The Fundamental Principle of Data Science
|
|
|
- Arnold Lewis
- 6 years ago
- Views:
Transcription
1 The Fundamental Principle of Data Science Harry Crane Department of Statistics Rutgers May 7, 2018 Web : Project : researchers.one Contact Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
2 Overview: Foundations of Data Science Burning Questions What is Data Science? What is a Foundation? What is the purpose of a Foundation? Acknowledgments FPDS based on ongoing collaboration with Ryan Martin. Also very grateful for conversations with Glenn Shafer, Dimitris Tsementzis, and Snow Zhang. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
3 Main References H. Crane. (2018). Logic of probability and conjecture. H. Crane. (2018). Probabilistic Foundations of Statistical Network Analysis. Chapman Hall. Released May 2018 Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
4 Foundations of Probability Axioms/ Principles encodes Formal expresses Basic discerns Theory Concept Pre-formal Intuition Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
5 Foundations of Probability Axioms/ Principles encodes Formal expresses Basic discerns Theory Concept Pre-formal Intuition Frequentist Probability Kolmogorov s Axioms Frequentist Probability Frequencies Chance/ Randomness Bayesian Probability de Finetti s coherence Subjective Probability Prices of fair bets Credence Dempster Shafer Probability Axioms of Belief Functions D Shafer Theory Measure of evidence Degrees of Belief Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
6 Foundations of Statistics Axioms/ Principles encodes Formal expresses Basic supports Theory Concept Inference Methods Frequentist Statistics Kolmogorov s Axioms Frequentist Probability Frequencies P-values, Conf intervals, etc. Bayesian Statistics de Finetti s coherence Subjective Probability Prices of fair bets Bayes Factor, Posterior inf Inferential Models Axioms of Belief Functions D Shafer Theory Measure of evidence Valid prob. inference Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
7 Instances of Data Science Data Method Inference Rule Decision/ Application Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
8 Instances of Data Science Data Method Inference Rule Decision/ Application Hypothesis testing Set of Observations P-value Reject H 0 Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
9 Instances of Data Science Data Method Inference Rule Decision/ Application Hypothesis testing Set of Observations P-value Reject H 0 Interaction data Network science Network analysis Advertising campaign Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
10 Instances of Data Science Data Method Inference Rule Decision/ Application Hypothesis testing Set of Observations P-value Reject H 0 Interaction data Network science Network analysis Clustering/Classification Advertising campaign Cloud of Points Clustering Fraud Detection Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
11 The F-word Data Method Inference Rule Decision Methods Hypothesis tests, Bayesian methods, confidence distributions, fiducial distributions, machine learning are not Foundations. Foundations cannot only be based on statistics and probability. Algorithmic methods (clustering, network science, machine learning) are data science. Inferential Models based on Dempster Shafer theory not strictly probability but still statistics/data science. Foundations of Data Science should handle all kinds of (complex) data not just sets, sequences, tables, arrays, etc. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
12 The F-word Data Method Inference Rule Decision Methods Hypothesis tests, Bayesian methods, confidence distributions, fiducial distributions, machine learning are not Foundations. Foundations cannot only be based on statistics and probability. Algorithmic methods (clustering, network science, machine learning) are data science. Inferential Models based on Dempster Shafer theory not strictly probability but still statistics/data science. Foundations of Data Science should handle all kinds of (complex) data not just sets, sequences, tables, arrays, etc. Role of a Foundation: Internal (Logical): Provides a standard of coherence by which to assess methods. Frequentist statistics: ad hoc principles/rules of thumb. Bayesian statistics: coherence/dutch book argument. Inferential Models (Martin & Liu): validity as standard of rigor. External (Scientific): Provides a standard of falsification for inferences. Inferences are meaningful/scientific only if they are falsifiable. Any method for data science should have built-in criteria under which inferences would be falsified. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
13 Principles of Data Science Data Method Inference Rule Decision Basic Concepts of Data Science: 1 Fundamental Principle of Data Science (Crane Martin): A Method is a protocol which takes any piece of data to an inference about every assertion A. Method : Data A Inference(A). 2 Identical data structures should lead to identical inferences: D = Data D = A Method D (A) = Inference(A) Method D (A). How to make these principles rigorous? Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
14 Simple Example: Hypothesis testing (frequentist) Data Method Inference Rule Decision If the data provides sufficient evidence in favor of A, then inferences should support A. Hypothesis testing: H 0 : null hypothesis Method: compute a P-value based on the data D: P = Pr(D H 0 ). Interpret P < α as evidence against H 0 (i.e., reject H 0 ). Inference based on P: { reject H0, P < α, Method D (H 0 ) = inconclusive, otherwise. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
15 Simple Example: Hypothesis testing (Bayes) Data Method Inference Rule Decision If the data provides sufficient evidence in favor of A, then inferences should support A. Hypothesis testing: H 0 : null hypothesis H 1 : alternative hypothesis Method: compute a Bayes factor based on the data D: BF = Pr(D H 1) Pr(D H 0 ). Interpret BF as measure of evidence in favor of H 1 over H 0 (denoted as A). Inference based on P (from Kass & Raftery, 1995): inconclusive, BF < 3.2, substantial, 3.2 BF 10, Method D (A) = strong, 10 < BF 100, decisive, BF > 100. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
16 Formalization: Data as Structure Data Method Inference Rule Decision Informal: Data is observable structure, including measurements, relations between measurements (interactions, dependencies), etc. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
17 Formalization: Data as Structure Data Method Inference Rule Decision Informal: Data is observable structure, including measurements, relations between measurements (interactions, dependencies), etc. Formal: Represent Data as a topological space whose points consist of observable information (observable data ) and the relations between them. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
18 Formalization: The structure of evidence Data Method Inference Rule Decision Informal: Inferences are based on the body of evidence in support of a claim. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
19 Formalization: The structure of evidence Data Method Inference Rule Decision Informal: Inferences are based on the body of evidence in support of a claim. Formal: Represent the body of evidence for A as a topological space Evid(A) whose points are pieces of evidence in favor of A and paths are relations between pieces of evidence. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
20 Formalization: The logical structure of inference Data Method Inference Rule Decision Informal: An inference is a judgment about evidential strength (based on data). Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
21 Formalization: The logical structure of inference Data Method Inference Rule Decision Informal: An inference is a judgment about evidential strength (based on data). Formal: Represent Inference(A) as the disjoint union of spaces whose points are evidence in favor of A, in favor of A, or that the outcome is inconclusive. Inference(A) Evid(A) Evid( A) Inconclusive(A). Can account for strength of evidence (strong, suggestive) but simple case here. Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
22 The Logic of Evidence: Syntax (Crane, 2018) (1) For any assertion A there is a space of evidence for that assertion: A Space Evid(A) Space (2) If A has been proven, then there is evidence to support A: A Space a A evid A (a) Evid(A) (3) If A implies B, then evidence for A implies evidence for B: A Space B Space f : A B imp(f ) : Evid(A) Evid(B) (4) There is no evidence for a contradiction ( ): Evid( ) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
23 The Logic of Evidence: Syntax (Crane, 2018) A Space (1) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
24 The Logic of Evidence: Syntax (Crane, 2018) A Space Evid(A) Space (1) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
25 The Logic of Evidence: Syntax (Crane, 2018) A Space Evid(A) Space (1) a A evid A (a) Evid(A) (2) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
26 The Logic of Evidence: Syntax (Crane, 2018) A Space Evid(A) Space (1) a A evid A (a) Evid(A) (2) A Space B Space f : A B (3) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
27 The Logic of Evidence: Syntax (Crane, 2018) A Space Evid(A) Space (1) a A evid A (a) Evid(A) (2) A Space B Space f : A B imp(f ) : Evid(A) Evid(B) (3) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
28 The Logic of Evidence: Full Picture (Crane, 2018) A Space Evid(A) Space (1) a A evid A (a) Evid(A) (2) A Space B Space f : A B imp(f ) : Evid(A) Evid(B) (3) Evid( ) (4) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
29 Consequences of these axioms The following can be derived as theorems from the rules (1)-(4). Evid(A B) Evid(A) Evid(B) Evid(A) Evid(B) Evid(A B) Inferences supporting A and B (jointly) support inferences for A and for B (individually), which in turn supports inferences for A or inferences for B, which supports inference for A or B. (Arrows do not reverse in general.) Universal/Existential quantification: Evid ( a A, B(a)) a A, Evid(B(a)). Inference that B(a) holds for all a A supports inference that B(a) holds for every a A. a A, Evid(B(a)) Evid ( a A, B(a)). Inferring B(a) for some a A supports inference that there exists a : A such that B(a) holds. Derived inference rules: Evid(A) (A B) Evid(A B) Inferring A and proving that A implies B supports inference in A and B. A Evid(A B) Evid(A B). Harry Crane Knowing (Rutgers) A and inferring Foundations that of AData implies Science B supports inference BFF5: in MayA7, 2018 and B. 21 / 26
30 Formalization in Coq/UniMath: Definition of Evid as a homotopy type A Space Evid(A) Space (1) Written in Coq using UniMath library: A Space a A evid A (a) Evid(A) (2) A Space B Space f : A B imp(f ) : Evid(A) Evid(B) (3) Evid( ) (4) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
31 Formalization in Coq/UniMath: Proofs of basic properties Theorems mentioned earlier: Written in Coq using UniMath library: Evid(A B) Evid(A) Evid(B) Evid(A) Evid(A B) Evid(B) Evid(A B) Evid(A) Evid(B) Evid(A B) Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
32 Interpretation of Evid: Semantics Can these properties be interpreted in a familiar statistics/data science context? Can interpret the space of evidence in terms of Probability functions on fixed set Ω. Spaces of probability measures (Giry monad in Category Theory). Similarly, finitely additive probabilities as in 1 and 2. Dempser Shafer belief functions with fixed frame of discernment Ω. Spaces of Dempster Shafer functions. Immediate connections to Maximum likelihood estimation Hypothesis testing: P-values, Bayes factors Inferential Models Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
33 Next Steps There are many... Develop the theory further. Shore up the semantics in terms of existing approaches. Formalize existing statistics, machine learning, algorithmic approaches in this framework. Many more... Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
34 References H. Crane. (2018). Logic of probability and conjecture. H. Crane. (2017). Probability as Shape. Talk at BFF4. Video available at H. Crane. (2018). Probabilistic Foundations of Statistical Network Analysis. Chapman Hall. D. Tsementzis. (2018). A meaning explanation for HoTT. Synthese, to appear. Web : Project : researchers.one Contact Harry Crane (Rutgers) Foundations of Data Science BFF5: May 7, / 26
Homotopy Probability Theory in the Univalent Foundations
Homotopy Probability Theory in the Univalent Foundations Harry Crane Department of Statistics Rutgers April 11, 2018 Harry Crane (Rutgers) Homotopy Probability Theory Drexel: April 11, 2018 1 / 33 Motivating
Model Theory in the Univalent Foundations
Model Theory in the Univalent Foundations Dimitris Tsementzis January 11, 2017 1 Introduction 2 Homotopy Types and -Groupoids 3 FOL = 4 Prospects Section 1 Introduction Old and new Foundations (A) (B)
Reasoning with Uncertainty
Reasoning with Uncertainty Representing Uncertainty Manfred Huber 2005 1 Reasoning with Uncertainty The goal of reasoning is usually to: Determine the state of the world Determine what actions to take
Chapter Three. Hypothesis Testing
3.1 Introduction The final phase of analyzing data is to make a decision concerning a set of choices or options. Should I invest in stocks or bonds? Should a new product be marketed? Are my products being
Principles of Statistical Inference
Principles of Statistical Inference Nancy Reid and David Cox August 30, 2013 Introduction Statistics needs a healthy interplay between theory and applications theory meaning Foundations, rather than theoretical
De Finetti s ultimate failure. Krzysztof Burdzy University of Washington
De Finetti s ultimate failure Krzysztof Burdzy University of Washington Does philosophy matter? Global temperatures will rise by 1 degree in 20 years with probability 80%. Reading suggestions Probability
Symmetric Probability Theory
Symmetric Probability Theory Kurt Weichselberger, Munich I. The Project p. 2 II. The Theory of Interval Probability p. 4 III. The Logical Concept of Probability p. 6 IV. Inference p. 11 Kurt.Weichselberger@stat.uni-muenchen.de
(1) Introduction to Bayesian statistics
Spring, 2018 A motivating example Student 1 will write down a number and then flip a coin If the flip is heads, they will honestly tell student 2 if the number is even or odd If the flip is tails, they
Discussion of Dempster by Shafer. Dempster-Shafer is fiducial and so are you.
Fourth Bayesian, Fiducial, and Frequentist Conference Department of Statistics, Harvard University, May 1, 2017 Discussion of Dempster by Shafer (Glenn Shafer at Rutgers, www.glennshafer.com) Dempster-Shafer
Uncertainty and Rules
Uncertainty and Rules We have already seen that expert systems can operate within the realm of uncertainty. There are several sources of uncertainty in rules: Uncertainty related to individual rules Uncertainty
From syntax to semantics of Dependent Type Theories - Formalized
RDP 2015, Jun. 30, 2015, WCMCS, Warsaw. From syntax to semantics of Dependent Type Theories - Formalized by Vladimir Voevodsky from the Institute for Advanced Study in Princeton, NJ. I will be speaking
Lecture 10: Introduction to reasoning under uncertainty. Uncertainty
Lecture 10: Introduction to reasoning under uncertainty Introduction to reasoning under uncertainty Review of probability Axioms and inference Conditional probability Probability distributions COMP-424,
Principles of Statistical Inference
Principles of Statistical Inference Nancy Reid and David Cox August 30, 2013 Introduction Statistics needs a healthy interplay between theory and applications theory meaning Foundations, rather than theoretical
Reasoning Under Uncertainty
Reasoning Under Uncertainty Chapter 14&15 Part Kostas (1) Certainty Kontogiannis Factors E&CE 457 Objectives This unit aims to investigate techniques that allow for an algorithmic process to deduce new
LOGIC OF PROBABILITY AND CONJECTURE
LOGIC OF PROBABILITY AND CONJECTURE HARRY CRANE Abstract. I introduce a formalization of probability in intensional Martin-Löf type theory (MLTT) and homotopy type theory (HoTT) which takes the concept
Naive Bayes classification
Naive Bayes classification Christos Dimitrakakis December 4, 2015 1 Introduction One of the most important methods in machine learning and statistics is that of Bayesian inference. This is the most fundamental
Bayesian Statistics. State University of New York at Buffalo. From the SelectedWorks of Joseph Lucke. Joseph F. Lucke
State University of New York at Buffalo From the SelectedWorks of Joseph Lucke 2009 Bayesian Statistics Joseph F. Lucke Available at: https://works.bepress.com/joseph_lucke/6/ Bayesian Statistics Joseph
Uncertain Inference and Artificial Intelligence
March 3, 2011 1 Prepared for a Purdue Machine Learning Seminar Acknowledgement Prof. A. P. Dempster for intensive collaborations on the Dempster-Shafer theory. Jianchun Zhang, Ryan Martin, Duncan Ermini
Uncertainty. Chapter 13
Uncertainty Chapter 13 Outline Uncertainty Probability Syntax and Semantics Inference Independence and Bayes Rule Uncertainty Let s say you want to get to the airport in time for a flight. Let action A
Estimation of reliability parameters from Experimental data (Parte 2) Prof. Enrico Zio
Estimation of reliability parameters from Experimental data (Parte 2) This lecture Life test (t 1,t 2,...,t n ) Estimate θ of f T t θ For example: λ of f T (t)= λe - λt Classical approach (frequentist
Bayesian Reasoning. Adapted from slides by Tim Finin and Marie desjardins.
Bayesian Reasoning Adapted from slides by Tim Finin and Marie desjardins. 1 Outline Probability theory Bayesian inference From the joint distribution Using independence/factoring From sources of evidence
STAT 499/962 Topics in Statistics Bayesian Inference and Decision Theory Jan 2018, Handout 01
STAT 499/962 Topics in Statistics Bayesian Inference and Decision Theory Jan 2018, Handout 01 Nasser Sadeghkhani a.sadeghkhani@queensu.ca There are two main schools to statistical inference: 1-frequentist
Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14 Introduction One of the key properties of coin flips is independence: if you flip a fair coin ten times and get ten
A union of Bayesian, frequentist and fiducial inferences by confidence distribution and artificial data sampling
A union of Bayesian, frequentist and fiducial inferences by confidence distribution and artificial data sampling Min-ge Xie Department of Statistics, Rutgers University Workshop on Higher-Order Asymptotics
Machine Learning. Bayes Basics. Marc Toussaint U Stuttgart. Bayes, probabilities, Bayes theorem & examples
Machine Learning Bayes Basics Bayes, probabilities, Bayes theorem & examples Marc Toussaint U Stuttgart So far: Basic regression & classification methods: Features + Loss + Regularization & CV All kinds
Introduction to Imprecise Probability and Imprecise Statistical Methods
Introduction to Imprecise Probability and Imprecise Statistical Methods Frank Coolen UTOPIAE Training School, Strathclyde University 22 November 2017 (UTOPIAE) Intro to IP 1 / 20 Probability Classical,
Bios 6649: Clinical Trials - Statistical Design and Monitoring
Bios 6649: Clinical Trials - Statistical Design and Monitoring Spring Semester 2015 John M. Kittelson Department of Biostatistics & nformatics Colorado School of Public Health University of Colorado Denver
Machine Learning
Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University January 13, 2011 Today: The Big Picture Overfitting Review: probability Readings: Decision trees, overfiting
Inferential models: A framework for prior-free posterior probabilistic inference
Inferential models: A framework for prior-free posterior probabilistic inference Ryan Martin Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago rgmartin@uic.edu
Probability Theory Review
Probability Theory Review Brendan O Connor 10-601 Recitation Sept 11 & 12, 2012 1 Mathematical Tools for Machine Learning Probability Theory Linear Algebra Calculus Wikipedia is great reference 2 Probability
M B S E. Representing Uncertainty in SysML 2.0. Chris Paredis Director. Model-Based Systems Engineering Center
2016 Copyright Georgia Tech. All Rights Reserved. MBSE Center 1 M B S E Model-Based Systems Engineering Center Representing Uncertainty in SysML 2.0 Chris Paredis Director Model-Based Systems Engineering
Probability is related to uncertainty and not (only) to the results of repeated experiments
Uncertainty probability Probability is related to uncertainty and not (only) to the results of repeated experiments G. D Agostini, Probabilità e incertezze di misura - Parte 1 p. 40 Uncertainty probability
Belief functions: A gentle introduction
Belief functions: A gentle introduction Seoul National University Professor Fabio Cuzzolin School of Engineering, Computing and Mathematics Oxford Brookes University, Oxford, UK Seoul, Korea, 30/05/18
Data Mining Chapter 4: Data Analysis and Uncertainty Fall 2011 Ming Li Department of Computer Science and Technology Nanjing University
Data Mining Chapter 4: Data Analysis and Uncertainty Fall 2011 Ming Li Department of Computer Science and Technology Nanjing University Why uncertainty? Why should data mining care about uncertainty? We
Bayesian Models in Machine Learning
Bayesian Models in Machine Learning Lukáš Burget Escuela de Ciencias Informáticas 2017 Buenos Aires, July 24-29 2017 Frequentist vs. Bayesian Frequentist point of view: Probability is the frequency of
Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 10
EECS 70 Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 10 Introduction to Basic Discrete Probability In the last note we considered the probabilistic experiment where we flipped
BFF Four: Are we Converging?
BFF Four: Are we Converging? Nancy Reid May 2, 2017 Classical Approaches: A Look Way Back Nature of Probability BFF one to three: a look back Comparisons Are we getting there? BFF Four Harvard, May 2017
Probability Basics. Robot Image Credit: Viktoriya Sukhanova 123RF.com
Probability Basics These slides were assembled by Eric Eaton, with grateful acknowledgement of the many others who made their course materials freely available online. Feel free to reuse or adapt these
Computer Proof Assistants and Univalent Foundations of Mathematics
Nov. 16, 2014, CMA2014, Kuwait. Computer Proof Assistants and Univalent Foundations of Mathematics by Vladimir Voevodsky from the Institute for Advanced Study in Princeton, NJ. Kepler s Conjecture. In
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Bayesian Learning. Tobias Scheffer, Niels Landwehr
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning Tobias Scheffer, Niels Landwehr Remember: Normal Distribution Distribution over x. Density function with parameters
Hypothesis Testing. Rianne de Heide. April 6, CWI & Leiden University
Hypothesis Testing Rianne de Heide CWI & Leiden University April 6, 2018 Me PhD-student Machine Learning Group Study coordinator M.Sc. Statistical Science Peter Grünwald Jacqueline Meulman Projects Bayesian
Bayesian Statistics as an Alternative for Analyzing Data and Testing Hypotheses Benjamin Scheibehenne
Bayesian Statistics as an Alternative for Analyzing Data and Testing Hypotheses Benjamin Scheibehenne http://scheibehenne.de Can Social Norms Increase Towel Reuse? Standard Environmental Message Descriptive
Two hours. Note that the last two pages contain inference rules for natural deduction UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE
COMP 0 Two hours Note that the last two pages contain inference rules for natural deduction UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Mathematical Techniques for Computer Science Date: Friday
Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
CS 4649/7649 Robot Intelligence: Planning
CS 4649/7649 Robot Intelligence: Planning Probability Primer Sungmoon Joo School of Interactive Computing College of Computing Georgia Institute of Technology S. Joo (sungmoon.joo@cc.gatech.edu) 1 *Slides
Basic Probabilistic Reasoning SEG
Basic Probabilistic Reasoning SEG 7450 1 Introduction Reasoning under uncertainty using probability theory Dealing with uncertainty is one of the main advantages of an expert system over a simple decision
A gentle introduction to imprecise probability models
A gentle introduction to imprecise probability models and their behavioural interpretation Gert de Cooman gert.decooman@ugent.be SYSTeMS research group, Ghent University A gentle introduction to imprecise
For True Conditionalizers Weisberg s Paradox is a False Alarm
For True Conditionalizers Weisberg s Paradox is a False Alarm Franz Huber Abstract: Weisberg (2009) introduces a phenomenon he terms perceptual undermining He argues that it poses a problem for Jeffrey
Frequentist Statistics and Hypothesis Testing Spring
Frequentist Statistics and Hypothesis Testing 18.05 Spring 2018 http://xkcd.com/539/ Agenda Introduction to the frequentist way of life. What is a statistic? NHST ingredients; rejection regions Simple
Pengju XJTU 2016
Introduction to AI Chapter13 Uncertainty Pengju Ren@IAIR Outline Uncertainty Probability Syntax and Semantics Inference Independence and Bayes Rule Wumpus World Environment Squares adjacent to wumpus are
Imprecise Probability
Imprecise Probability Alexander Karlsson University of Skövde School of Humanities and Informatics alexander.karlsson@his.se 6th October 2006 0 D W 0 L 0 Introduction The term imprecise probability refers
Semantic Foundations for Probabilistic Programming
Semantic Foundations for Probabilistic Programming Chris Heunen Ohad Kammar, Sam Staton, Frank Wood, Hongseok Yang 1 / 21 Semantic foundations programs mathematical objects s1 s2 2 / 21 Semantic foundations
Bayesian Inference: What, and Why?
Winter School on Big Data Challenges to Modern Statistics Geilo Jan, 2014 (A Light Appetizer before Dinner) Bayesian Inference: What, and Why? Elja Arjas UH, THL, UiO Understanding the concepts of randomness
Structural Foundations for Abstract Mathematics
May 5, 2013 What no foundation can give us: Certainty What foundations need not give us: Ontological reduction What set theory gives us: Common language in which all mathematics can be encoded:,,,... Dispute
Chapter 12: Estimation
Chapter 12: Estimation Estimation In general terms, estimation uses a sample statistic as the basis for estimating the value of the corresponding population parameter. Although estimation and hypothesis
Course Introduction. Probabilistic Modelling and Reasoning. Relationships between courses. Dealing with Uncertainty. Chris Williams.
Course Introduction Probabilistic Modelling and Reasoning Chris Williams School of Informatics, University of Edinburgh September 2008 Welcome Administration Handout Books Assignments Tutorials Course
Probabilistic Machine Learning. Industrial AI Lab.
Probabilistic Machine Learning Industrial AI Lab. Probabilistic Linear Regression Outline Probabilistic Classification Probabilistic Clustering Probabilistic Dimension Reduction 2 Probabilistic Linear
Dempster-Shafer Theory and Statistical Inference with Weak Beliefs
Submitted to the Statistical Science Dempster-Shafer Theory and Statistical Inference with Weak Beliefs Ryan Martin, Jianchun Zhang, and Chuanhai Liu Indiana University-Purdue University Indianapolis and
Bayesian Networks BY: MOHAMAD ALSABBAGH
Bayesian Networks BY: MOHAMAD ALSABBAGH Outlines Introduction Bayes Rule Bayesian Networks (BN) Representation Size of a Bayesian Network Inference via BN BN Learning Dynamic BN Introduction Conditional
Probability and Statistics
Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT
9/12/17. Types of learning. Modeling data. Supervised learning: Classification. Supervised learning: Regression. Unsupervised learning: Clustering
Types of learning Modeling data Supervised: we know input and targets Goal is to learn a model that, given input data, accurately predicts target data Unsupervised: we know the input only and want to make
Statistical Inference
Statistical Inference Classical and Bayesian Methods Class 6 AMS-UCSC Thu 26, 2012 Winter 2012. Session 1 (Class 6) AMS-132/206 Thu 26, 2012 1 / 15 Topics Topics We will talk about... 1 Hypothesis testing
Probabilistic Reasoning
Course 16 :198 :520 : Introduction To Artificial Intelligence Lecture 7 Probabilistic Reasoning Abdeslam Boularias Monday, September 28, 2015 1 / 17 Outline We show how to reason and act under uncertainty.
Computational Cognitive Science
Computational Cognitive Science Lecture 9: A Bayesian model of concept learning Chris Lucas School of Informatics University of Edinburgh October 16, 218 Reading Rules and Similarity in Concept Learning
Introduction to Bayesian Inference
Introduction to Bayesian Inference p. 1/2 Introduction to Bayesian Inference September 15th, 2010 Reading: Hoff Chapter 1-2 Introduction to Bayesian Inference p. 2/2 Probability: Measurement of Uncertainty
A Logic for Reasoning about Evidence
Journal of Artificial Intelligence Research 26 (2006) 1 34 Submitted 11/05; published 05/06 A Logic for Reasoning about Evidence Joseph Y. Halpern Cornell University, Ithaca, NY 14853 USA Riccardo Pucella
CMPT Machine Learning. Bayesian Learning Lecture Scribe for Week 4 Jan 30th & Feb 4th
CMPT 882 - Machine Learning Bayesian Learning Lecture Scribe for Week 4 Jan 30th & Feb 4th Stephen Fagan sfagan@sfu.ca Overview: Introduction - Who was Bayes? - Bayesian Statistics Versus Classical Statistics
Deciding, Estimating, Computing, Checking
Deciding, Estimating, Computing, Checking How are Bayesian posteriors used, computed and validated? Fundamentalist Bayes: The posterior is ALL knowledge you have about the state Use in decision making:
Deciding, Estimating, Computing, Checking. How are Bayesian posteriors used, computed and validated?
Deciding, Estimating, Computing, Checking How are Bayesian posteriors used, computed and validated? Fundamentalist Bayes: The posterior is ALL knowledge you have about the state Use in decision making:
Grundlagen der Künstlichen Intelligenz
Grundlagen der Künstlichen Intelligenz Uncertainty & Probabilities & Bandits Daniel Hennes 16.11.2017 (WS 2017/18) University Stuttgart - IPVS - Machine Learning & Robotics 1 Today Uncertainty Probability
Bayesian Inference. STA 121: Regression Analysis Artin Armagan
Bayesian Inference STA 121: Regression Analysis Artin Armagan Bayes Rule...s! Reverend Thomas Bayes Posterior Prior p(θ y) = p(y θ)p(θ)/p(y) Likelihood - Sampling Distribution Normalizing Constant: p(y
Statistical Machine Learning Lecture 1: Motivation
1 / 65 Statistical Machine Learning Lecture 1: Motivation Melih Kandemir Özyeğin University, İstanbul, Turkey 2 / 65 What is this course about? Using the science of statistics to build machine learning
Machine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 11 Maximum Likelihood as Bayesian Inference Maximum A Posteriori Bayesian Gaussian Estimation Why Maximum Likelihood? So far, assumed max (log) likelihood
Uncertainty. Introduction to Artificial Intelligence CS 151 Lecture 2 April 1, CS151, Spring 2004
Uncertainty Introduction to Artificial Intelligence CS 151 Lecture 2 April 1, 2004 Administration PA 1 will be handed out today. There will be a MATLAB tutorial tomorrow, Friday, April 2 in AP&M 4882 at
STAT 515 fa 2016 Lec Statistical inference - hypothesis testing
STAT 515 fa 2016 Lec 20-21 Statistical inference - hypothesis testing Karl B. Gregory Wednesday, Oct 12th Contents 1 Statistical inference 1 1.1 Forms of the null and alternate hypothesis for µ and p....................
Structure learning in human causal induction
Structure learning in human causal induction Joshua B. Tenenbaum & Thomas L. Griffiths Department of Psychology Stanford University, Stanford, CA 94305 jbt,gruffydd @psych.stanford.edu Abstract We use
COHERENCE AND PROBABILITY. Rosangela H. Loschi and Sergio Wechsler. Universidade Federal de Minas Gerais e Universidade de S~ao Paulo ABSTRACT
COHERENCE AND PROBABILITY Rosangela H. Loschi and Sergio Wechsler Universidade Federal de Minas Gerais e Universidade de S~ao Paulo ABSTRACT A process of construction of subjective probability based on
Classification & Information Theory Lecture #8
Classification & Information Theory Lecture #8 Introduction to Natural Language Processing CMPSCI 585, Fall 2007 University of Massachusetts Amherst Andrew McCallum Today s Main Points Automatically categorizing
For True Conditionalizers Weisberg s Paradox is a False Alarm
For True Conditionalizers Weisberg s Paradox is a False Alarm Franz Huber Department of Philosophy University of Toronto franz.huber@utoronto.ca http://huber.blogs.chass.utoronto.ca/ July 7, 2014; final
Machine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University August 30, 2017 Today: Decision trees Overfitting The Big Picture Coming soon Probabilistic learning MLE,
AIM HIGH SCHOOL. Curriculum Map W. 12 Mile Road Farmington Hills, MI (248)
AIM HIGH SCHOOL Curriculum Map 2923 W. 12 Mile Road Farmington Hills, MI 48334 (248) 702-6922 www.aimhighschool.com COURSE TITLE: Statistics DESCRIPTION OF COURSE: PREREQUISITES: Algebra 2 Students will
Confidence Distribution
Confidence Distribution Xie and Singh (2013): Confidence distribution, the frequentist distribution estimator of a parameter: A Review Céline Cunen, 15/09/2014 Outline of Article Introduction The concept
Bayesian vs frequentist techniques for the analysis of binary outcome data
1 Bayesian vs frequentist techniques for the analysis of binary outcome data By M. Stapleton Abstract We compare Bayesian and frequentist techniques for analysing binary outcome data. Such data are commonly
DIFFERENT APPROACHES TO STATISTICAL INFERENCE: HYPOTHESIS TESTING VERSUS BAYESIAN ANALYSIS
DIFFERENT APPROACHES TO STATISTICAL INFERENCE: HYPOTHESIS TESTING VERSUS BAYESIAN ANALYSIS THUY ANH NGO 1. Introduction Statistics are easily come across in our daily life. Statements such as the average
Rigorous Science - Based on a probability value? The linkage between Popperian science and statistical analysis
/3/26 Rigorous Science - Based on a probability value? The linkage between Popperian science and statistical analysis The Philosophy of science: the scientific Method - from a Popperian perspective Philosophy
Whaddya know? Bayesian and Frequentist approaches to inverse problems
Whaddya know? Bayesian and Frequentist approaches to inverse problems Philip B. Stark Department of Statistics University of California, Berkeley Inverse Problems: Practical Applications and Advanced Analysis
Examine characteristics of a sample and make inferences about the population
Chapter 11 Introduction to Inferential Analysis Learning Objectives Understand inferential statistics Explain the difference between a population and a sample Explain the difference between parameter and
Machine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University January 14, 2015 Today: The Big Picture Overfitting Review: probability Readings: Decision trees, overfiting
Rigorous Science - Based on a probability value? The linkage between Popperian science and statistical analysis
/9/27 Rigorous Science - Based on a probability value? The linkage between Popperian science and statistical analysis The Philosophy of science: the scientific Method - from a Popperian perspective Philosophy
Dynamics of Inductive Inference in a Uni ed Model
Dynamics of Inductive Inference in a Uni ed Model Itzhak Gilboa, Larry Samuelson, and David Schmeidler September 13, 2011 Gilboa, Samuelson, and Schmeidler () Dynamics of Inductive Inference in a Uni ed
Uncertainty and Bayesian Networks
Uncertainty and Bayesian Networks Tutorial 3 Tutorial 3 1 Outline Uncertainty Probability Syntax and Semantics for Uncertainty Inference Independence and Bayes Rule Syntax and Semantics for Bayesian Networks
Statistical Methods in Particle Physics. Lecture 2
Statistical Methods in Particle Physics Lecture 2 October 17, 2011 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2011 / 12 Outline Probability Definition and interpretation Kolmogorov's
The Limitation of Bayesianism
The Limitation of Bayesianism Pei Wang Department of Computer and Information Sciences Temple University, Philadelphia, PA 19122 pei.wang@temple.edu Abstract In the current discussion about the capacity
Tests about a population mean
October 2 nd, 2017 Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 1: Descriptive statistics Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter 8: Confidence
Rigorous Science - Based on a probability value? The linkage between Popperian science and statistical analysis
Rigorous Science - Based on a probability value? The linkage between Popperian science and statistical analysis The Philosophy of science: the scientific Method - from a Popperian perspective Philosophy
Probability and Information Theory. Sargur N. Srihari
Probability and Information Theory Sargur N. srihari@cedar.buffalo.edu 1 Topics in Probability and Information Theory Overview 1. Why Probability? 2. Random Variables 3. Probability Distributions 4. Marginal
Three grades of coherence for non-archimedean preferences
Teddy Seidenfeld (CMU) in collaboration with Mark Schervish (CMU) Jay Kadane (CMU) Rafael Stern (UFdSC) Robin Gong (Rutgers) 1 OUTLINE 1. We review de Finetti s theory of coherence for a (weakly-ordered)
A survey of the theory of coherent lower previsions
A survey of the theory of coherent lower previsions Enrique Miranda Abstract This paper presents a summary of Peter Walley s theory of coherent lower previsions. We introduce three representations of coherent
Conditional Degree of Belief
Conditional Degree of Belief Jan Sprenger July 25, 2016 Abstract This paper investigates the semantics and epistemology of conditional degree of belief. It defends the thesis that conditional degrees of
The Curry-Howard Isomorphism
The Curry-Howard Isomorphism Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) The Curry-Howard Isomorphism MFES 2008/09