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2023/2024

Modelling and Inference

Code: 104392 ECTS Credits: 6
Degree Type Year Semester
2503740 Computational Mathematics and Data Analytics OB 2 1

Contact

Name:
Rosario Delgado de la Torre
Email:
rosario.delgado@uab.cat

Teaching groups languages

You can check it through this link. To consult the language you will need to enter the CODE of the subject. Please note that this information is provisional until 30 November 2023.


Prerequisites

Good knowledge of the contents of the courses taken during the first year of the bachelor's degree is considered very important, especially those of probability and calculus.


Objectives and Contextualisation

This is the first course in the Bachelor's degree that focuses on Statistical Inference, a branch of statistics that uses data from a "representative" sample to acquire information about a population. The course is required throughout the Bachelor's degree, as it covers different concepts and techniques that serve as the basis for many of the topics introduced in upcoming courses within the Bachelor's. In particular, the course will start with a brief introduction to statistics, followed by a chapter on parameter estimation (both point and based on confidence intervals), and finally chapters on frequentist-based significance tests and an introduction to classical linear regression models.

To protect everyone's safety, in-person teaching and evaluable activities will be adjusted in accordance with health authority recommendations.


Competences

  • Calculate and reproduce certain mathematical routines and processes with ease.
  • Formulate hypotheses and think up strategies to confirm or refute them.
  • Relate new mathematical objects with other known objects and deduce their properties.
  • Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  • Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
  • Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  • Use computer applications for statistical analysis, numerical and symbolic computation, graphic visualisation, optimisation and other to experiment and solve problems.
  • Using criteria of quality, critically evaluate the work carried out.

Learning Outcomes

  1. Analyse data using inference techniques for one or two samples.
  2. Choose the appropriate statistical software to analyse the data through inference techniques.
  3. Describe the basic properties of timestamp and interval estimators.
  4. Identify statistical distributions.
  5. Identify statistical inference as an instrument of prognosis and prediction.
  6. Interpret obtained results and provide conclusions that refer to the experimental hypothesis.
  7. Recognise the usefulness of Bayesian methods and apply these appropriately.
  8. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  9. Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
  10. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  11. Understabd the distinct methods of data collection.
  12. Use statistical software to manage databases.
  13. Use statistical software to obtain the summary indexes of study variables.
  14. Use the properties of distribution function.
  15. Use the properties of the density function.
  16. Using criteria of quality, critically evaluate the work carried out.
  17. Validate and manage information to carry out statistical processing on this.

Content

Preliminaries of Probability (reminder): Probability and random variables. Law concept. Discrete-valued distributions. Density and probability functions. Expectation and variance. Moment generating function. Examples.

Topic 1. Introduction to Statistics.

1. Descriptive statistics and inferential statistics.

1.1. Basic concepts in inference: statistical population and sample; parameters, statistics and estimators.

1.2. Statistical models: parametric and non-parametric.

2. Most common statistics: the sample moments. The order statistics.

3. Distribution of some statistics.

3.1. From a sample of a Normal population: Fisher's theorem.

3.2. The Central Limit Theorem: asymptotic normality of sample moments and proportion.

Topic 2: Point estimation.

1. Point estimators: definition and properties.

1.1. Bias

1.2. Comparison of estimators without bias. Relative efficiency

1.3. Comparison of estimators with bias: the mean square error.

1.4. Consistency of an estimator.

2. Methods to obtain estimators.

2.1. Method of moments.

2.2. Method of maximum likelihood (MLE)

2.2.1. Invariance of the likelihood.

2.2.2. Score function and Fisher information.

2.2.3. Cramer-Rao inequality.

2.2.4. Properties of the MLE.

2.2.5. Delta method.

2.2.6. Numerical procedures for determining MLE.

Topic3. Estimation by confidence intervals.

1. Concept of confidence region and interval.

2. The "pivot" method for the construction of confidence intervals.

3. Confidence intervals for the parameters of a population.

3.1. For the mean of a Normal population with known and unknown deviations.

3.2. For the variance of a Normal population with known and unknown means.

3.3. Asymptotic confidence intervals: Wald, Score and LRT.

4. Confidence intervals for the parameters of two populations.

4.1. Confidence intervals with independent samples.

4.2. Confidence intervals for the difference of means of two Normal populations with paired data.

Topic 4: Significance tests.

1. Introduction.

1.1. Type I and II errors.

1.2. Power function.

1.3. Tests consistency.

1.4. p-values.

1.5. Duality between confidence intervals and significance tests.

2. Tests for the parameters of a population.

2.1. For the mean of a Normal population with known and unknown deviations.

2.2. Asymptotic tests for the mean of a population when the sample is large.

2.3. For the variance of a Normal population.

3. Tests for the parameters of two populations.

3.1. Hypothesis tests with independent samples.

3.2. Tests of hypotheses with paired data.

Topic 5. Simple linear regression model.

1. Purpose of the model.

2. Ordinary least squares (OLS) estimators.

3. Inferencebased on the linear regression model.

4. Predictions.

IMPORTANT: In teaching, the gender perspective involves reviewing androcentric biases and questioning the assumptions and hidden gender stereotypes. This revision involves including the contents of the subjectthe knowledge produced by scientific women, often forgotten, seeking the recognition of their contributions,as well as that of their works in the bibliographical references. 


Methodology

The course is organized into lecture, exercise and lab sessions.

In lectures, we will introduce the concepts and techniques outlined in the course program. Given that the content is mostly based on the standard topics of an introduction to statistical inference course, the recommended bibliography can be used to follow the course. Lecture slides and related material will be available in Moodle. The exercise sessions are intended to work through and understand statistical concepts. Each exercise will be available in Moodle. The goal of the lab sessions is to learn how to apply the methods given in lectures using the statistical software R, as well as how to evaluate the findings. 

IMPORTANT: To work more comfortably with R, it is recommended to use the RStudio interface: it is free, "Open source" and works with Windows, Mac and Linux. https://www.rstudio.com/

OBSERVATION: The gender perspective in teaching goes beyond the contents of the subjects, since it also implies a revision of the teaching methodologies and of the interactions between the students and the teaching staff, both in the classroom and outside. In this sense, participatory teaching methodologies, where an egalitarian, less hierarchical environment is generated in the classroom, avoiding stereotyped examples in gender and sexist vocabulary, with the aim of developing critical reasoning and respect for the diversity and plurality of ideas, people and situations, tend to be more favorable to the integration and full participationof the students in the classroom, and therefore their effective implementation in this subject will besought.

Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.


Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Practical classes 10 0.4 1, 16, 2, 5, 6, 10, 8, 9, 12, 13, 17
Problems class 12 0.48 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14
Theory classes 27 1.08 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14
Type: Autonomous      
Exams 15 0.6 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14
Problems resolution 33 1.32 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14
Workshop resolution 23 0.92 1, 16, 11, 3, 2, 4, 5, 6, 10, 8, 9, 7, 15, 14, 12, 13, 17

Assessment

During the lecture sessions the basic concepts of the subject will be introduced and a wide set of examples will be presented. In the problems and practices sessions, exercises will be solved and practices with R will be carried out. Classroom attendance is recommended to have an idea about the course in general, as well as the exercises and practices.

 

Assessment:

The student's grade will be the weighted average of the following activities:

PAC1: partial exam, which accounts for 25% of the grade.
PAC2: delivery of the answer sheets of the computer practices with R carried out in the classroom, which represents 15% of the grade.
Final exam: which will consist of some conceptual questions in the form of short questions and some problems in which you will have to solve a series of exercises similar to those that have been worked on in class sessions. This exam represents the remaining 60% of the grade.

Important: if the grade for any of these activities does not reach 3 out of 10, it will count as 0 in the calculation of the final grade.

Recovery: if this grade does not reach 5, the student has the right to another opportunity to pass the subject through the recovery exam. In this exam, 85% of the grade corresponding to the final exam and PAC1 can be recovered. The practical part with R (PAC2) is not recoverable. In no case can the recovery exam be used to raise the grade if the student has already passed the subject with the first exam.

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Single evaluation:

The student who has taken advantage of the single evaluation modality must take a final exam that will consist of some conceptual questions in the form of short questions and some problems in which they will have to solve a series of exercises similar to those that have been worked on in class sessions. When they finish, they will deliver, inaddition to the exam, the answer sheets for the computer practices with R carried out throughout the course.

The student's grade will be the weighted average of the two previous activities, where the final exam will account for 85% of the mark, and the evaluation of the answer sheets of the computer practices with R the remaining 15%.

Important: if the grade for any of these activities does not reach 3 out of 10, it will count as 0 in the calculation of the final grade.

If this qualification does not reach 5, the student has the right to another opportunity to pass the subject through the recovery exam that will be held on the date set by the coordination of the degree. In this exam it will be possible to recover 85% of the mark corresponding to the final exam. The practical part with R is not recoverable. In no case can the recovery exam be used to raise the grade if the student has already passed the subject with the first exam.


Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Final exam 0,60 10 0.4 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14
Grading exercises 0,15 12 0.48 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 7, 15, 14
Mid-term exam 0,25 8 0.32 1, 16, 11, 3, 2, 4, 5, 6, 10, 8, 9, 15, 14, 12, 13, 17

Bibliography

  1. Berger, R.L., Casella, G.: Statistical Inference. Duxury Advanced Series. 2002.
  2. Daalgard, P.: Introductory Statistics with R. Springer. 2008.
  3. Daniel, W.W.: Biostatistics. Wiley. 1974.
  4. DeGroot, M. H.: Schervish, M.J. Probability and Statistics. Pearson Academic. 2010.
  5. Delgado, R.: Probabilidad y Estadística con aplicaciones. 2018. https://www.amazon.es/Probabilidad-Estad%C3%ADstica-aplicaciones-Rosario-Delgado/dp/1983376906
  6. Peña, D.: Estadística. Fundamentos de estadística. Alianza Universidad. 2001.
  7. R Tutorial. An introduction to Statistics. https://cran.r-project.org/manuals.html. juny 2019.
  8. Silvey, S.D.: Statistical Inference. Chapman&Hall. 1975.
  9. Held, Sabanés and Bové (2013): Applied Statistical Inference: Likelihood and Bayes. Springer
  10. Pawitan (2013): In all Likelihood: Statistical Modelling and Inference using Likelihood. Oxford University Press
  11. Young, Smith (2005): Essentials of Statistical Inference. Cambridge University Press
  12. Cox, D.R. and Hinkley, D.V. (1979). Theoretical Statistics. 1st Edition, Chapman and Hall/CRC

Software

R Core Team (2021). R: A language and environment for statistical computing. R
  Foundation for Statistical Computing, Vienna, Austria. URL
  https://www.R-project.org/.