of the FI score. As the FI score raised, groups of interconnected deficits increased and how the nodes are connected changed. CONCLUSIONS: Frailty phenomenon can be modeled using a Bayesian network. Using the full sample, the most central nodes were self-report of health (most connected node) and difficulty walking a block, and all deficits related to mobility were very interconnected. When frailty levels are considered, the most connected nodes differ, but are related with vitality, mainly at lower frailty levels. We derived that not all deficits are equally related since clusters of very related deficits and non-connected deficits were obtained, which might be considered in the construction of the FI score. Further research should aim to identify the nature of all observed interactions, which might allow the development of specific interventions to mitigate the consequences of frailty in older adults.