Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulli στ-shift rule, which is members of Wolfram’s class II, and it was said to be simple as periodic before. It is worthwhile to study dynamical behaviors of rule 58 and whether it possesses chaotic attractors or not. It is shown that there exist two Bernoulli-measure attractors of rule 58. The dynamical properties of topological entropy and topological mixing of rule 58 are exploited on these two subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems, we divide determinative block systems into two subsets. In addition, it is shown that rule 58 possesses rich and complicated dynamical behaviors in the space of bi-infinite sequences. Furthermore, we prove that four rules of global equivalence class ε43 of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations, and we find that these attractors are not global attractors. The Lameray diagram is used to show clearly the iterative process of an attractor

The Bayesian Analysis of Complex, High-Dimensional Models: Can It Be CODA?

We consider the Bayesian analysis of a few complex, high-dimensional models and show that intuitive priors, which are not tailored to the fine details of the model and the estimated parameters, produce estimators which perform poorly in situations in which good, simple frequentist estimators exist. The models we consider are: stratified sampling, the partial linear model, linear and quadratic functionals of white noise and estimation with stopping times. We present a strong version of Doob's consistency theorem which demonstrates that the existence of a uniformly $\sqrt{n}$-consistent estimator ensures that the Bayes posterior is $\sqrt{n}$-consistent for values of the parameter in subsets of prior probability 1. We also demonstrate that it is, at least, in principle, possible to construct Bayes priors giving both global and local minimax rates, using a suitable combination of loss functions. We argue that there is no contradiction in these apparently conflicting findings.Comment: Published in at http://dx.doi.org/10.1214/14-STS483 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Switching Costs and the Gittins Index

The Theorem of Gittins and Jones (1974) is, perhaps, the single most powerful result in the literature on Bandit problems. This result establishes that in independent-armed Bandit problems with geometric discounting over an infinite horizon, all optimal strategies may be obtained by solving a family of simple optimal stopping problems that associate with each arm an index known as the dynamic allocation index or, more popularly, as the Gittins index. Importantly, the Gittins index of an arm depends solely on the characteristics of that arm and the rate of discounting, and is otherwise completely independent of the problem under consideration. These features simplify significantly the task of characterizing optimal strategies in this class of problems

Switching Costs and the Gittins Index

The Theorem of Gittins and Jones (1974) is, perhaps, the single most powerful result in the literature on Bandit problems. This result establishes that in independent-armed Bandit problems with geometric discounting over an infinite horizon, all optimal strategies may be obtained by solving a family of simple optimal stopping problems that associate with each arm an index known as the dynamic allocation index or, more popularly, as the Gittins index. Importantly, the Gittins index of an arm depends solely on the characteristics of that arm and the rate of discounting, and is otherwise completely independent of the problem under consideration. These features simplify significantly the task of characterizing optimal strategies in this class of problems

Voting in the bicameral Congress: large majorities as a signal of quality

publication-status: Acceptedtypes: ArticleWe estimate a model of voting in Congress that allows for dispersed information about the quality of proposals in an equilibrium context. In equilibrium, the Senate only approves House bills that receive the support of a supermajority of members of the lower chamber. We estimate this endogenous supermajority rule to be about four-fifths on average across policy areas. Our results indicate that the value of information dispersed among legislators is significant, and that in equilibrium a large fraction of House members' (40–50%) votes following their private information. Finally, we show that the probability of a type I error in Congress (not passing a good bill) is on average about twice as high as the probability of a type II error (passing a low-quality bill)