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These proceedings aim to collect the ideas presented, discussed, and disputed at the 40th Workshop on Bayesian Inference and Maximum Entropy, MaxEnt 2021. Skilling and Knuth seek to rebuild the foundations of quantum mechanics from probability theory, and Caticha competes in that endeavour with a very different entropy-based approach. Costa connects entropy with general relativity, Pessoa reports new insights on ecology and Yousefi derives classical density functional theory, both through the maximum entropy principle. Von Toussaint, Preuss, Albert, Rath, Ranftl and Kvas report the latest developments in regression and surrogate-based inference with applications to optimization and inverse problems in plasma physics, biomechanics and geodesy. Van Soom presents new priors for phonetics, Stern et al. propose a new haphazard sampling method, and Kelter uncovers two measure theoretic iss phonetics ues with hypothesis testing.

Maximum entropy method

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This book describes a useful tool for solving linear inverse problems subject to convex constraints. The method of maximum entropy in the mean automatically takes care of the constraints. It consists of a technique for transforming a large dimensional inverse problem into a small dimensional non-linear variational problem. A variety of mathematical aspects of the maximum entropy method are explored as well.

Inverse problems (Differential equations) --- Maximum entropy method.

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Bayesian statistical decision theory --- Maximum entropy methods --- Congresses.

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Bayesian statistical decision theory --- Maximum entropy method --- Congresses --- Industrial applications

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This book is a collection of introductory, interdisciplinary articles and lectures covering the fundamentals of the maximum entropy approach, a powerful new technique that provides a much needed extension of the established principles of rational inference in the sciences. Maximum entropy allows the interpretation of incomplete and "noisy" data, providing a description of the underlying physical systems. It has found application in both practical and theoretical studies ranging from image enhancement to nuclear physics, and from statistical mechanics to economics. The work explores these applications with specific problems of data analysis taken from the physical sciences. It will interest all physical scientists who deal with data and its interpretation, including statisticians and statistical physicists.

Statistical physics --- Mathematical physics --- Maximum entropy method --- Bayesian statistical decision theory --- Bayesian statistical decision theory. --- Maximum entropy method.