The paper examines relationships between the conditional Shannon entropy and the expectation of `α-norm for joint probability distributions. More precisely, we investigate the tight bounds of the expectation of `α-norm with a fixed conditional Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the conditional Shannon entropy and several informati...

Conditional expectation is one of the most useful tools of probability. The Radon-Nikodym theorem enables us to construct conditional expectations. Definition 1. A measure µ on (Ω, F) is σ-finite iff there is A n ∈ F satisfying A n ↑ Ω where µ(A n) < ∞ for all n. Definition 2. Let µ, ν be measures on (Ω, F). Then ν is absolutely continuous with respect to µ (ν << µ) iff for all A ∈ F, µ(A) = 0 ...

The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on ...

Suppose we want to derandomize A—that is, give a deterministic variant of A which succeeds with probabilty 1 on every input. Sometimes we can do this using the method of conditional expectation. We can think of A as a binary tree which, given x, branches on the sampled value of each random bit Ri in turn. Paths in this tree correspond to different possible random strings R1, . . . , Rm that cou...

A general methodology is proposed for the explanation of variability in a quantity of interest x in terms of covariates z = (z1, . . . ,zL). It provides the conditional mean x̄(z) as a sum of components, where each component is represented as a product of non-parametric one-dimensional functions of each covariate zl that are computed through an alternating projection procedure. Both x and the zl...

The ‘expectation–conditional maximization either’ (ECME) algorithm has proven to be an effective way of accelerating the expectation–maximization algorithm for many problems. Recognizing the limitation of using prefixed acceleration subspaces in the ECME algorithm, we propose a dynamic ECME (DECME) algorithm which allows the acceleration subspaces to be chosen dynamically. The simplest DECME im...

The behavior of the conditional expectation E{X I Y) under a small perturbation Z of the conditioning random variable Y is analyzed. We show that if Y and Z are independent then E(XIY + EZ) converges to E{XIY) in mean as E --* 0 for all integrable X, provided the distribution of Y is absolutely continuous. We also show that the limit is E{X 1 Y, Z} rather than E(X 1 Y}, i.e., there is no stabil...