# Browsing by Subject "entropy"

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Item Open Access Coalescing random walks on n-block Markov chains(2014-01-11) Lan, KathleenFix a discrete-time Markov chain $(V,P)$ with finite state space $V$ and transition matrix $P$. Let $(V_n,P_n)$ be the Markov chain on n-blocks induced by $(V,P)$, which we call the n-block process associated with the base chain $(V,P)$. We study coalescing random walks on mixing n-block Markov chains in discrete time. In particular, we are interested in understanding the asymptotic behavior of $\mathbb{E} C_n$, the expected coalescence time for $(V_n,P_n)$, as $n\to\infty$. Define the quantity $L=-\log\lambda$, where $\lambda$ is the Perron eigenvalue of the matrix $Q$ that has entries $Q_{i,j}=P_{i,j}^2$. We prove the existence of four limits and show that all of them are equal to $L$: $\lim\limits_{n\to\infty}\frac{1}{n}\log\mathbb{E} C_n$, $\lim\limits_{n\to\infty}\frac{1}{n}\log m_n^*$, $\lim\limits_{n\to\infty} \frac{1}{n}\log \bar{m}_n$, and $\lim\limits_{n\to\infty} -\frac{1}{n}\log\Delta_n$, where $m_n^*$ and $\bar{m}_n$ are the maximum and average meeting times for $(V_n, P_n)$ respectively. We establish the inequalities $0Item Open Access Configurational entropy measurements in extremely supercooled liquids that break the glass ceiling.(Proceedings of the National Academy of Sciences of the United States of America, 2017-10-10) Berthier, Ludovic; Charbonneau, Patrick; Coslovich, Daniele; Ninarello, Andrea; Ozawa, Misaki; Yaida, ShoLiquids relax extremely slowly on approaching the glass state. One explanation is that an entropy crisis, because of the rarefaction of available states, makes it increasingly arduous to reach equilibrium in that regime. Validating this scenario is challenging, because experiments offer limited resolution, while numerical studies lag more than eight orders of magnitude behind experimentally relevant timescales. In this work, we not only close the colossal gap between experiments and simulations but manage to create in silico configurations that have no experimental analog yet. Deploying a range of computational tools, we obtain four estimates of their configurational entropy. These measurements consistently confirm that the steep entropy decrease observed in experiments is also found in simulations, even beyond the experimental glass transition. Our numerical results thus extend the observational window into the physics of glasses and reinforce the relevance of an entropy crisis for understanding their formation.Item Open Access Hosts of avian brood parasites have evolved egg signatures with elevated information content.(Proc Biol Sci, 2015-07-07) Caves, Eleanor M; Stevens, Martin; Iversen, Edwin S; Spottiswoode, Claire NHosts of brood-parasitic birds must distinguish their own eggs from parasitic mimics, or pay the cost of mistakenly raising a foreign chick. Egg discrimination is easier when different host females of the same species each lay visually distinctive eggs (egg 'signatures'), which helps to foil mimicry by parasites. Here, we ask whether brood parasitism is associated with lower levels of correlation between different egg traits in hosts, making individual host signatures more distinctive and informative. We used entropy as an index of the potential information content encoded by nine aspects of colour, pattern and luminance of eggs of different species in two African bird families (Cisticolidae parasitized by cuckoo finches Anomalospiza imberbis, and Ploceidae by diederik cuckoos Chrysococcyx caprius). Parasitized species showed consistently higher entropy in egg traits than did related, unparasitized species. Decomposing entropy into two variation components revealed that this was mainly driven by parasitized species having lower levels of correlation between different egg traits, rather than higher overall levels of variation in each individual egg trait. This suggests that irrespective of the constraints that might operate on individual egg traits, hosts can further improve their defensive 'signatures' by arranging suites of egg traits into unpredictable combinations.Item Open Access The Verification of Probabilistic Forecasts in Decision and Risk Analysis(2009) Jose, Victor RichmondProbability forecasts play an important role in many decision and risk analysis applications. Research and practice over the years have shown that the shift towards distributional forecasts provides a more accurate and appropriate means of capturing risk in models for these applications. This means that mathematical tools for analyzing the quality of these forecasts, may it come from experts, models or data, become important to the decision maker. In this regard, strictly proper scoring rules have been widely studied because of their ability to encourage assessors to provide truthful reports. This dissertation contributes to the scoring rule literature in two main areas of assessment - probability forecasts and quantile assessments.

In the area of probability assessment, scoring rules typically studied in the literature, and commonly used in practice, evaluate probability assessments relative to a default uniform measure. In many applications, the uniform baseline used to represent some notion of ignorance is inappropriate. In this dissertation, we generalize the power and pseudospherical family of scoring rules, two large parametric families of commonly-used scoring rules, by incorporating the notion of a non-uniform baseline distribution for both the discrete and continuous cases. With an appropriate normalization and choice of parameters, we show that these new families of scoring rules relate to various well-known divergence measures from information theory and to well-founded decision models when framed in an expected utility maximization context.

In applications where the probability space considered has an ordinal ranking between states, an important property often considered is sensitivity to distance. Scoring rules with this property provide higher scores to assessments that allocate higher probability mass to events “closer” to that which occurs based on some notion of distance. In this setting, we provide an approach that allows us to generate new sensitive to distance strictly proper scoring rules from well-known strictly proper binary scoring rules. Through the use of the weighted scoring rules, we also show that these new scores can incorporate a specified baseline distribution, in addition to being strictly proper and sensitive to distance.

In the inverse problem of quantile assessment, scoring rules have not yet been well-studied and well-developed. We examine the differences between scoring rules for probability and quantile assessments, and demonstrate why the tools that have been developed for probability assessments no longer encourage truthful reporting when used for quantile assessments. In addition, we shed light on new properties and characterizations for some of these rules that could guide decision makers trying to choosing an appropriate scoring rule.