Bayesian Statistics and Questionnaire Data Analysis

General Objectives.

• Know and understand the main elements and concepts of the Bayesian inference (likelihood function, prior densities etc.).

• Know and understand the main concepts and techniques for the classical questionnaires analysis and item response theory (IRT).

• Know and understand Bayesian versions of IRT.

Synthetic General Description.

The course aims at introducing the methodology and practical techniques of

• Main elements of the Bayesian inference

• The design and (frequentist and Bayesian) analysis of questionnaires

This knowledge is relevant in several applications. Bayesian inference has found application in a wide range of activities, including machine learning, engineering, philosophy, medicine, etc. Moreover, questionnaires are frequently used by researchers from various fields to gather opinions, attitudes, and behaviors, and to identify trends or correlations. Researchers use questionnaire analysis methods to interpret the collected responses. It involves extracting valuable insights and patterns from the data gathered, enabling researchers or analysts to draw conclusions and make informed decisions based on the results Questionnaires analysis and item response theory (IRT) are nice example of application since extends the knowldge of the students from regression and classification, to another kind of application, which has several points of similarities to a classification problem handled by logistic-type regression but at the same time is completely different. Connection with parallel logistic regressors are also discussed. Bayesian versions of IRT are described.

Expected Learning Results

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- Knowledge and understanding:

The course aims at introducing the methodology and practical techniques for the design of questionnaires and Bayesian statistics. This knowledge is relevant in several areas.

The course will give the main concepts and techniques for the design of questionnaires and data analysis of collected data. On completion, students will acquire knowledge about: a) Elements of Bayesian inference b) design of a statistical survey; c) techniques for questionnaire design; d) methods for statistical analysis of collected data and for providing statistical reports.

- Applying knowledge and understanding:

On completion, students will be able: a) to perform Bayesian statistics; b) to design a statistical survey; c) to analyze collected data through suitable statistical methods and models; d) to provide a statistical report for summarizing the main results

- Making judgments:

On completion, students will able how to choose a suitable statistical model, apply sound statistical methods, and perform the analyses using statistical software Matlab, and/or R.

- Basic of mathematics and statistics

- Basic elements of Data Analysis and Statistical Learning

- More specifically, as examples: concept of probability density and likelihood function, properties of the estimators, unbiasness and consistency.

1. Introduction to Bayesian inference. (3 CFU)

   1. Description of the main actors: likelihood function, priors, marginal likelihood and posterior density.

   2. Examples of application.

   3. Benefits and drawbacks – practical application: Monte Carlo methods (MCMC, inportance sampling etc.)

   4. Labs in Matlab/Octave (or R).

2. Statistical Analyses of Questionnaire Data. (3 CFU)

   1. Design of Questionnaires: types of measures and Questions. Types of error in surveys. Evaluating survey questions; methods of Data Collection; introduction to Survey Research Methods.

   2. Classical theory; factorial analysis, Latent Class Analysis.

   3. Modern theory: Item Response Theory (IRT); likelihood of IRT – connections with parallel logistic regressors and Bayesian versions.

   4. Labs in Matlab/Octave (or R).

• Bayesian Data Analysis — Gelman, Carlin, Stern, Dunson, Vehtari & Rubin (2013)
• Bayesian Theory — José M. Bernardo & Adrian F. M. Smith (1994)
• The Bayesian Choice — Christian P. Robert (2007)
• Monte Carlo Statistical Methods — Christian Robert & George Casella
• Gaussian Processes for Machine Learning — Carl E. Rasmussen & Christopher K. I. Williams
• Random Number Generation and Monte Carlo Methods — James E. Gentle
• L. Martino, D. Luengo, J. Míguez, "Independent Random Sampling Methods", Springer, 2018.    DOI: 10.1007/978-3-319-72634-2

• L. Martino, V. Elvira. "Metropolis Sampling", Wiley StatsRef: Statistics Reference Online, 2017. arXiv:1704.04629    

•  V. Elvira, L. Martino, "Advances in Importance Sampling", Wiley StatsRef: Statistics Reference Online, 2020. arXiv:2102.05407

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• Falissard B. (2012), Analysis of Questionnaire Data with R, CRC Press, Boca Raton.

• Bartolucci F., Bacci S., Gnaldi M. (2016), Statistical Analysis of Questionnaires, CRC Press, Boca Raton.

• Fowler F. J. (2009), Survey Research Methods, SAGE Publications, Thousand Oaks, California.

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• Lecture notes and slides

Evaluation by (a) a research project on a topic decided jointly by student and Professor - (b) oral exam.

Scale marks:

• Failed: the student does not know the basic concept of the course and has completed less than 40% of the required assignemnts

• 18-20: the student has a basic knowledge of the topics of the course but he has great difficulties in applying them to practical exercises and problem solving pipelines.

• 21-24: the student has a basic knowledge of the topics of the course and he is able to solve simple problemns and exercises with some guidance from the teacher.

• 25-27: the student has a good knowledge of the topics of the course and can complete the assignemnt in autonomy with minor errors

• 28-30 e lode: The student has full knowledge of the topics of the course and is able to complete in autonomy assignemnts making connections and with only very minimal occasional mistakes.

- Explain what is a likelihood function.

- Say what is a Monte Carlo method.

- Describe the relationship between logistic regression and an item response theory model.

- Describe differences between MCMC and importance sampling algorithms.