t In a large electorate it is natural to consider voters’ preference profiles as frequency distributions over the set of all possible preferences. We assume coherence in voters’ preferences resulting in accumulation of voters preferences. We show that such distributions can be studied via superpositions of simpler so called unimodal distributions. At these, it is shown that all well-known rules choose the mode
as the outcome. We provide a set of sufficient conditions for a rule to have this trait of choosing the mode under unimodal distributions. Further we show that Condorcet consistent rules, Borda rule, plurality rule are robust under tail-perturbations of unimodal distributions.
Characterizing the Distortion of Some Simple Euclidean EmbeddingsDon Sheehy
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
The document introduces linear logic and provides examples of proofs in linear logic using natural deduction. It discusses key concepts in linear logic including linear implication (-○), contexts, the restriction that each hypothesis can only be used once, and the introduction and extraction rules for linear implication. It also covers other linear logic connectives like falsehood, AND, OR, storage (!A), and provides examples of proofs using these connectives. Finally, it discusses computations in linear logic through typed lambda calculus and a linear Lisp machine that avoids garbage collection by not allowing data sharing.
Prosody is an essential component of human speech. Prosody, broadly, describes all of the production qualities of speech that are not involved in conveying lexical information. Where the words are “what is said”, prosody is “how it is said”. Prosody of speech, plays an important role not only in communicating the syntax, semantics and pragmatics of spoken language, but also in conveying information about the speaker and their internal state (e.g. emotion or fatigue).
Understanding prosody is critical to understanding speech communication. Spoken language processing (SLP) technology that approaches human levels of competence will necessarily include automatic analysis of prosody. Despite the importance of prosody in spoken communication, researchers are often unable to reliably incorporate prosodic information into applications. One explanation is a lack of compact, consistent, and universal representations of prosodic information. This talk will describe the state of the art in prosodic analysis and its use in spoken language processing with a focus on the development of new representations of prosody.
This document discusses rule-based systems and logic programming. It contains the following key points in 3 sentences:
The document introduces rule-based systems that represent knowledge as IF-THEN rules and facts, and describes forward chaining which applies rules to derive new facts from initial facts, and backward chaining which starts with a goal and looks for rules to prove the goal. It explains Horn clause logic and how Prolog implements backward chaining using Horn clauses. It also discusses how forward chaining can be used to dynamically add facts to a knowledge base and apply rules to derive new facts.
This document discusses inference in first-order logic and various proof strategies. It begins by describing a general proof procedure that uses binary resolution and represents proofs as trees. It then discusses different proof strategies like unit preference, set of support strategy, input resolution, linear resolution, and SLD-resolution. SLD-resolution is described as a sound and complete proof procedure for definite clauses. The document also introduces the concepts of non-monotonic reasoning and default reasoning, describing both non-monotonic logic and default logic as approaches to modeling this type of reasoning.
This document discusses constraint satisfaction problems (CSPs) and backtracking search algorithms for solving CSPs. It defines a CSP as consisting of variables with domains of possible values, and constraints specifying allowed value combinations. Backtracking search involves trying value assignments for variables and backtracking when constraints are violated. The document presents examples of CSP problems and maps coloring CSP. It also describes techniques like minimum remaining values and least constraining value heuristics to improve search efficiency in backtracking.
Asslam o alaikum dear students, my name is Nadeem Altaf. I am from Pakistan. I am a student & there isan topic about Graeco Latin Square Design and Other designs
Characterizing the Distortion of Some Simple Euclidean EmbeddingsDon Sheehy
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
The document introduces linear logic and provides examples of proofs in linear logic using natural deduction. It discusses key concepts in linear logic including linear implication (-○), contexts, the restriction that each hypothesis can only be used once, and the introduction and extraction rules for linear implication. It also covers other linear logic connectives like falsehood, AND, OR, storage (!A), and provides examples of proofs using these connectives. Finally, it discusses computations in linear logic through typed lambda calculus and a linear Lisp machine that avoids garbage collection by not allowing data sharing.
Prosody is an essential component of human speech. Prosody, broadly, describes all of the production qualities of speech that are not involved in conveying lexical information. Where the words are “what is said”, prosody is “how it is said”. Prosody of speech, plays an important role not only in communicating the syntax, semantics and pragmatics of spoken language, but also in conveying information about the speaker and their internal state (e.g. emotion or fatigue).
Understanding prosody is critical to understanding speech communication. Spoken language processing (SLP) technology that approaches human levels of competence will necessarily include automatic analysis of prosody. Despite the importance of prosody in spoken communication, researchers are often unable to reliably incorporate prosodic information into applications. One explanation is a lack of compact, consistent, and universal representations of prosodic information. This talk will describe the state of the art in prosodic analysis and its use in spoken language processing with a focus on the development of new representations of prosody.
This document discusses rule-based systems and logic programming. It contains the following key points in 3 sentences:
The document introduces rule-based systems that represent knowledge as IF-THEN rules and facts, and describes forward chaining which applies rules to derive new facts from initial facts, and backward chaining which starts with a goal and looks for rules to prove the goal. It explains Horn clause logic and how Prolog implements backward chaining using Horn clauses. It also discusses how forward chaining can be used to dynamically add facts to a knowledge base and apply rules to derive new facts.
This document discusses inference in first-order logic and various proof strategies. It begins by describing a general proof procedure that uses binary resolution and represents proofs as trees. It then discusses different proof strategies like unit preference, set of support strategy, input resolution, linear resolution, and SLD-resolution. SLD-resolution is described as a sound and complete proof procedure for definite clauses. The document also introduces the concepts of non-monotonic reasoning and default reasoning, describing both non-monotonic logic and default logic as approaches to modeling this type of reasoning.
This document discusses constraint satisfaction problems (CSPs) and backtracking search algorithms for solving CSPs. It defines a CSP as consisting of variables with domains of possible values, and constraints specifying allowed value combinations. Backtracking search involves trying value assignments for variables and backtracking when constraints are violated. The document presents examples of CSP problems and maps coloring CSP. It also describes techniques like minimum remaining values and least constraining value heuristics to improve search efficiency in backtracking.
Asslam o alaikum dear students, my name is Nadeem Altaf. I am from Pakistan. I am a student & there isan topic about Graeco Latin Square Design and Other designs
This document contains examples and exercises related to logic and propositions. It introduces topics like propositional logic, truth tables, predicates, quantification, and preconditions/postconditions. Some key points:
- Propositions are statements that can be either true or false. Truth tables can be used to determine the truth value of compound propositions.
- Predicates are statements with variables that can be evaluated as true or false by assigning values to the variables.
- Quantifiers like "for all" (universal) and "there exists" (existential) are used to specify the scope of predicates being true.
- Preconditions and postconditions describe the valid inputs and expected outputs of programs to verify
This first lecture describes what EMT is. Its history of evolution. Main personalities how discovered theories relating to this theory. Applications of EMT . Scalars and vectors and there algebra. Coordinate systems. Field, Coulombs law and electric field intensity.volume charge distribution, electric flux density, gauss's law and divergence
1. The document discusses filters, including analog filters that process continuous-time signals and digital filters that process discrete-time signals. It also discusses different types of filters like lowpass, highpass, bandstop, and bandpass filters.
2. Active filters are described as overcoming some of the drawbacks of passive filters by using op-amps instead of inductors. While they have some disadvantages like limited bandwidth, active filters are commonly used for voice and data communications due to economic and performance advantages over passive filters.
3. Bode plots are introduced as important tools for analyzing the frequency response of filters by plotting magnitude or phase versus frequency on logarithmic scales. Corner frequency is defined as the frequency where the
Constraint satisfaction problems (CSPs) define states as assignments of variables to values from their domains, with constraints specifying allowable combinations. Backtracking search assigns one variable at a time using depth-first search. Improved heuristics like most-constrained variable selection and least-constraining value choice help. Forward checking and constraint propagation techniques like arc consistency detect inconsistencies earlier than backtracking alone. Local search methods like min-conflicts hill-climbing can also solve CSPs by allowing constraint violations and minimizing them.
This document discusses convolutional codes. It defines basic concepts like constraint length and generator polynomials that define convolutional codes. It describes representations of convolutional codes using state diagrams and trellis diagrams. It also discusses decoding convolutional codes using the Viterbi algorithm, which finds the most likely path through the trellis. The document concludes by discussing properties of convolutional codes like free distance, which is the minimum Hamming distance between codewords.
Elements of Inference covers the following concepts and takes off right from where we left off in the previous slide https://www.slideshare.net/GiridharChandrasekar1/statistics1-the-basics-of-statistics.
Population Vs Sample (Measures)
Probability
Random Variables
Probability Distributions
Statistical Inference – The Concept
The document summarizes an edge-based finite-volume scheme for solving partial differential equations on triangular grids that achieves third-order accuracy without using curved elements. It presents a new boundary closure formula that allows the scheme to maintain third-order accuracy at boundary nodes. Numerical results are shown for the Burgers' equation and advection-diffusion equation on irregular triangular grids, demonstrating third-order convergence of errors at both interior and boundary nodes using the new boundary formula.
Variational inference is a technique for estimating Bayesian models that provides similar precision to MCMC at a greater speed, and is one of the main areas of current research in Bayesian computation. In this introductory talk, we take a look at the theory behind the variational approach and some of the most common methods (e.g. mean field, stochastic, black box). The focus of this talk is the intuition behind variational inference, rather than the mathematical details of the methods. At the end of this talk, you will have a basic grasp of variational Bayes and its limitations.
This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.
This document discusses theorem proving in propositional and first-order logic. It covers topics like:
- Propositional logic, including syntax, semantics, and techniques like truth tables, natural deduction, resolution, and DPLL algorithms.
- Converting propositional formulas to conjunctive normal form.
- Applications of theorem proving like hardware and software verification.
- First-order logic, including its syntax of terms, formulas, and semantics based on interpretations and variable assignments in a domain.
The document provides an overview of fundamental concepts and techniques in automated theorem proving at the propositional and first-order levels. It also mentions applications and references for further study.
This document discusses analysis of variance (ANOVA) and chi-square tests. It covers F tests, one-way and two-way ANOVA, assumptions of ANOVA, and how to perform chi-square goodness of fit and independence tests. Examples are provided for variance ratio F tests, one-way ANOVA, two-way ANOVA, and chi-square tests. Limitations of chi-square tests include requiring independent observations and minimum expected frequencies of 5 per cell.
This document discusses theorem proving and its applications. It covers several topics:
1. Theorem proving can be useful for mathematics, hardware verification, safety-critical systems, automated planning, and cryptanalysis. Early implementations included proving evenness of sums.
2. Propositional logic is a simple but important logic that is decidable and can model many problems. Techniques for checking tautologies include truth tables, natural deduction, and converting to SAT problems solved by SAT solvers.
3. Modern SAT solvers use techniques like conflict-driven clause learning and non-chronological backtracking to efficiently handle large problems from applications like electronic design automation.
Lec10: Medical Image Segmentation as an Energy Minimization ProblemUlaş Bağcı
Enhancement, Noise Reduction, and Signal Processing • MedicalImageRegistration • MedicalImageSegmentation • MedicalImageVisualization • Machine Learning in Medical Imaging • Shape Modeling/Analysis of Medical Images Deep Learning in Radiology Fuzzy Connectivity (FC) – Affinity functions • Absolute FC • Relative FC (and Iterative Relative FC) • Successful example applications of FC in medical imaging • Segmentation of Airway and Airway Walls using RFC based method
Energyfunctional
– Data and Smoothness terms
• GraphCut – Min cut
– Max Flow
• ApplicationsinRadiologyImages
1. The document discusses various set operations including union, intersection, difference, symmetric difference, complement, and disjoint sets.
2. Key properties and definitions of each operation are provided, such as the formal definition of union as the set of elements in either set and the definition of intersection as the set of elements common to both sets.
3. Examples are used to illustrate each operation, and proofs of set identities are demonstrated using set builder notation, membership tables, and showing subsets.
The document describes a randomized complete block design (RCBD) experimental method. RCBD involves comparing treatments (e.g. fertilizers) applied to experimental units (e.g. corn crops) grouped into blocks (e.g. fields). Treatments are randomly assigned to experimental units within each block. RCBD controls for variability between blocks (e.g. differences in soil between fields) to isolate the effect of treatments. It provides more precise results than a completely randomized design when blocks are homogeneous within and heterogeneous between.
Extended Isolation Forest was proposed to address inconsistencies in anomaly scores produced by standard Isolation Forest. It uses hyperplanes with random slopes for branching at each split in the isolation trees, rather than restricting splits to be axis-parallel. This allows it to better represent data structure and density, producing score maps free of artifacts. Empirical results on both synthetic and real data demonstrated Extended Isolation Forest provides more reliable, robust anomaly scores with lower variance compared to standard Isolation Forest.
Okay, here are the steps:
1) Given:
2) Transform into spherical unit vectors:
3) Write in terms of spherical components:
So the vector components in spherical coordinates are:
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
More Related Content
Similar to Frequency Based Analysis of Voting Rules
This document contains examples and exercises related to logic and propositions. It introduces topics like propositional logic, truth tables, predicates, quantification, and preconditions/postconditions. Some key points:
- Propositions are statements that can be either true or false. Truth tables can be used to determine the truth value of compound propositions.
- Predicates are statements with variables that can be evaluated as true or false by assigning values to the variables.
- Quantifiers like "for all" (universal) and "there exists" (existential) are used to specify the scope of predicates being true.
- Preconditions and postconditions describe the valid inputs and expected outputs of programs to verify
This first lecture describes what EMT is. Its history of evolution. Main personalities how discovered theories relating to this theory. Applications of EMT . Scalars and vectors and there algebra. Coordinate systems. Field, Coulombs law and electric field intensity.volume charge distribution, electric flux density, gauss's law and divergence
1. The document discusses filters, including analog filters that process continuous-time signals and digital filters that process discrete-time signals. It also discusses different types of filters like lowpass, highpass, bandstop, and bandpass filters.
2. Active filters are described as overcoming some of the drawbacks of passive filters by using op-amps instead of inductors. While they have some disadvantages like limited bandwidth, active filters are commonly used for voice and data communications due to economic and performance advantages over passive filters.
3. Bode plots are introduced as important tools for analyzing the frequency response of filters by plotting magnitude or phase versus frequency on logarithmic scales. Corner frequency is defined as the frequency where the
Constraint satisfaction problems (CSPs) define states as assignments of variables to values from their domains, with constraints specifying allowable combinations. Backtracking search assigns one variable at a time using depth-first search. Improved heuristics like most-constrained variable selection and least-constraining value choice help. Forward checking and constraint propagation techniques like arc consistency detect inconsistencies earlier than backtracking alone. Local search methods like min-conflicts hill-climbing can also solve CSPs by allowing constraint violations and minimizing them.
This document discusses convolutional codes. It defines basic concepts like constraint length and generator polynomials that define convolutional codes. It describes representations of convolutional codes using state diagrams and trellis diagrams. It also discusses decoding convolutional codes using the Viterbi algorithm, which finds the most likely path through the trellis. The document concludes by discussing properties of convolutional codes like free distance, which is the minimum Hamming distance between codewords.
Elements of Inference covers the following concepts and takes off right from where we left off in the previous slide https://www.slideshare.net/GiridharChandrasekar1/statistics1-the-basics-of-statistics.
Population Vs Sample (Measures)
Probability
Random Variables
Probability Distributions
Statistical Inference – The Concept
The document summarizes an edge-based finite-volume scheme for solving partial differential equations on triangular grids that achieves third-order accuracy without using curved elements. It presents a new boundary closure formula that allows the scheme to maintain third-order accuracy at boundary nodes. Numerical results are shown for the Burgers' equation and advection-diffusion equation on irregular triangular grids, demonstrating third-order convergence of errors at both interior and boundary nodes using the new boundary formula.
Variational inference is a technique for estimating Bayesian models that provides similar precision to MCMC at a greater speed, and is one of the main areas of current research in Bayesian computation. In this introductory talk, we take a look at the theory behind the variational approach and some of the most common methods (e.g. mean field, stochastic, black box). The focus of this talk is the intuition behind variational inference, rather than the mathematical details of the methods. At the end of this talk, you will have a basic grasp of variational Bayes and its limitations.
This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.
This document discusses theorem proving in propositional and first-order logic. It covers topics like:
- Propositional logic, including syntax, semantics, and techniques like truth tables, natural deduction, resolution, and DPLL algorithms.
- Converting propositional formulas to conjunctive normal form.
- Applications of theorem proving like hardware and software verification.
- First-order logic, including its syntax of terms, formulas, and semantics based on interpretations and variable assignments in a domain.
The document provides an overview of fundamental concepts and techniques in automated theorem proving at the propositional and first-order levels. It also mentions applications and references for further study.
This document discusses analysis of variance (ANOVA) and chi-square tests. It covers F tests, one-way and two-way ANOVA, assumptions of ANOVA, and how to perform chi-square goodness of fit and independence tests. Examples are provided for variance ratio F tests, one-way ANOVA, two-way ANOVA, and chi-square tests. Limitations of chi-square tests include requiring independent observations and minimum expected frequencies of 5 per cell.
This document discusses theorem proving and its applications. It covers several topics:
1. Theorem proving can be useful for mathematics, hardware verification, safety-critical systems, automated planning, and cryptanalysis. Early implementations included proving evenness of sums.
2. Propositional logic is a simple but important logic that is decidable and can model many problems. Techniques for checking tautologies include truth tables, natural deduction, and converting to SAT problems solved by SAT solvers.
3. Modern SAT solvers use techniques like conflict-driven clause learning and non-chronological backtracking to efficiently handle large problems from applications like electronic design automation.
Lec10: Medical Image Segmentation as an Energy Minimization ProblemUlaş Bağcı
Enhancement, Noise Reduction, and Signal Processing • MedicalImageRegistration • MedicalImageSegmentation • MedicalImageVisualization • Machine Learning in Medical Imaging • Shape Modeling/Analysis of Medical Images Deep Learning in Radiology Fuzzy Connectivity (FC) – Affinity functions • Absolute FC • Relative FC (and Iterative Relative FC) • Successful example applications of FC in medical imaging • Segmentation of Airway and Airway Walls using RFC based method
Energyfunctional
– Data and Smoothness terms
• GraphCut – Min cut
– Max Flow
• ApplicationsinRadiologyImages
1. The document discusses various set operations including union, intersection, difference, symmetric difference, complement, and disjoint sets.
2. Key properties and definitions of each operation are provided, such as the formal definition of union as the set of elements in either set and the definition of intersection as the set of elements common to both sets.
3. Examples are used to illustrate each operation, and proofs of set identities are demonstrated using set builder notation, membership tables, and showing subsets.
The document describes a randomized complete block design (RCBD) experimental method. RCBD involves comparing treatments (e.g. fertilizers) applied to experimental units (e.g. corn crops) grouped into blocks (e.g. fields). Treatments are randomly assigned to experimental units within each block. RCBD controls for variability between blocks (e.g. differences in soil between fields) to isolate the effect of treatments. It provides more precise results than a completely randomized design when blocks are homogeneous within and heterogeneous between.
Extended Isolation Forest was proposed to address inconsistencies in anomaly scores produced by standard Isolation Forest. It uses hyperplanes with random slopes for branching at each split in the isolation trees, rather than restricting splits to be axis-parallel. This allows it to better represent data structure and density, producing score maps free of artifacts. Empirical results on both synthetic and real data demonstrated Extended Isolation Forest provides more reliable, robust anomaly scores with lower variance compared to standard Isolation Forest.
Okay, here are the steps:
1) Given:
2) Transform into spherical unit vectors:
3) Write in terms of spherical components:
So the vector components in spherical coordinates are:
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
Similar to Frequency Based Analysis of Voting Rules (20)
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
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and the revised UWWTD (Urban Waste Water Treatment Directive)”
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With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
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The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
4. Motivation
• Think about voting with a large electorate: nation wide
elections.
• Voters’ preferences are diverse.
• But there is some coherence between these voters’
preferences.
• Coherence is expressed in one or several accumulations of
voters’ preferences.
3
5. Motivation
Preferences p q p+q
abc 100 80 180
acb 80 60 140
bac 80 100 180
cab 60 20 80
bca 60 80 140
cba 20 60 80
Visual
• Preference profiles are represented by frequency
distributions. Each accumulation in the distribution can
be seen as an agglomeration of preferences around a
local mode.
• The whole profile: addition or superposition of such
(local) unimodal distributions yielding a multimodal
distribution.
• Unimodal distributions =⇒ multimodal distributions.
4
6. Motivation
• Admissible set of preferences: the set of linear orders over
all candidates. A profile is a frequency distribution on this
set of linear orders.
• This set is structured by the Kemeny distance.
• Kemeny distance: minimal path-length needed to convert
one order into another by swapping consecutively ordered
candidates.
eg.
5
7. Motivation
• A unimodal distribution: mode has highest frequency and
further frequency decreases with the distance from the
mode.
• Multimodal distributions: additions of such unimodal
distributions.
• Choosing the mode as the collective order at unimodal
distribution.
• Multimodal distributions: natural to infer the outcome
from the intersection of the local modes.
6
8. Connection with Existing Literature
• Some empirical papers: impartial culture or uniform
distribution.
• Gehrlein (2006), Riker (1982, p. 122): calculates the
probability of a majority cycle occurring. Cycle
• Criticism of the impartial culture.
• Grofman et al. (2003): impartial culture is a worst-case
scenario.
7
9. Connection with Existing Literature
• Other probabilistic models: Multinomial likelihood models
(Gillett, 1976, 1978), Dual Culture (Gehrlein; 1978), Maximal
Culture Condition (Fishburn and Gehrlein, 1977).
• Message: when voters’ preferences are homogeneous,
there is an increased likelihood that a Pairwise Majority
Rule Winner exists meaning that Condorcet cycles are
avoided.
• Our results: at unimodal distributions not only pairwise
majority cycles are absent, but also all well-known rules
have the same collective order: the mode.
8
10. Connection with Existing Literature
• Merlin et. al. (2004): probability of conflicts in a U.S.
presidential type of election.
• Main difference: distributions are structured by the
Kemeny distance – A structure that therewith stems from
the preferences.
9
11. Our Approach
• Starting with observing the behaviour of several
well-known rules like Condorcet consistent rules, Scoring
rules, Elimination rules, under unimodal distributions.
• Common trait: choose the mode at unimodal distributions.
• We provide sufficient conditions for choosing the mode at
unimodal distribution.
C B E
10
12. Our Approach
• Investigate the robustness of this result under
perturbations.
• We show that even under this tail-perturbed distributions
Condorcet consistent rules, Borda rule and plurality rule
choose the mode as the outcome.
P
11
14. Model
• N: finite but large set of voters, and A: finite set of
candidates, |A| ≥ 3.
• R ⊆ A × A: complete, anti-symmetric and transitive
relations on A.
• complete: for any a, b ∈ A either a ≽ b or b ≽ a.
• anti-symmetric: a ≽ b and b ≽ a implies a ≈ b.
• transitive: a ≽ b and b ≽ c implies a ≽ c.
• L: set of these linear orders on A.
• τxyR: the preference relation where the positions of x and
y are swapped.
• Ex: τadabcd = dbca.
12
15. Model
• p: voter i’s preference (linear order) p(i) ∈ L. p ∈ LN.
• Ties in the collective orders → collective orders are weak
orders, i.e. complete and transitive orders on A.
• W: the set of all these weak orders.
13
16. Model
• Lxy: set of linear orders R at which x is strictly preferred to
y, that is Lxy = {R ∈ L|(x, y) ∈ R}.
• Ex: For 3 candidates a, b, c, Lab = {abc, acb, cab} and
Lba = {cba, bca, bac}.
• The collective decision is formalized by preference rule.
• F: a function that assigns to every profile p in LN a
collective preference F(p) in W.
14
17. Frequency Distribution
• A frequency distribution: a function representing the
number of times each linear order appears in a profile.
• Given a profile p and a preference R in L, then f(R, p)
denotes the number of voters with preference R at profile
p, that is
f(R, p) = |{i ∈ N | p(i) = R}|.
Preferences p
abc 100
acb 80
bac 80
cab 60
bca 60
cba 20
• f(bac, p) = 80.
15
18. Frequency Distribution
• There is a metric space over L induced by the Kemeny
distance function d. For two preferences R1 and R2,
d(R1, R2) = 1
2|(R1 R2) ∪ (R2 R1)|. If R1 and R2 are linear
orders, then d(R1, R2) = |(R1 R2)|.
• Ex: d(abc, bca)
= |({(a, b), (a, c), (b, c)} {(b, a), (c, a), (b, c)})|
= |{(a, b), (a, c)}|
= 2
back
16
19. • A profile p is called unimodal if there exists a preference
R, the mode, such that for every two preferences R1 and R2
in L, if d(R, R1) < d(R, R2), then we have f(R1, p) > f(R2, p).
• Symmetric unimodal: if in addition f(R1, p) = f(R2, p)
whenever d(R, R1) = d(R, R2).
17
30. Intuition Behind the Analysis
R R τabR
. . .
. a b
. . .
a . .
. . .
. . .
b . .
. b a
. . .
. . .
• R ↔ τabR. R ∈ Lab and τabR ∈ Lba.
• In pairwise comparison a will beat b.
• The mode consists of all pairs which in pairwise
comparisons beat each other: the Condorcet order.
27
31. Intuition Behind the Analysis
• The same argument =⇒ at any score rule a gets a higher
score than b and that b is eliminated before a.
• Mode: “positional” order + order of “elimination”.
28
33. Monotonicity
p
a b d c a …
b c b d c …
c d c b d …
d a a a b …
q
a d d c a …
d c b d c …
c b c b d …
b a a a b …
• Monotonicity: (d, b) ∈ F(p) implies (d, b) ∈ F(q).
• In this way monotonicity is defined pairwise but it is
sensitive to the positions of candidates. It is therefore
satisfied by many rules.
30
34. Monotonicity
p
a b d c a …
b c b d c …
c d c b d …
d a a a b …
q
d d d c a …
a c b d c …
c b c b d …
b a a a b …
• Monotonicity: (d, b) ∈ F(p) does not imply (d, b) ∈ F(q).
• In this way monotonicity is defined pairwise but it is
sensitive to the positions of candidates. It is therefore
satisfied by many rules.
30
35. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p)
31
36. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
31
37. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
31
38. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
• positive discrimination: (a, b) ∈ F(p)
31
39. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
• positive discrimination: (a, b) ∈ F(p), (b, c) ∈ F(p),
31
40. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
• positive discrimination: (a, b) ∈ F(p), (b, c) ∈ F(p),
(a, c) ∈ F(p).
31
41. First Result
Theorem
Let p be a unimodal profile with mode R. Then F(p) = R for a
rule F from LN to W in each of the following two cases:
1. F is positively discriminating;
2. F is anonymous, neutral, monotone and discriminating.
32
44. Tail-perturbed Distributions
There is a linear order, say R∗, and a real number ν such that
1. Frequencies are constant at any given distance from R∗.
So, f(R1, p) = f(R2, p) whenever d(R1, R∗) = d(R2, R∗) for all
linear orders R1 and R2.
33
45. Tail-perturbed Distributions
There is a linear order, say R∗, and a real number ν such that
1. Frequencies are constant at any given distance from R∗.
So, f(R1, p) = f(R2, p) whenever d(R1, R∗) = d(R2, R∗) for all
linear orders R1 and R2.
2. For linear orders R1, R2 and R3, with
d(R1, R∗) < d(R2, R∗) ν < d(R3, R∗) we have
f(R1, p) > f(R2, p) > f(R3, p).
33
46. Tail-perturbed Distributions
• δ =
(m
2
)
: diameter distance of L.
• ρ = 1
2δ: radius distance.
• ρ ν: perturbations only occur in the second half of the
set of linear orders.
• Ex: For an election with 4 (i.e.m = 4) candidates,
δ =
(4
2
)
= 6 and ρ = 3.
34
47. Condorcet Consistent Rules
• Condorcet consistent rules depend on pairwise majority
comparisons of the candidates.
• may yield cycles and break cycles differently.
• If, however, at a certain profile pairwise majority
comparisons yield a complete, strict and transitive order
from overall winner (the Condorcet winner) to overall loser
(the Condorcet loser), then this is the outcome of all these
rules at that profile.
C
35
49. Perturbation and Condorcet Consistent Rules
• Frequency doesn’t only depend on the distance to R∗.
• First half: frequency declines with the distance to R∗.
Second half: frequency at distance, say δ − k, is bounded
by the minimum frequency at the ’opposite’ distance k.
36
50. Perturbation and Condorcet Consistent Rules
We prove that R∗ is the Condorcet order if frequencies satisfy
min
R∈Lk
xy
f(R) + min
R∈Lδ−k
xy
f(R) > max
R∈Lk
yx
f(R) + max
R∈Lδ−k
yx
f(R)
for all 0 ≤ k < ρ.
37
51. Borda Rule
• Borda rule is based on Borda score.
• Borda score of a candidate x is the number of candidates
below x in a preference R, summed over all preferences.
• In the collective order we rank the candidates according to
their Borda score.
B
38
53. Perturbation and Borda Rule
• Frequency depends on the distance to R∗.
• At distances larger than radius distance, say δ − k,
frequency is lower than at the ’opposite’ distance k. 39
54. Perturbation and Borda Rule
We prove that R∗ is the chosen order by Borda rule if
frequencies satisfy
f(k) > f(δ − k) for all 0 ≤ k < ρ.
40
55. Plurality Rule
• Plurality rule is based on Plurality score.
• That is the number of times a candidate is at the top of
voter’s preferences.
• In the collective order we rank the candidates according to
their Plurality score.
P
41
57. Perturbation and Plurality Rule
• Frequency depends on the distance to R∗.
• First half: ν = 1
2
(m−1
2
)
+ m − 3
2 > ρ = 1
2
(m
2
)
.
42
58. Perturbation and Plurality Rule
We prove that R∗ is the chosen order by Plurality rule if the
distribution is ν-tail perturbed unimodal, with
ν = 1
2
(m−1
2
)
+ m − 3
2
43
60. Multimodal Distribution
• For discriminating collective decision rules: the outcome
at the union of two unimodal distributions is in the
intersection of the two modes of these unimodal
distributions.
• Ex: Intersection of abc and bac: set of concordant pairs:
{(a, c), (b, c)}.
• Generalizes to more than two unimodal distributions.
• Difficult situations: empty intersection.
45
62. Conclusion
• Recognized a common trait in all well-known collective
decision rules.
• Extended this result outside the domain of unimodal
distribution.
• As Borda rule and Plurality rule are “opposite extremes” in
the class of score rules, we expect that a large subclass of
score rules also choose the mode at such ν−tail
perturbed unimodal distribution.
• Predictability of the outcome of multimodal distributions.
46
64. Pairwise Majority Cycle
p
a b c
b c a
c a b
1 1 1
Candidate
Candidate
a b c
a - 2 1
b 1 - 2
c 2 1 -
Table 1: Occurrence of a Cycle
(a, b) : (b, c) : (c, a): Cycle
q
a a c
b c a
c b b
1 1 1
Candidate
Candidate
a b c
a - 3 2
b 0 - 1
c 1 2 -
Table 2: Non-occurrence of a Cycle
(a, b) : (a, c) : (c, b) =⇒ acb back
48
65. Condorcet Consistent Rules
Copeland Rule:
p
a a c
b c a
c b b
1 1 1
Candidate
Candidate
a b c
a - 3 2
b 0 - 1
c 1 2 -
Candidate Score
a 2
b 0
c 1
F(p) = acb.
(a, b)(a, c)(c, b) =⇒ acb.
back back to CR
49
66. Borda Rule
Borda Rule:
p
a a c
c c b
b d d
d b a
1 1 1
Candidate
Candidate
a b c d Sum
a - 2 2 2 6
b 1 - 0 2 3
c 1 3 - 3 7
d 1 1 0 - 2
Candidate Score Rank
a 6 2
b 3 3
c 7 1
d 2 4
(a, b) : (a, c) : (a, d) : (b, d) : (c, b) : (c, d) =⇒ acbd.
F(p) = cabd.
back back to BR
50
67. Plurality Rule
Plurality Rule:
p
b c d
a a a
c b b
d d c
1 1 1
Candidate
Candidate
a b c d
a - 2 2 2
b 1 - 2 2
c 1 1 - 2
d 1 1 1 -
Candidate Score Rank
a 0 2
b 1 1
c 1 1
d 1 1
(a, b) : (a, c) : (a, d) : (b, c) : (b, d) : (c, d) =⇒ abcd.
F(p) = (b − c − d)a.
back back to PR
51
68. Elimination Rules
Coombs Rule:
p
b c d
a a a
c b b
d d c
1 1 1
Round 1 Scores
a 3
b 3
c 2
d 1
d eliminated.
p
b c a
a a b
c b c
1 1 1
Round 2 Scores
a 3
b 2
c 1
c eliminated.
p
b a a
a b b
1 1 1
Round 3 Scores
a 2
b 1
b eliminated.
F(p) = abcd.
(a, b) : (a, c) : (a, d) : (b, c) : (b, d) : (c, d) =⇒ abcd. back 52