Relevance Vector Machines (RVMs) are really interesting models when contrasted with the highly geometrical (and popular) SVMs.
In the light of a question like How does a Support Vector Machine (SVM) work?, and how RVMs are substantially different to SVMs, e.g. What is the difference between Informative (IVM) and Relevance (RVM) vector machines, I think this is a good question to be made.
So, shortly, how do RVMs work?
Answers could tackle the assumptions made (if any), and the general optimization problem.
The main points are:
How does the RVM achieve greater sparsity and automatic parameter selection?
How is it formulated as a Gaussian Process and what makes it different?