minnorm approximation of maxent mindiv problems for
play

MinNorm approximation of MaxEnt/MinDiv problems for probability - PowerPoint PPT Presentation

Feb 19, 2023 •188 likes •789 views

MinNorm approximation of MaxEnt/MinDiv problems for probability tables Patrick Bogaert and Sarah Gengler UCL Rebuilding probability tables UCL Rebuilding probability tables Limited number of samples Poor estimates UCL Rebuilding

  1. MinNorm approximation of MaxEnt/MinDiv problems for probability tables Patrick Bogaert and Sarah Gengler UCL

  2. Rebuilding probability tables UCL

  3. Rebuilding probability tables • Limited number of samples  Poor estimates UCL

  4. Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? UCL

  5. Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? Rewriting information as equality / inequality constraints  UCL

  6. Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? Rewriting information as equality / inequality constraints  UCL

  7. Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? Rewriting information as equality / inequality constraints  • Equality constraints MaxEnt  • Inequality constraints Minimum divergence ( MinDiv )  UCL

  8. Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? Rewriting information as equality / inequality constraints  • Equality constraints MaxEnt  • Inequality constraints Minimum divergence ( MinDiv )   Need for an efficient methodology to rebuild probability tables from both equality and inequality constraints UCL

  9. The MaxEnt problem UCL

  10. The MaxEnt problem • Equality constraints UCL

  11. The MaxEnt problem • Equality constraints • Entropy maximized subject to the equality constraints UCL

  12. The MaxEnt problem • Equality constraints • Entropy maximized subject to the equality constraints UCL

  13. The MaxEnt problem • Equality constraints • Entropy maximized subject to the equality constraints  Sequence of MinNorm problems for solving the MaxEnt problem UCL

  14. MinNorm as an approximation of MaxEnt UCL

  15. MinNorm as an approximation of MaxEnt • Taylor series of ln p i around p i = k i UCL

  16. MinNorm as an approximation of MaxEnt • Taylor series of ln p i around p i = k i • Truncating at degree one and summing over i UCL

  17. MinNorm as an approximation of MaxEnt • Taylor series of ln p i around p i = k i • Truncating at degree one and summing over i • In particular, if k i = 1/n UCL

  18. MinNorm as an approximation of MaxEnt • For any other choice of the k i ‘s, by completing the square UCL

  19. MinNorm as an approximation of MaxEnt • For any other choice of the k i ‘s, by completing the square • Summing over i Where UCL

  20. The MinDiv problem UCL

  21. The MinDiv problem • Divergence or Kullback-Leibler distance UCL

  22. The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing UCL

  23. The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing  UCL

  24. The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing   UCL

  25. The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing    Sequence of MinNorm problems for solving the MinDiv problem UCL

  26. The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing    Sequence of MinNorm problems for solving the MinDiv problem Both Equality and Inequality constraints can be processed together by MinNorm approximations UCL

  27. MinNorm as an approximation of MinDiv UCL

  28. MinNorm as an approximation of MinDiv • Taylor series around p i = k i and completing the square UCL

  29. MinNorm as an approximation of MinDiv • Taylor series around p i = k i and completing the square • Summing over i Where UCL

  30. Application in drainage classes mapping UCL

  31. Application in drainage classes mapping • Categorical data are found in a wide variety of applications UCL

  32. Application in drainage classes mapping UCL

  33. Application in drainage classes mapping • Categorical data are found in a wide variety of applications • 90 % of variables collected in soil surveys are categorical • Soil drainage, an important criterion in rating soils for various uses UCL

  34. Application in drainage classes mapping UCL

  35. Application in drainage classes mapping UCL

  36. Application in drainage classes mapping UCL

  37. Application in drainage classes mapping UCL

  38. Application in drainage classes mapping UCL

  39. Application in drainage classes mapping UCL

  40. Application in drainage classes mapping UCL

  41. Application in drainage classes mapping UCL

  42. Integrating the lithological map : 4 cases UCL

  43. Integrating the lithological map : 4 cases UCL

  44. Integrating the lithological map : 4 cases UCL

  45. Spatial prediction UCL

  46. Integrating the lithological map : 4 cases UCL

  47. Spatial prediction UCL

  48. Integrating the lithological map : 4 cases UCL

  49. Conclusions UCL

  50. Conclusions UCL

  51. Conclusions  MinNorm Approximations UCL

  52. Conclusions  MinNorm Approximations UCL

  53. Conclusions  MinNorm Approximations UCL

  54. Conclusions  MinNorm Approximations UCL

  55. Conclusions  MinNorm Approximations UCL

  56. Conclusions  MinNorm Approximations UCL

  57. Conclusions  MinNorm Approximations UCL

  58. Thank you for your attention UCL

  59. References UCL

Recommend

Overview MAXENT-Modeling: A framework for Discrete MAXENT-Models and RMs IRT-Modeling?

Overview MAXENT-Modeling: A framework for Discrete MAXENT-Models and RMs IRT-Modeling?

Overview MAXENT-Modeling: A framework for Discrete MAXENT-Models and RMs IRT-Modeling? Conceptual foundation and relationships of the MAXENT-framework The canonical MAXENT-distribution is structurally similar to Dipl.-Psych. Georg

366 views • 7 slides
6. Approximation and fitting norm approximation least-norm problems regularized

6. Approximation and fitting norm approximation least-norm problems regularized

Convex Optimization Boyd & Vandenberghe 6. Approximation and fitting norm approximation least-norm problems regularized approximation robust approximation 61 Norm approximation minimize Ax b ( A R m

877 views • 20 slides
Approximation Algorithms for Geometric Proximity Problems: Introduction Background Approximation

Approximation Algorithms for Geometric Proximity Problems: Introduction Background Approximation

Approximation Algorithms for Geometric Proximity Problems: Introduction Background Approximation Part I: Approximating Euclidean MSTs Prior Work Preliminaries WSPD+MST Lower Bound In Theory David M. Mount Simple+Slow Smart+Sloppy

1.41k views • 126 slides
From Maxent to Machine Learning and Back T. Sears ANU March 2007 T. Sears (ANU) From Maxent to

From Maxent to Machine Learning and Back T. Sears ANU March 2007 T. Sears (ANU) From Maxent to

From Maxent to Machine Learning and Back T. Sears ANU March 2007 T. Sears (ANU) From Maxent to Machine Learning and Back Maxent 2007 1 / 36 50 Years Ago . . . The principles and mathematical methods of statistical mechanics are seen to

1.24k views • 96 slides
Online and Approximation Algorithms for Bin-Packing and Knapsack Problems Xin Han Iwama Lab,

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems Xin Han Iwama Lab,

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems Xin Han Iwama Lab, School of Informatics, Kyoto University, Kyoto 606-8501, Japan Online and Approximation Algorithms for Bin-Packing and Knapsack Problems p.1/57

1.04k views • 57 slides
Chance constrained problems: penalty reformulation and performance of sample approximation

Chance constrained problems: penalty reformulation and performance of sample approximation

Chance constrained problems: penalty reformulation and performance of sample approximation technique Martin Branda, Jitka Dupa cov a Charles University in Prague Faculty of Mathematics and Physics SP XII Conference August 16-20, 2010,

793 views • 44 slides
Chance constrained problems: reformulation using penalty functions and sample approximation

Chance constrained problems: reformulation using penalty functions and sample approximation

Chance constrained problems: reformulation using penalty functions and sample approximation technique Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics 25th

577 views • 32 slides
6. Approximation and fitting Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao

6. Approximation and fitting Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao

6. Approximation and fitting Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao Tong University 2018 SJTU Ying Cui 1 / 31 Outline Norm approximation Least-norm problems Regularized approximation Robust approximation SJTU

688 views • 31 slides
On the finite element approximation of 4 th order singularly perturbed eigenvalue problems

On the finite element approximation of 4 th order singularly perturbed eigenvalue problems

On the finite element approximation of 4 th order singularly perturbed eigenvalue problems Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with D. Savvidou (UCY) and H.-G. Roos (TU-Dresden) The

962 views • 48 slides
Approximation algorithms Some optimisation problems are hard, little chance of finding

Approximation algorithms Some optimisation problems are hard, little chance of finding

Approximation algorithms Some optimisation problems are hard, little chance of finding poly-time algorithm that computes optimal solution largest clique smallest vertex cover largest independent set But: We can calculate a

701 views • 56 slides
Applications (I) Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Norm

Applications (I) Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Norm

Applications (I) Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Norm Approximation Basic Norm Approximation Penalty Function Approximation Approximation with Constraints Least-norm Problems Regularized

868 views • 59 slides
Numerical approximation for optimal control problems via MPC and HJB Giulia Fabrini Konstanz

Numerical approximation for optimal control problems via MPC and HJB Giulia Fabrini Konstanz

Numerical approximation for optimal control problems via MPC and HJB Giulia Fabrini Konstanz Women In Mathematics 15 May, 2018 G. Fabrini (University of Konstanz) Numerical approximation for OCP 1 / 33 Outline Outline Introduction and

729 views • 33 slides
Max-min and min-max approximation problems for normal matrices revisited Petr Tich Czech

Max-min and min-max approximation problems for normal matrices revisited Petr Tich Czech

Max-min and min-max approximation problems for normal matrices revisited Petr Tich Czech Academy of Sciences, University of West Bohemia joint work with Jrg Liesen TU Berlin January 30, SNA 2014, Nymburk, Czech Republic 1 Bounding

564 views • 17 slides
Approximation Algorithms for Geometric Proximity Problems: Preliminaries Introduction Convex

Approximation Algorithms for Geometric Proximity Problems: Preliminaries Introduction Convex

Approximation Algorithms for Geometric Proximity Problems: Preliminaries Introduction Convex Part II: Approximating Convex Bodies Approximations Canonical Form Quadtree-based Gold Standard New Approach David M. Mount Our Results

1.17k views • 96 slides
Self-similar solutions to extension and approximation problems Robert Young New York University

Self-similar solutions to extension and approximation problems Robert Young New York University

Self-similar solutions to extension and approximation problems Robert Young New York University (joint with Larry Guth and Stefan Wenger) April 2019 Parts of this work were supported by NSF grant DMS 1612061, the Sloan Foundation, and the

1.32k views • 54 slides
Self-similar solutions to extension and approximation problems Robert Young New York University

Self-similar solutions to extension and approximation problems Robert Young New York University

Self-similar solutions to extension and approximation problems Robert Young New York University (joint with Larry Guth and Stefan Wenger) June 2019 Parts of this work were supported by NSF grant DMS 1612061, the Sloan Foundation, and the

1.03k views • 55 slides
Local coderivatives and approximation of Hodge Laplace problems Ragnar Winther Department of

Local coderivatives and approximation of Hodge Laplace problems Ragnar Winther Department of

Local coderivatives and approximation of Hodge Laplace problems Ragnar Winther Department of Mathematics University of Oslo Norway based on joint work with Jeonghun J. Lee, UT, Austin 1 The mixed finite element method Elliptic problem:

814 views • 58 slides
Optimal trajectory approximation by cubic splines on fed-batch control problems A. Ismael F. Vaz 1

Optimal trajectory approximation by cubic splines on fed-batch control problems A. Ismael F. Vaz 1

Optimal trajectory approximation by cubic splines on fed-batch control problems A. Ismael F. Vaz 1 Eugnio C. Ferreira 2 Alzira M.T. Mota 3 1 Production and Systems Department Minho University aivaz@dps.uminho.pt 2 IBB-Institute for

644 views • 52 slides
Approximation of optimal control problems for conservation laws via vanishing viscosity and

Approximation of optimal control problems for conservation laws via vanishing viscosity and

Introduction Another approach Necessary conditions Approximation of optimal control problems for conservation laws via vanishing viscosity and relaxation Andrea Marson University of Padova Work in progress with F . Ancona (Padova) and M.

1.41k views • 94 slides
GTI Approximation Algorithms A. Ada, K. Sutner Carnegie Mellon University Spring 2018

GTI Approximation Algorithms A. Ada, K. Sutner Carnegie Mellon University Spring 2018

GTI Approximation Algorithms A. Ada, K. Sutner Carnegie Mellon University Spring 2018 Optimization Problems 1 Traveling Salesman Problem Minimizing Cost 3 There are lots of combinatorial problems that take the following form: a set I

450 views • 15 slides
Duality in a maximum generalized entropy model Shinto Eguchi Osamu Komori Atsumi Ohara MaxEnt

Duality in a maximum generalized entropy model Shinto Eguchi Osamu Komori Atsumi Ohara MaxEnt

MaxEnt, 2014 21-26 September 2014 Chateau Clos Luce, Amboise Duality in a maximum generalized entropy model Shinto Eguchi Osamu Komori Atsumi Ohara MaxEnt in ecology Presence data obtained by an ecological study

443 views • 30 slides
Local approximation algorithms for scheduling problems in sensor networks Patrik Flor een,

Local approximation algorithms for scheduling problems in sensor networks Patrik Flor een,

Local approximation algorithms for scheduling problems in sensor networks Patrik Flor een, Petteri Kaski, Topi Musto, Jukka Suomela Helsinki Institute for Information Technology HIIT Department of Computer Science University of Helsinki

195 views • 17 slides
Switching problems and related BSDE approximation Romuald ELIE CEREMADE, Universit

Switching problems and related BSDE approximation Romuald ELIE CEREMADE, Universit

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps Switching problems and related BSDE approximation Romuald ELIE CEREMADE, Universit Paris-Dauphine Joint work with J.-F.

394 views • 28 slides
Adaptive Approximation for Multivariate Linear Problems with Inputs Lying in a Cone Fred J.

Adaptive Approximation for Multivariate Linear Problems with Inputs Lying in a Cone Fred J.

Adaptive Approximation for Multivariate Linear Problems with Inputs Lying in a Cone Fred J. Hickernell Department of Applied Mathematics Center for Interdisciplinary Scientific Computation Illinois Institute of Technology hickernell@iit.edu

440 views • 30 slides

More recommend