A detailed algorithmic explanation is required for how a network of chemical reactions can generate the sophisticated behavior displayed by living cells. Though several previous works have shown that reaction networks are computationally universal and can in principle implement any algorithm, there is scope for constructions that map well onto biological reality, make efficient use of the computational potential of the native dynamics of reaction networks, and make contact with statistical mechanics. We describe a new reaction network scheme for solving a large class of statistical problems. Specifically we show how reaction networks can implement information projection, and consequently a generalized Expectation-Maximization algorithm, to solve maximum likelihood estimation problems in partially-observed exponential families on categorical data. Our scheme can be thought of as an algorithmic interpretation of E. T. Jaynes’s vision of statistical mechanics as statistical inference.
The belief propagation scheme of Napp and Adams @cite_15 shows how reaction networks can implement sum-product algorithms from probabilistic graphical models to compute marginals of joint distributions. There is some formal similarity between the reaction networks of Napp and Adams, and our EM reaction network. In particular, their scheme also has two sets of species, sum'' and product'' species, and two subnetworks. In each subnetwork, one set of species changes in number, and the other set appears catalytically. We speculate that this may be because message passing algorithms are themselves a special case of the EM algorithm @cite_22 @cite_38 , in which case possibly the Napp-Adams scheme may be related in some as-yet-undiscovered way to our EM scheme.
@cite_38 @cite_15 @cite_22
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One of the central themes in Sum-Product networks (SPNs) is the interpretation of sum nodes as marginalized latent variables (LVs). This interpretation yields an increased syntactic or semantic structure, allows the application of the EM algorithm and to efficiently perform MPE inference. In literature, the LV interpretation was justified by explicitly introducing the indicator variables corresponding to the LVs’ states. However, as pointed out in this paper, this approach is in conflict with the completeness condition in SPNs and does not fully specify the probabilistic model. We propose a remedy for this problem by modifying the original approach for introducing the LVs, which we call SPN augmentation. We discuss conditional independencies in augmented SPNs, formally establish the probabilistic interpretation of the sum-weights and give an interpretation of augmented SPNs as Bayesian networks. Based on these results, we find a sound derivation of the EM algorithm for SPNs. Furthermore, the Viterbi-style algorithm for MPE proposed in literature was never proven to be correct. We show that this is indeed a correct algorithm, when applied to selective SPNs, and in particular when applied to augmented SPNs. Our theoretical results are confirmed in experiments on synthetic data and 103 real-world datasets. In this paper, we study the problem of reaching a consensus among all the agents in the networked control systems (NCS) in the presence of misbehaving agents. A reputation-based resilient distributed control algorithm is first proposed for the leader-follower consensus network. The proposed algorithm embeds a resilience mechanism that includes four phases (detection, mitigation, identification, and update), into the control process in a distributed manner. At each phase, every agent only uses local and one-hop neighbors' information to identify and isolate the misbehaving agents, and even compensate their effect on the system. We then extend the proposed algorithm to the leaderless consensus network by introducing and adding two recovery schemes (rollback and excitation recovery) into the current framework to guarantee the accurate convergence of the well-behaving agents in NCS. The effectiveness of the proposed method is demonstrated through case studies in multirobot formation control and wireless sensor networks. Recent work on molecular programming has explored new possibilities for computational abstractions with biomolecules, including logic gates, neural networks, and linear systems. In the future such abstractions might enable nanoscale devices that can sense and control the world at a molecular scale. Just as in macroscale robotics, it is critical that such devices can learn about their environment and reason under uncertainty. At this small scale, systems are typically modeled as chemical reaction networks. In this work, we develop a procedure that can take arbitrary probabilistic graphical models, represented as factor graphs over discrete random variables, and compile them into chemical reaction networks that implement inference. In particular, we show that marginalization based on sum-product message passing can be implemented in terms of reactions between chemical species whose concentrations represent probabilities. We show algebraically that the steady state concentration of these species correspond to the marginal distributions of the random variables in the graph and validate the results in simulations. As with standard sum-product inference, this procedure yields exact results for tree-structured graphs, and approximate solutions for loopy graphs. Motivated by recent progress in using restricted Boltzmann machines as preprocess-ing algorithms for deep neural network, we revisit the mean-field equations (belief-propagation and TAP equations) in the best understood such machine, namely the Hopfield model of neural networks, and we explicit how they can be used as iterative message-passing algorithms, providing a fast method to compute the local polariza-tions of neurons. In the "retrieval phase" where neurons polarize in the direction of one memorized pattern, we point out a major difference between the belief propagation and TAP equations : the set of belief propagation equations depends on the pattern which is retrieved, while one can use a unique set of TAP equations. This makes the latter method much better suited for applications in the learning process of restricted Boltzmann machines. In the case where the patterns memorized in the Hopfield model are not independent, but are correlated through a combinatorial structure, we show that the TAP equations have to be modified. This modification can be seen either as an alteration of the reaction term in TAP equations, or, more interestingly, as the consequence of message passing on a graphical model with several hidden layers, where the number of hidden layers depends on the depth of the correlations in the memorized patterns. This layered structure is actually necessary when one deals with more general restricted Boltzmann machines.