Recall that in Theorem 2.1, we analyzed empirical risk minimization with a finite hypothesis class F , i.e., |F| < +∞. Here, we will prove results for possibly infinite hypothesis classes. Although the PAC-Bayes framework is far more general, we will concentrate of the prediction problem as before, i.e., (∀f ∈ F) f : X → Y. Also, note that Theorem 2.1 could have been stated in a more general fa...

Recall that in Theorem 2.1, we analyzed empirical risk minimization with a finite hypothesis class F , i.e., |F| < +∞. Here, we will prove results for possibly infinite hypothesis classes. Although the PAC-Bayes framework is far more general, we will concentrate of the prediction problem as before, i.e., (∀f ∈ F) f : X → Y. Also, note that Theorem 2.1 could have been stated in a more general fa...

We show that the uniform estimate in the Calabi-Yau theorem easily follows from the local stability of the complex Monge-Ampère equation.

We establish a version of Bochner Theorem due to S. Boylan for Banach spaces with a basis.

The Geometric Theory of Ignorance produces posterior distributions from priors and likelihoods without invoking Bayes Theorem at all. The standard bayesian posteriors minimize the ignorance action with parameters δ = ν = 0 when the truth t is replaced by the empirical distribution of n independent observations. Different values for the parameters produce new ways for processing the data obtaini...