Statistical Methods in Applied Computer Science

Stefan Arnborg, KTH http://www.nada.kth.se/~stefan

Statistical Methods in Applied Computer Science

DD2447, DD3342, spring 2011

SYLLABUS Common statistical models and their use: Bayesian, testing, and fiducial statistical philosophy Hypothesis choice Parametric inference Non-parametric inference Elements of regression Clustering Graphical statistical models Prediction and retrodiction Chapman-Kolmogoroff formulation Evidence theory, estimation and combination of evidence. Support Vector Machines and Kernel methods Vovk/Gammerman hedged prediction technology Stochastic simulation, Markov Chain Monte Carlo. Variational Bayes

LEARNING GOALS After successfully taking this course, you will be able to: -motivate the use of uncertainty management and statistical methodology in computer science applications, as well as the main methods in use, -account for algorithms used in the area and use the standard tools, -critically evaluate the applicability of these methods in new contexts, and design new applications of uncertainty management, -follow research and development in the area.

GRADING DD2447: Bologna grades Grades are E-A during 2009. 70% of homeworks and a very short oral discussion of them gives grade C. Less gives F-D. For higher grades, essentially all homeworks should be turned in on time. Alternative assignments will be substituted for those homeworks you miss. For grade B you must pass one Master's test, for grade A you must do two Master's tests or a project with some research content. DD3342: Pass/Fail Research level project, or deeper study of part of course

Applications of Uncertainty everywhere

Medical Imaging/Research (Schizophrenia) Land Use Planning

Environmental Surveillance and Prediction Finance and Stock

Marketing into Google Robot Navigation and Tracking

Security and Military Performance Tuning

…

Some Master’s Projects using this syllabus (subset)

•  Recommender system for Spotify •  Behavior of mobile phone users •  Recommender system for book club •  Recommender for job search site •  Computations in evolutionary genetics •  Gene hunting •  Psychiatry: genes, anatomy, personality •  Command and control: Situation awareness •  Diagnosing drilling problems •  Speech, Music, …

Aristotle: Logic Logic as a semi-formal system was created by Aristotle, probably inspired by current practice in mathematical arguments. There is no record of Aristotle himself applying logic, but probably the Elements of Euclid derives from Aristotles illustrations of the logical method.

Which role has logic in Computer Science??

Visualization •  Visualize data in such a way that the

important aspects are obvious - A good visualization strikes you as a punch between your eyes (Tukey, 1970)

•  Pioneered by Florence Nightingale, first female member of Royal Statistical Society, inventor of pie charts and performance metrics

Probabilistic approaches

•  Bayes: Probability conditioned by observation •  Cournot: An event with very small probability

will not happen. •  Vapnik-Chervonenkis: VC-dimension and PAC,

distribution-independence •  Kolmogorov/Vovk: A sequence is random if it

cannot be compressed

Peirce: Abduction and uncertainty

Aristotles induction , generalizing from particulars, is considered invalid by strict deductionists. Peirce made the concept clear, or at least confused on a higher level. Abduction is verification by finding a plausible explanation. Key process in scientific progress.

Sherlock Holmes: common sense inference

Techniques used by Sherlock are modeled on Conan Doyle’s professor in medical school, who followed the methodological tradition of Hippocrates and Galen. Abductive reasoning, first spelled out by Peirce, is found in 217 instances in Sherlock Holmes adventures - 30 of them in the first novel, ‘A study in Scarlet’.

Thomas Bayes, amateur mathematician

If we have a probability model of the world we know how to compute probabilities of events. But is it possible to learn about the world from events we see? Bayes’ proposal was forgotten but rediscovered by Laplace.

An alternative to Bayes’ method - hypothesis testing - is based on

’Cournot’s Bridge’: an event with very small

probability will not happen

Antoine Augustine Cournot (1801--1877)���Pioneer in stochastic processes, market theory���and structural post-modernism. Predicted demise of academic system due to discourses of administration and excellence(cf Readings).

Kolmogorov and randomness Andrei Kolmogorov(1903-1987) is the mathematician best known for shaping probability theory into a modern axiomatized theory. His axioms of probability tells how probability measures are defined, also on infinite and infinite-dimensional event spaces and complex product spaces. Kolmogorov complexity characterizes a random string by the smallest size of a description of it. Used to explain Vovk/Gammerman scheme of hedged prediction. Also used in MDL ���(Minimum Description Length) inference.

Normative claim of Bayesianism

•  EVERY type of uncertainty should be treated as probability

•  This claim is controversial and not universally accepted: Fisher(1922), Cramér, Zadeh, Dempster, Shafer, Walley(1999) …

•  Students encounter many approaches to uncertainty management and identify weaknessess in foundational arguments.

Foundations for Bayesian Inference •  Bayes method, first documented method

based on probability: Plausibility of event depends on observation, Bayes rule:

•  Bayes’ rule organizing principle for uncertainty •  Parameter and observation spaces can be extremely

complex, priors and likelihoods also. •  MCMC current approach -- often but not always

applicable (difficult when posterior has many local maxima separated by low density regions)

•  Variational Bayes –approximate posterior by factorized function – result also approximate.

Showcase application: PET-camera

f (! | D)" f (D | !) f (! )

Camera geometry&noise film scene regularity

and also any other camera or imaging device …

PET camera

D: film, count by detector pair j X: radioactivity in voxel i a: camera geometry

likelihood

prior

Inference about Y gives posterior, its mean is often a good picture

Sinogram and reconstruction

Tumour

Fruit Fly Drosophila family (Xray)

Introduction

GOMOS (Global Ozone Monitoring by Occultation ofStars)

The Royal Statistical SocietyLondon 10 December 2003

Markov chain Monte Carlo methods for highdimensional inversion in remote sensing

Heikki Haario1, Marko Laine1, Markku Lehtinen2

Eero Saksman3 and Johanna Tamminen4

1 University of Helsinki, Finland2 University of Oulu, Sodankylä, Finland

3 University of Jyväskylä, Finland4 Finnish Meteorological Institute, Helsinki, Finland

The Royal Statistical SocietyLondon 10 December 2003

!* WIRED on Total Information Awareness! WIRED (Dec 2, 2002) article "Total Info System Totally Touchy"! discusses the Total Information Awareness system.! !~~~ Quote:!"People have to move and plan before committing a terrorist act. Our!hypothesis is their planning process has a signature." !Jan Walker, Pentagon spokeswoman, in Wired, Dec 2, 2002. !!"What's alarming is the danger of false !positives based on incorrect data," !

!Herb Edelstein, in Wired, Dec 2, 2002. !

Combination of evidence f (! | D)" f (D | !) f (! )

f (! |{d1,d2}) " f (d1 | ! ) f (d2 | ! ) f (! )

In Bayes’ method, evidence is likelihood for observation.

Particle filter- general tracking

Chapman Kolmogorov version of Bayes’ rule

f (!t | Dt ) " f (dt | !t)# f (!t |!t$1) f (!t$1 | Dt$1 )d!t$1

Berry and Linoff have eloquently stated their preferences with ���the often quoted sentence: "Neural networks are a good choice for most classification problems when the results of the model are more important than understanding how the model works".��� “Neural networks typically give the right answer”

1950-1980: The age of rationality. Let us describe the world with a mathematical model and compute the best way to manage it!! This is a large Bayesian Network, a popular statistical model

Ed Jaynes devoted a large part of his career to promote Bayesian inference. He also championed the use of Maximum Entropy in physics Outside physics, he received resistance from people who had already invented other methods. Why should statistical mechanics say anything about our daily human world??

Robust Bayes •  Priors and likelihoods are convex sets of probability

distributions (Berger, de Finetti, Walley,...): imprecise probability:

•  Every member of posterior is a ’parallell combination’ of one member of likelihood and one member of prior.

•  For decision making: Jaynes recommends to use that member of posterior with maximum entropy (Maxent estimate).

f (! | D)" f (D | !) f (! )F(! | D) " F(D | ! )F(! )

SVM and Kernel method Based on Vapnik-Chervonenkis learning theory Separate classes by wide margin hyperplane classifier, or enclose data points between close parallell hyperplanes for regression Possibly after non-linear mapping to highdimensional space Assumption is only point exchangeability

Classify with hyperplanes

Frank Rosenblatt (1928 – 1971) Pioneering work in classifying by hyperplanes in high-dimensional spaces. Criticized by Minsky-Papert, since real classes are not normally linearly separable. ANN research taken up again in 1980:s, with non-linear mappings to get improved separation. Predecessor to SVM/kernel methods

Find parallel hyperplanes Classification Red: true separating plane. Blue: wide margin separation in sample Classify by plane between blue planes

SVM and Kernel method

Vovk/Gammerman Hedged predictions

•  Based on Kolmogorov complexity or non-conformance measure

•  In classification, each prediction comes with confidence

•  Asymptotically, misclassifications appear independently and with probability 1-confidence.

•  Only assumption is exchangeability