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We consider estimation problems, in which the estimand, X, and observation, Y , take values in measurable spaces. Regular conditional versions of the forward and inverse Bayes formula are shown to have dual variational characterisations involving the minimisation of an apparent information, and the maximisation of a compatible information. These both have(More)
We investigate the information theoretic properties of Kalman-Bucy filters in continuous time, developing notions of information supply, storage and dissipation. By introducing a concept of energy, we develop a physical analogy in which the unobservable signal describes a statistical mechanical system interacting with a heat bath. The abstract 'universe'(More)
This paper is the first in a two-part study of a variational Bayesian method and its application to a problem of reliable communication. The variational method expresses a Bayesian posterior distribution as the unique minimizer of a quantity dubbed apparent information. This has the same nature as the free energy of statistical mechanics. The minimum(More)