Belief Propagation Algorithms: From Matching Problems to Network Discovery in Cancer Genomics – We review belief propagation algorithms inspired by the study of phase transitions in combinatorial optimization problems. In particular, we present rigorous results on convergence of such algorithms for matching and associated bargaining problems on networks. We also present a belief propagation algorithm for the prize- collecting Steiner tree problem, for which rigorous convergence results are not yet known. Finally, we show how this algorithm can be used to discover pathways in cancer genomics, and to suggest possible drug targets for cancer therapy. These methods give us the ability to share information across multiple patients to help reconstruct highly patient-specific networks.
Message Passing Inference with Chemical Reaction 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.
Information-theoretic lower bounds for distributed statistical estimation with communication constraints – We establish minimax risk lower bounds for distributed statistical estimation given a budget B of the total number of bits that may be communicated. Such lower bounds in turn reveal the minimum amount of communication required by any procedure to achieve the classical optimal rate for statistical estimation. We study two classes of protocols in which machines send messages either independently or interactively. The lower bounds are established for a variety of problems, from estimating the mean of a population to estimating parameters in linear regression or binary classification.