The stability of conditional Markov processes and Markov chains in random environments

We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of $\sigma$-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365--393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space.Comment: Published in at http://dx.doi.org/10.1214/08-AOP448 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Borell-Ehrhard Game

A precise description of the convexity of Gaussian measures is provided by sharp Brunn-Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoretic mechanism: a minimax variational principle for Brownian motion. As an application, we obtain a Gaussian improvement of Barthe's reverse Brascamp-Lieb inequality.Comment: 23 page

Chaining, Interpolation, and Convexity

We show that classical chaining bounds on the suprema of random processes in terms of entropy numbers can be systematically improved when the underlying set is convex: the entropy numbers need not be computed for the entire set, but only for certain "thin" subsets. This phenomenon arises from the observation that real interpolation can be used as a natural chaining mechanism. Unlike the general form of Talagrand's generic chaining method, which is sharp but often difficult to use, the resulting bounds involve only entropy numbers but are nonetheless sharp in many situations in which classical entropy bounds are suboptimal. Such bounds are readily amenable to explicit computations in specific examples, and we discover some old and new geometric principles for the control of chaining functionals as special cases.Comment: 21 pages; final version, to appear in J. Eur. Math. So

The stability of quantum Markov filters

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.Comment: Final versio

The Big Squeeze: A Social and Political History of the Controversial Mammogram

[Excerpt] The Big Squeeze: A Social and Political History of the Controversial Mammogram chronicles the often turbulent history of screening mammography since its introduction in the early 1970s. This book makes five key points. First, it shows how pivotal decisions during mammography\u27s initial roll-out made it all but inevitable that the test would never be far from controversy. Second, it describes how, at several key points in its history, the establishment of a culture of mammography screening was greatly aided by concurrent social and political forces and movements. Third, it illustrates how politics came to dominate the debate, eventually achieving primacy over science itself. Fourth, The Big Squeeze describes the collateral economy that developed around screening. As mammography was aggressively promoted in the late 1980s to early 1990s, utilization rates rapidly increased. As this occurred, the mundane mammogram became the little pink engine that could, and did, drive the growth of a vast screening-dependent secondary economy. Finally, mammography\u27s burden, overdiagnosis, is considered in the last chapter. Overdiagnosis, the screening detection of cancers that would never otherwise have come to light in the individual\u27s lifetime, is an important yet woefully underdiscussed risk of mammography. This phenomenon is more significant than that, however. Overdiagnosis helped make fighting breast cancer the most favored disease cause and mammography the most favored weapon in the fight

Commensurations of Out(F_n)

Let \Out(F_n) denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup \Gamma<\Out(F_n), the group \Aut(\Gamma) is isomorphic to the normalizer of $\Gamma$ in \Out(F_n). We prove that $\Gamma$ is {\em co-Hopfian} : every injective homomorphism $\Gamma\to \Gamma$ is surjective. Finally, we prove that the abstract commensurator \Comm(\Out(F_n)) is isomorphic to \Out(F_n).Comment: Revised version, 43 pages. To appear in Publ. Math. IHE

Mapping tori of free group automorphisms are coherent

The mapping torus of an endomorphism \Phi of a group G is the HNN-extension G*_G with bonding maps the identity and \Phi. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.Comment: 17 pages, published versio