arXiv subjects

Tom LaGatta

Publications and source records attributed to Tom LaGatta.

11 recordsLinked to original sources

arXiv2022

The Mathematics of Evolution: The Price Equation, Natural Selection, and Environmental Change

George Price introduced his famous equation to study selective and environmental effects in discrete populations. We extend Price's framework to the measurable and quantum cases, decomposing all evolutionary processes into selective and environmental components. We also extend Fisher's fundamental theorem, showing that selective change of relative fitness equals variance of relative fitness. We introduce novel selective and environmental entropy functionals. Selective entropy is non-positive, representing biological negentropy, and environmental entropy is non-negative, representing physical entropy. Environmental entropy further decomposes into dispersion and mixing entropies. We prove four novel Laws of Natural Selection, showing that selection consistently acts to increase selection, but can be disrupted by environmental change. We apply convex analysis to variance and entropy functionals and their selective changes, and equilibrium processes arise to optimize these inequalities. These laws are inspired by but distinct from the classical Thermodynamic Laws. Our Zeroth Law is a refinement of Fisher's theorem, showing that variance of relative fitness is bounded below by a quantity depending on the child-bearing population. Our First Law shows that selective acceleration of relative fitness is also bounded below, depending on the variance. This is a non-conservative, selective version of the Thermodynamic First Law. Our Second Law shows that the selective change of selective entropy and its selective acceleration are similarly bounded by non-positive constants. This is a formal, rigorous version of the Thermodynamic Second Law. Our Third Law shows that for a class of equilibrium processes, selective change of environmental entropy vanishes, and otherwise may vary in an open window around zero. This is a selective version of the Third Law of Thermodynamics.

math.PR↗

arXiv2019

Conscientious Classification: A Data Scientist's Guide to Discrimination-Aware Classification

Recent research has helped to cultivate growing awareness that machine learning systems fueled by big data can create or exacerbate troubling disparities in society. Much of this research comes from outside of the practicing data science community, leaving its members with little concrete guidance to proactively address these concerns. This article introduces issues of discrimination to the data science community on its own terms. In it, we tour the familiar data mining process while providing a taxonomy of common practices that have the potential to produce unintended discrimination. We also survey how discrimination is commonly measured, and suggest how familiar development processes can be augmented to mitigate systems' discriminatory potential. We advocate that data scientists should be intentional about modeling and reducing discriminatory outcomes. Without doing so, their efforts will result in perpetuating any systemic discrimination that may exist, but under a misleading veil of data-driven objectivity.

stat.ML↗

arXiv2016

Group Incentives and Rational Voting

Our model describes competition between groups driven by the choices of self-interested voters within groups. Within a Poisson voting environment, parties observe aggregate support from groups and can allocate prizes or punishments to them. In a tournament style analysis, the model characterizes how contingent allocation of prizes based on relative levels of support affects equilibrium voting behavior. In addition to standard notions of pivotality, voters influence the distribution of prizes across groups. Such prize pivotality supports positive voter turnout even in non-competitive electoral settings. The analysis shows that competition for a prize awarded to the most supportive group is only stable when two groups actively support a party. However, competition among groups to avoid punishment is stable in environments with any number of groups. We conclude by examining implications for endogenous group formation and how politicians structure the allocation of rewards and punishments.

math.PR↗

arXiv2015

Random geometry and the Kardar-Parisi-Zhang universality class

We consider a model of a quenched disordered geometry in which a random metric is defined on ${\mathbb R}^2$, which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius $R$ scales as $R^χ$, with a fluctuation exponent $χ\simeq 1/3$, while the lateral spread of the minimizing geodesic between two points at a distance $L$ grows as $L^ξ$, with wandering exponent value $ξ\simeq 2/3$. Results on related first-passage percolation (FPP) problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang (KPZ) universality class of surface kinetic roughening, with $ξ$ and $χ$ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy-Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW-GUE statistics with good accuracy in arrival times.

cond-mat.stat-mech↗

arXiv2014

A Structural Approach to Coordinate-Free Statistics

We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, we construct a version of the Ordinary Least Squares estimator in absolute generality. Our Gauss-Markov theorem demonstrates that OLS is a "best linear unbiased estimator", extending the classical result. We construct a stochastic version of the OLS estimator, which is a continuous disintegration exactly for the class of "uncorrelated implies independent" (UII) measures. As a consequence, Gaussian measures always exhibit continuous disintegrations through continuous linear maps, extending a theorem of the first author. Applying this framework to some problems in machine learning, we prove a useful representation theorem for covariance tensors, and show that OLS defines a good kriging predictor for vector-valued arrays on general index spaces. We also construct a support-vector machine classifier in this setting. We hope that our article shines light on some deeper connections between probability theory, statistics and machine learning, and may serve as a point of intersection for these three communities.

math.PR↗

arXiv2013

Geodesics of Random Riemannian Metrics

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.

math.PR↗

arXiv2013

Geodesics of Random Riemannian Metrics: Supplementary Material

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the setting of this article. In Appendix C, we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix D, we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix E, we present some analytic estimates which we use in the article. In Appendix F, we present the construction of the conditional mean operator for Gaussian measures. In Appendix G, we describe Fermi normal coordinates, which we use in our construction of the bump metric.

math.PR↗

arXiv2013

Elections, Protest, and Alternation of Power

Despite many examples to the contrary, most models of elections assume that rules determining the winner will be followed. We present a model where elections are solely a public signal of the incumbent popularity, and citizens can protests against leaders that do not step down from power. In this minimal setup, rule-based alternation of power as well as "semi-democratic" alternation of power independent of electoral rules can both arise in equilibrium. Compliance with electoral rules requires there to be multiple equilibria in the protest game, where the electoral rule serves as a focal point spurring protest against losers that do not step down voluntarily. Such multiplicity is possible when elections are informative and citizens not too polarized. Extensions to the model are consistent with the facts that protests often center around accusations of electoral fraud and that in the democratic case turnover is peaceful while semi-democratic turnover often requires citizens to actually take to the streets.

physics.soc-ph↗

arXiv2011

Dissertation: Geodesics of Random Riemannian Metrics

We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on $\mathbb R^d$. We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one. In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction $v$, the geodesic starting from the origin in the direction $v$ is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.

math.PR↗

arXiv2011

Continuous Disintegrations of Gaussian Processes

The goal of this paper is to understand the conditional law of a stochastic process once it has been observed over an interval. To make this precise, we introduce the notion of a continuous disintegration: a regular conditional probability measure which varies continuously in the conditioned parameter. The conditioning is infinite-dimensional in character, which leads us to consider the general case of probability measures in Banach spaces. Our main result is that for a certain quantity $M$ based on the covariance structure, the finiteness of M is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition is quite reasonable: for the familiar case of stationary processes, M = 1.

math.PR↗

arXiv2010

A Shape Theorem for Riemannian First-Passage Percolation

Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.

math.PR↗