THE GEOMETRY of SLOPPINESS 1. Introduction Mathematical Models

Total Page:16

File Type:pdf, Size:1020Kb

THE GEOMETRY of SLOPPINESS 1. Introduction Mathematical Models THE GEOMETRY OF SLOPPINESS EMILIE DUFRESNE, HEATHER A HARRINGTON, AND DHRUVA V RAMAN Abstract. The use of mathematical models in the sciences often involves the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. In this paper, we develop a pre- cise mathematical foundation for sloppiness as initially introduced and define rigorously key concepts, such as `model manifold', in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric. This opens up the possibility of alternative quantification of sloppiness, beyond the standard use of the Fisher Information Matrix, which assumes that parameter space is equipped with the usual Euclidean metric and the measurement error is infinitesimal. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models. 1. Introduction Mathematical models describing physical, biological, and other real-life phe- nomena contain parameters whose values must be estimated from data. Over the past decade, a powerful framework called \sloppiness" has been developed that relies on Information Geometry [1] to study the uncertainty in this proce- dure [10, 17, 56, 57, 58, 55]. Although the idea of using the Fisher Information to quantify uncertainty is not new (see for example [20, 45]), the study of sloppi- ness gives rise to a particular observation about the uncertainty of the procedure and has potential implications beyond parameter estimation. Specifically, sloppi- ness has enabled advances in the field of systems biology, drawing connections to sensitivity [25, 19, 24], experimental design [4, 37, 25], identifiability [47, 55, 13], robustness [17], and reverse engineering [19, 14]. Sethna, Transtrum and co-authors identified sloppiness as a universal property of highly parameterised mathematical models [61, 56, 54, 25]. More recently a non-local version of sloppiness has emerged, called predictive sloppiness [33]. However, the precise interpretation of sloppiness remains a matter of active discussion in the literature [4, 26, 29]. This paper's main contribution is to serve as a first step towards a unified math- ematical framework for sloppiness rooted in algebra and geometry. While our work does not synthesizes the entirety of the field, we provides some of the mathematical elements needed to formalize sloppiness as it was initially introduced. We extend the concept beyond time dependent models, in particular, to statistical models. We rigorously define the concepts and building blocks for the theory of sloppiness. Our approach requires techniques from many fields including algebra, geometry, and statistics. We illustrate each new concept with a simple concrete example. The new mathematical foundation we provide for sloppiness is not limited by current computational tools and opens up the way to further work. Our general setup is a mathematical model M that describes the behavior of a variable x Rm depending on a parameter p P Rr. Our first step is to explain how each precise2 choice of perfect data z induces2 an⊆ equivalence relation on the ∼M;z Date: 3rd Aug, 2018. 1 2 DUFRESNE, HARRINGTON, AND RAMAN parameter space: two parameters are equivalent if they produce the same perfect data. We then characterize the various concepts of structural identifiability in terms 2 DUFRESNE,HARRINGTON,ANDRAMAN of the equivalence relation M;z. Roughly speaking, structural identifiability asks to what extent perfect data∼ determines the value of the parameters. See section 2. theory. We illustrate each new concept with a simple concrete example. The Assume that the perfect data z is a point of N for some N. The second crucial new mathematical foundations we provide forR sloppiness is not limited by current step needed in order to define sloppiness is a map φ from parameter space P to computational tools and opens up the way to further work. data space N giving the perfect data as a function φ(p) of the parameters known Our generalR set up is of a mathematical model M which describes the behavior as a \model manifold" in the literature [56, 57, 58, 55], which we rename as a of a variable x m depending on a parameter p P r.Ourfirststepisto model prediction map.R A model prediction map thus inducesR an injective function explain how each2 precise choice of perfect data z induces2 ✓ an equivalence relation on the set of equivalence classes (the set-theoretic quotient P= ), that is, the M,z) on the parameter space: two parameters are equivalentM;z if they produce equivalence( classes can be separated by N functions P . See∼ Section 3. the⇠ same perfect data. We then characterize the variousR concepts of structural The next step is to assume that the mathematical model! describes the phenome- identifiability in terms of the equivalence relation (M,z). non we are studying perfectly, but that the \real data"⇠ is corruptedN by measurement errorAssume and the we use can of finite identify sample the perfct size. Thatdata z is,with we assume a point that of R noisyfor some data arisesN.The second crucial step needed in order to define sloppiness is a map φ from parameter from a random process whoseN probability distribution then induces a premetric d on thespace parameterP to data space, space via theR Kullback-Leiblergiving the perfect divergence data as a ( function see start of of the Section paramaters 4 ). Thisknown premetric as a “modeld quantifies manifold” the proximity [48, 49, 50, between 47], which the two we parametersredefine as a in model parameter predic- tion map. A model prediction map φ: P RN induces an injective function on space via the discrepancy between the probability! distributions of the noisy data associatedthe set of to equivalence the two parameters. classes, that is, the equivalence classes can be separated by N functions P R. See Section 3. The aforementioned! premetric d has a tractable approximation in the limit of decreasingThe sloppiness measurement of a noise model using manifests the Fisher itself Information when dealing Matrix with (FIM).noisy data. In the We standardassume definition, that the mathematical a model is \sloppy" model when describes the condition the phenomenon number we of the are FIM studying is large,perfectly, that is, but there that are the several “real data” orders is of corrupted magnitude by between measurement its largest error and and/or smallest the use eigenvalues.of finite sample Multiscale size. Therefore sloppiness noisy (see [44]) data extends arises from this a concept random to process regimes whose of non- prob- infinitesimalability distribution noise. induces a premetric on the parameter space, via the Kullback- LeiblerWe conceptually divergence. extend This premetric the notion quantifies of sloppiness the proximity to a comparison between between parameters the in premetricparameterd and space a reference via the discrepancy metric on parameter between the space. probability We demonstrate distributions that of us- the noisy data associated to distinct parameters. ing the condition number of the FIM to measure sloppiness at a parameter p0, as is doneThe in aforementioned most of the sloppiness premetric literature has a tractable [10, 17, 56,approximation 57, 58, 55], in corresponds the limit of decreasing noise using the Fisher Information Matrix (FIM). In the standard defi- to comparing an approximation of d in an infinitesimal neighborhood of p0 to the nition, a model is “sloppy” when the condition number of the FIM is large, that is, standard Euclidean metric on Rr P . Note that considering the entire spectrum of thethere FIM, are as severalis done ordersnewer work of magnitude in⊃ the sloppiness between literature its largest (eg, and [61]) smallest corresponds eigenvalues. to performingWe conceptually a more refined extend comparisonthe notion of between sloppiness an approximationto a comparison of betweend in an thisinfini- pre- metric and a reference metric on parameter space. We demonstrater that using the tesimal neighborhood of p0 to the standard Euclidean metric on R P . Multiscale sloppiness,condition which number we of extend the FIM here to beyond measure its sloppiness original definition at a parameter [44]⊃ forp0 Euclidean, as is done parameterin mostspace of the and sloppiness Gaussian literature, measurement corresponds noise, avoids to comparing approximating an approximationd, and so of d in an infinitesimal neighborhood of p0 to the standard Euclidean metric on betterr reflects the sloppiness of models beyond| the infinitesimal scale. Finally, we describeR sup theP . intimateWe alsoput relationship into context between and generalise sloppiness the and quantification practical identifiability, of sloppiness thatintroduced is, whether in [36] generic which noisy avoids data approximating allows for boundedd and confidenceso better reflects regions the when slop- performingpiness of maximum models beyond likelihood the infinitesimal estimation. Seescale. Section Finally, 4. we describe the intimate relationshipThe following between diagram sloppiness illustrates and the practical main objects identifiability. discussed See in this Section paper: 4. The following diagram illustrates the main objects discussed in this paper: M a mathematical model P/ M,z x variable, belongs to X Rm ⇠ ✓ inj y observable, belongs to Y Rn ✓ φ model prediction map r φ N r R P Z R p parameter, belongs to P R ⊃ ⇢ ✓ z data, belongs to Z RN ✓ g g function giving y in terms of x Rm X Y Rn ⊃ ⇢ 2. Arelationonparameterspace A mathematical model M describes the behavior of a variable x X Rm 2 ✓ depending on a parameter p P Rr, with measurable output y = g(x) Y Rn. We further specify a choice2 of perfect⇢ data z produced for the parameter2 2 value p. We think of the perfect data z as belonging to the wider data space Z which THE GEOMETRY OF SLOPPINESS 3 2. An equivalence relation on parameter space A mathematical model M describes the behavior of a variable x X Rm 2 ⊆ depending on a parameter p P Rr, with measurable output y = g(x) Y Rn.
Recommended publications
  • Stable Components in the Parameter Plane of Transcendental Functions of Finite Type
    Stable components in the parameter plane of transcendental functions of finite type N´uriaFagella∗ Dept. de Matem`atiquesi Inform`atica Univ. de Barcelona, Gran Via 585 Barcelona Graduate School of Mathematics (BGSMath) 08007 Barcelona, Spain [email protected] Linda Keen† CUNY Graduate Center 365 Fifth Avenue, New York NY, NY 10016 [email protected], [email protected] November 11, 2020 Abstract We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the virtual center, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected.
    [Show full text]
  • Backpropagation TA: Yi Wen
    Backpropagation TA: Yi Wen April 17, 2020 CS231n Discussion Section Slides credits: Barak Oshri, Vincent Chen, Nish Khandwala, Yi Wen Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer Motivation Recall: Optimization objective is minimize loss Motivation Recall: Optimization objective is minimize loss Goal: how should we tweak the parameters to decrease the loss? Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer A Simple Example Loss Goal: Tweak the parameters to minimize loss => minimize a multivariable function in parameter space A Simple Example => minimize a multivariable function Plotted on WolframAlpha Approach #1: Random Search Intuition: the step we take in the domain of function Approach #2: Numerical Gradient Intuition: rate of change of a function with respect to a variable surrounding a small region Approach #2: Numerical Gradient Intuition: rate of change of a function with respect to a variable surrounding a small region Finite Differences: Approach #3: Analytical Gradient Recall: partial derivative by limit definition Approach #3: Analytical Gradient Recall: chain rule Approach #3: Analytical Gradient Recall: chain rule E.g. Approach #3: Analytical Gradient Recall: chain rule E.g. Approach #3: Analytical Gradient Recall: chain rule Intuition: upstream gradient values propagate backwards -- we can reuse them! Gradient “direction and rate of fastest increase” Numerical Gradient vs Analytical Gradient What about Autograd?
    [Show full text]
  • Decomposing the Parameter Space of Biological Networks Via a Numerical Discriminant Approach
    Decomposing the parameter space of biological networks via a numerical discriminant approach Heather A. Harrington1, Dhagash Mehta2, Helen M. Byrne1, and Jonathan D. Hauenstein2 1 Mathematical Institute, The University of Oxford, Oxford OX2 6GG, UK {harrington,helen.byrne}@maths.ox.ac.uk, www.maths.ox.ac.uk/people/{heather.harrington,helen.byrne} 2 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame IN 46556, USA {dmehta,hauenstein}@nd.edu, www.nd.edu/~{dmehta,jhauenst} Abstract. Many systems in biology (as well as other physical and en- gineering systems) can be described by systems of ordinary dierential equation containing large numbers of parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to iden- tify and characterize the steady-state solutions as the model parameters vary, a technically challenging problem in a high-dimensional parameter landscape. Rather than simply determining the number and stability of steady-states at distinct points in parameter space, we decompose the parameter space into nitely many regions, the number and structure of the steady-state solutions being consistent within each distinct region. From a computational algebraic viewpoint, the boundary of these re- gions is contained in the discriminant locus. We develop global and local numerical algorithms for constructing the discriminant locus and classi- fying the parameter landscape. We showcase our numerical approaches by applying them to molecular and cell-network models. Keywords: parameter landscape · numerical algebraic geometry · dis- criminant locus · cellular networks. 1 Introduction The dynamic behavior of many biophysical systems can be mathematically mod- eled with systems of dierential equations that describe how the state variables interact and evolve over time.
    [Show full text]
  • The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 4, October 2016, Pages 513–554 http://dx.doi.org/10.1090/bull/1544 Article electronically published on August 2, 2016 THE EMERGENCE OF GRAVITATIONAL WAVE SCIENCE: 100 YEARS OF DEVELOPMENT OF MATHEMATICAL THEORY, DETECTORS, NUMERICAL ALGORITHMS, AND DATA ANALYSIS TOOLS MICHAEL HOLST, OLIVIER SARBACH, MANUEL TIGLIO, AND MICHELE VALLISNERI In memory of Sergio Dain Abstract. On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein’s prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detec- tion of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole bina- ries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science.
    [Show full text]
  • Investigations of Structures in the Parameter Space of Three-Dimensional Turing-Like Patterns Martin Skrodzki, Ulrich Reitebuch, Eric Zimmermann
    Investigations of structures in the parameter space of three-dimensional Turing-like patterns Martin Skrodzki, Ulrich Reitebuch, Eric Zimmermann To cite this version: Martin Skrodzki, Ulrich Reitebuch, Eric Zimmermann. Investigations of structures in the parameter space of three-dimensional Turing-like patterns. AUTOMATA2021, Jul 2021, Marseille, France. hal- 03270664 HAL Id: hal-03270664 https://hal.archives-ouvertes.fr/hal-03270664 Submitted on 25 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Investigations of structures in the parameter space of three-dimensional Turing-like patterns∗ Martin Skrodzki,† Computer Graphics and Visualization, TU Delft [email protected] Ulrich Reitebuch, Institute of Mathematics, FU Berlin [email protected] Eric Zimmermann, Institute of Mathematics, FU Berlin [email protected] Abstract In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjec- ture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found two-fold versions of so-called \zero-" and \one-dimensional" structures as well as \two-dimensional" structures and use our experimental find- ings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.
    [Show full text]
  • A Practical Guide to Compact Infinite Dimensional Parameter Spaces
    A Practical Guide to Compact Infinite Dimensional Parameter Spaces∗ Joachim Freybergery Matthew A. Mastenz May 24, 2018 Abstract Compactness is a widely used assumption in econometrics. In this paper, we gather and re- view general compactness results for many commonly used parameter spaces in nonparametric estimation, and we provide several new results. We consider three kinds of functions: (1) func- tions with bounded domains which satisfy standard norm bounds, (2) functions with bounded domains which do not satisfy standard norm bounds, and (3) functions with unbounded do- mains. In all three cases we provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a k · ks-norm bound are compact under a norm k · kc. We illustrate how the choice of norms affects the parameter space, the strength of the conclusions, as well as other regularity conditions in two common settings: nonparametric mean regression and nonparametric instrumental variables estimation. JEL classification: C14, C26, C51 Keywords: Nonparametric Estimation, Sieve Estimation, Trimming, Nonparametric Instrumental Variables ∗This paper was presented at Duke, the 2015 Triangle Econometrics Conference, and the 2016 North American and China Summer Meetings of the Econometric Society. We thank audiences at those seminars as well as Bruce Hansen, Kengo Kato, Jack Porter, Yoshi Rai, and Andres Santos for helpful conversations and comments. We also thank the editor and the referees for thoughtful feedback which improved this paper. yDepartment of Economics, University of Wisconsin-Madison, William H. Sewell Social Science Building, 1180 Observatory Drive, Madison, WI 53706, USA. [email protected] zCorresponding author.
    [Show full text]
  • Interactive Parameter Space Partitioning for Computer Simulations
    TECHNICAL REPORT SFU-CMPT TR 2011-06 1 Paraglide: Interactive Parameter Space Partitioning for Computer Simulations Steven Bergner Michael Sedlmair Sareh Nabi-Abdolyousefi Ahmed Saad Torsten Moller¨ [email protected] [email protected] [email protected] [email protected] [email protected] Abstract—In this paper we introduce paraglide, a visualization system designed for interactive exploration of parameter spaces of multi-variate simulation models. To get the right parameter configuration, model developers frequently have to go back and forth between setting parameters and qualitatively judging the outcomes of their model. During this process, they build up a grounded understanding of the parameter effects in order to pick the right setting. Current state-of-the-art tools and practices, however, fail to provide a systematic way of exploring these parameter spaces, making informed decisions about parameter settings a tedious and workload-intensive task. Paraglide endeavors to overcome this shortcoming by assisting the sampling of the parameter space and the discovery of qualitatively different model outcomes. This results in a decomposition of the model parameter space into regions of distinct behaviour. We developed paraglide in close collaboration with experts from three different domains, who all were involved in developing new models for their domain. We first analyzed current practices of six domain experts and derived a set of design requirements, then engaged in a longitudinal user-centered design process, and finally conducted three in-depth case studies underlining the usefulness of our approach. F 1 LINKING FORMAL AND REAL SYSTEMS in the suitability of the modelling mechanisms and, thus, become a substantial aid in the research process.
    [Show full text]
  • Visualizing Likelihood Density Functions Via Optimal Region Projection
    Computers & Graphics 41 (2014) 62–71 Contents lists available at ScienceDirect Computers & Graphics journal homepage: www.elsevier.com/locate/cag Special Section on Uncertainty and Parameter Space Analysis in Visualization Visualizing likelihood density functions via optimal region projection Hal Canary a, Russell M. Taylor II d,n, Cory Quammen a, Scott Pratt b, Facundo A. Gómez b, Brian O'Shea b, Christopher G. Healey c a Department of Computer Science, University of North Carolina at Chapel Hill, United States b Department of Physics and Astronomy, Michigan State University, United States c Department of Computer Science, North Carolina State University, United States d Department of Computer Science, University of North Carolina at Chapel Hill, CB #3175, Sitterson Hall, Chapel Hill, NC 27599-3175, United States article info abstract Article history: Effective visualization of high-likelihood regions of parameter space is severely hampered by the large Received 15 July 2013 number of parameter dimensions that many models have. We present a novel technique, Optimal Received in revised form Percentile Region Projection, to visualize a high-dimensional likelihood density function that enables the 9 February 2014 viewer to understand the shape of the high-likelihood region. Optimal Percentile Region Projection has Accepted 26 February 2014 three novel components: first, we select the region of high likelihood in the high-dimensional space Available online 12 March 2014 before projecting its shadow into a lower-dimensional projected space. Second, we analyze features on Keywords: the surface of the region in the projected space to select the projection direction that shows the most Uncertainty interesting parameter dependencies.
    [Show full text]
  • On Curved Exponential Families
    U.U.D.M. Project Report 2019:10 On Curved Exponential Families Emma Angetun Examensarbete i matematik, 15 hp Handledare: Silvelyn Zwanzig Examinator: Örjan Stenflo Mars 2019 Department of Mathematics Uppsala University On Curved Exponential Families Emma Angetun March 4, 2019 Abstract This paper covers theory of inference statistical models that belongs to curved exponential families. Some of these models are the normal distribution, binomial distribution, bivariate normal distribution and the SSR model. The purpose was to evaluate the belonging properties such as sufficiency, completeness and strictly k-parametric. Where it was shown that sufficiency holds and therefore the Rao Blackwell The- orem but completeness does not so Lehmann Scheffé Theorem cannot be applied. 1 Contents 1 exponential families ...................... 3 1.1 natural parameter space ................... 6 2 curved exponential families ................. 8 2.1 Natural parameter space of curved families . 15 3 Sufficiency ............................ 15 4 Completeness .......................... 18 5 the rao-blackwell and lehmann-scheffé theorems .. 20 2 1 exponential families Statistical inference is concerned with looking for a way to use the informa- tion in observations x from the sample space X to get information about the partly unknown distribution of the random variable X. Essentially one want to find a function called statistic that describe the data without loss of im- portant information. The exponential family is in probability and statistics a class of probability measures which can be written in a certain form. The exponential family is a useful tool because if a conclusion can be made that the given statistical model or sample distribution belongs to the class then one can thereon apply the general framework that the class gives.
    [Show full text]
  • Parameters, Estimation, Likelihood Function a Summary of Concepts
    Math 2710 A Second Course in Statistics Parameters, estimation, likelihood function A summary of concepts needed for section 6.5: maximum likelihood estimators, from earlier sections 6.1-6.4 Suppose random variables X1,...,Xn from a random sample from a discrete distribution or a continuous distribution for which the p.f. or the p.d.f. is f(x θ), where the parameter θ belongs to some parameter space Ω. | Here the parameter θ describes everything which is unknown about the p.f. or p.d.f. of the data. The parameter θ is the object of statistical inference. The assumption that θ Ω,whereΩ is is some set of possible values, gives the precise description "to what degree"∈ θ is unknown: there may be limitations on θ,expressedin θ Ω, but otherwise θ is unknown. The set Ω is called the parameter space. Every statistical procedure∈ (tests, confidence intervals, estimates) enables statements and conclusions about θ, but always under the "a priori" assumption that θ Ω. For instance, we might test a null ∈ hypothesis θ Ω0,whereΩ0 Ω etc. ∈ ⊂ Example 1. Let X1,...,Xn be from a random sample from a Bernoulli distribution with unknown probability of success p. Here the p.f. for one observation is (where x is either 0 or 1) x 1 x Pr(Xi = x)=p (1 p) − . − Here the unknown parameter θ is θ = p and the parameter space may be Ω =(0, 1) (the open interval) or Ω =[0, 1] (the closed interval, where p =0and p =1are included). Thus the p.f.
    [Show full text]
  • Parameter Estimation Lesson
    Introduction to model parameter estimation Aaron A. King with contributions from Ottar Bjørnstad, Ben Bolker, John Drake, Pej Rohani, and Dave Smith October 30, 2017 Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/by-nc/3.0/. Please share and remix noncommercially, mentioning its origin. 1 Introduction This course will focus on the utility of models in understanding, predicting, and controlling infectious disease systems. Models play a central role in this work because they allow us to precisely and quantitatively express our ideas about the mechanisms of infectious disease trans- mission, immunity, and ecology. To the extent our understanding of the important mechanisms is correct, such models are extremely useful in the design of policy. On the other hand, mod- els are useful in uncovering the important mechanisms, inasmuch as we can compare model predictions to data. Specifically, if we can translate competing hypotheses into mathematical models, these can be compared in terms of their ability to correctly capture the patterns seen in data. In order to fairly compare competing models, we must first try to find out what is the best each model can do. Practically speaking, this means that we have to find the values of the models’ parameters that give the closest correspondence between model predictions and data. Parameter estimation can be important even when we are fairly confident in the ability of a single model to explain the dynamics. Not surprisingly, the all-important quantity R0 is frequently the focus of considerable parameter-estimation effort. Here, we’ll try our hand as estimating R0 and other model parameters from an epidemic curve using a couple of different methods.
    [Show full text]
  • Lecture 1: January 15 1.1 Decision Principles
    10-716: Advanced Machine Learning Spring 2019 Lecture 1: January 15 Lecturer: Pradeep Ravikumar Scribes: Amir Alavi, Zhaoqi Cheng, Mark Cheung Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. 1.1 Decision Principles 1.1.1 Basics 1. State of nature: θ 2 Θ, where θ is also called the parameter and Θ is called the parameter space (all possible states of nature). 2. Statistical investigation performed to obtain information about θ, the outcome is called X, whose distribution depends on θ) 3. Infer state of nature 4. Take action/decision (in general, assumptions needed) denoted by a and A is the action space. In general, A 6= Θ. Loss is a function of the parameter and action space i.e., L :Θ × A ! R 1.1.2 3 Sources of info • Model-based X ∼ PΘ() • Prior Knowledge of θ • Loss/gain from taking action (based on estimation) 1.1.2.1 Examples • E1 A lady claims to be able to tell whether milk was added before tea (or vice versa). In all 10 trials, she correctly determines which was poured first. • E2 A drunk friend claims to be able to predict the outcome of a flip of a fair coin. In all 10 trials, he is correct. State of nature: θ: probability person answering correctly 1 Null hypothesis (person guessing) H0: Θ = 2 −10 Assuming 50% chance of guessing X ∼ Bin(10; θ), PH0 (x) = 2 , we would reject the hypothesis that the person guesses correctly.
    [Show full text]