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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

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The spectre of climate change that is unfolding now is undeniably a cumulative impact of anthropogenic interference in the climate system over the last two centuries. The science is clear and the policy community is being increasingly convinced and galvanised into action to address this emergent challenge in light of the associated economic and human dimensions. The impacts of climate change ranging from sea level rise, melting ice caps and glaciers, severe weather events, drought, flooding, warming, subtle changes in ecosystems – will impinge on every aspect of society and economic life. The costs of inaction will more than outweigh the costs of action. There is only a narrow window of opportunity to redress the situation. The Intergovernmental Panel on Climate Change (IPCC) in its Fourth Assessment Report has