## Learning objectives

Course Objectives:

The objective of the course is to provide the student with the ability to understand the basic rules of machine learning and, in particular:

- the most common statistical tests in classification among different categories

- the structure of the optimal classifier and its error analysis

- the most common feature extraction methods from input data

- the most common statistical estimators in machine learning

The abilities in applying the above-mentioned knowledge are in particular

in the:

- design and performance analysis of classifiers in machine learning

- selection of the most appropriate features to discriminate input categories

## Prerequisites

Pre-requisites:

Entry-level courses in linear algebra and probability theory, such as those

normally offered in the corresponding 3-year Laurea course, are necessary

pre-requisites for this course.

## Course unit content

CONTENTS:

PART 1: Fundamentals (Bononi):

Basic probability refresher. Bayesian binary and M-ary classification. MAP and Minimax classifiers. Performance and ROC. Gaussian case and linear discriminant rules.

Bayesian estimation (regression). Maximum likelihood, MMSE, MMAE estimators. Linear suboptimal estimators.

Supervised learning. Generative versus discriminative approaches. Plug-in learning. Bayesian learning. Minimum empirical risk learning. Nonparametric probability density estimation.

linear data reduction for feature extraction.

## Full programme

Syllabus of PART 1:

Fundamentals (Bononi)

(2H every lecture)

_____________________________________________

Lec. 1. Introduction

- Problem statement and definitions

- Examples of machine learning problems

- Glossary of equivalent terms in Radar detecton theory, hypothesis testing and machine learning

Lec. 2. Probability refresher

- Axioms, conditional probability, total probability law, Bayes law, double conditioning, chain rule, independence and conditional independence of events.

- Discrete random variables (RV): expectation, conditional expectation. Pairs of RVs. Sum rule. Iterated expectation. Vectors of RVs. An extended example.

Lec. 3. Probability refresher

- Random vectors:

expectation, covariance and its properties, spectral decomposition of covariance matrix, whitening.

- Continuous RV.

Parallels with discrete RVs. Functions of RVs. Mixed RVs. Continuous random vectors.

- Appendix: differentiation rules for vectors and matrices.

Lec. 4.

- Gaussian RVs and their linear transformations. Mahalanobis distance.

Classification:

- Bayesian prediction: introduction, loss function, conditional risk, argmin/argmax rules

- Bayes classification: introduction

Lec. 5. Classification

- 0/1 loss -> maximum a posteriori (MAP) classifier. Binary MAP. Decision regions.

- Classifier performance.

- Likelihood ratio tests and receiver operating curve (ROC)

- Minimax rule

Lec. 6. Classification

- Binary Gaussian classification

- Homoscedastic case: linear discriminant analysis

- Heteroscedastic case: Bhattacharrya bound

- Bayes classification with discrete features

- Classification with missing data (composite hypothesis testing)

Lec. 7. Estimation

- Bayesian estimation: introduction

- Quadratic loss: minimum mean square error (MMSE) estimator = regression curve

- L1 loss: minimum mean absolute error (MMAE) estimator

- 0/1 loss: MAP estimator, and maximum likelihood (ML) in uniform prior.

- Regression for vector Gaussian case

- ML estimation for Gaussian observations

Lec. 8. Estimation

- ML for multinomial

- Conjugate priors in MAP estimation

- Estimation accuracy and ML properties, Cramer Rao bounds.

Suboptimal (non Bayesian) estimation:

- LMMSE estimation (linear regression)

- LMMSE derivation with LDU decomposition

Lec. 9. Estimation

- LMMSE examples

- Generalized linear regression

- Example: polynomial regression

- Sample LMMSE

- Generalized sample LMMSE.

Lec. 10. Learning

- Supervised learning: introduction

- Generative vs discriminatie approaches

- Example: logistic model

- Plug-in learning

ML fitting of logistic model: logistic regression

Example: handwritten digit recognition.

- Bayesian Learning

Lec. 11.

Learning:

- Empirical risk minimization

Nonparametric density estimation:

- Parzen window estimator

- kNN estimator

Lec. 12. linear data reduction

- Principal component analysis (PCA)

- Fisher linear classifier

## Bibliography

Suggested Reading:

[1] C. W. Therrien, "Decision, estimation and classification" Wiley, 1989

[2] R. O. Duda, P. E. Hart, D. G. Stork, "Pattern classification", 2nd Ed., Wiley, 2001

[3] D. Barber "Bayesian Reasoning and Machine Learning" Cambridge University Press, 2012.

[4] C. M. Bishop "Pattern Recognition and Machine Learning", Springer, 2006.

[5] T. Hastie, R. Tibshirani, J. Friedman, "The Elements of Statistical Learning: Data mining, inference, and prediction", Springer, 2008.

## Teaching methods

Teaching methods:

Classroom teaching, 42 hours.

In-class problem solving, 6 hours.

## Assessment methods and criteria

Exams:

Part 1, Bononi: Oral only, to be scheduled on an individual basis. When

ready, please contact the instructor by email at alberto.bononi[AT]unipr.

it and by specifying the requested date. The exam consists of solving

some exercises and explaining theoretical details connected with them,

for a total time of about 1 hour. You can bring your summary of important

formulas in an A4 sheet to consult if you so wish.

Part 2, Cagnoni: A practical project will be assigned, whose results will be

presented and discussed by the student both as a written report and as

an oral presentation.

## Other information

Further information:

1) Office Hours

Bononi: Monday 11:30-13:30 (Scientific Complex, Building 2, floor 2,

Room 2/19T).

Cagnoni: by appointment (Scientific Complex, Building 1, floor 2, email

cagnoni[AT]ce.unipr.it).

2) web site of course:

www.tlc.unipr.it/bononi/didattica/ML/ML.html

To get userid and password, please send an email to

alberto.bononi[AT]unipr.it from your account nome@studenti.unipr.it.