In the previous chapter, Kanamori and Shimodaira provided generalization error estimators which can be used for model selection and active learning. The accuracy of these estimators is theoretically guaranteed in terms of the expectation over realizations of training input-output samples. In practice, we are only given a single realization of training samples. Therefore, ideally, we want to hav...

We define the min-min expectation selection problem (resp. max-min expectation selection problem) to be that of selecting k out of n given discrete probability distributions, to minimize (resp. maximize) the expectation of the minimum value resulting when independent random variables are drawn from the selected distributions. We assume each distribution has finitely many atoms. Let d be the num...

We present static analyses for probabilistic loops using expectation invariants. Probabilistic loops are imperative while-loops augmented with calls to random value generators. Whereas, traditional program analysis uses Floyd-Hoare style invariants to over-approximate the set of reachable states, our approach synthesizes invariant inequalities involving the expected values of program expression...

We will develop some inequalities which let us bound the probability of a random variable taking a value very far from its expectation. This is the most basic inequality we will use. This is useful if the only thing we know about a random variable is its expectation. It will also be useful to derive other inequalities later.

The various proposals for FQHE quasi-electron trial wave functions are reconsidered. In a short-range model for the electronic interaction, the energy expectation values of four different trial wave functions are calculated at filling factor 1 3 for up to ten electrons in the disk geometry. Jain's trial wave function displays the lowest energy expectation value.