This course serves as an introduction to foundational aspects of modern statistical data analysis. Frequentist and Bayesian inference are presented from the perspective of probabilistic modelling. The course will consist of three main parts:

1) Probability foundations: probability spaces, random variables, distribution of a random variable, expectation and covariance, important limit theorems and inequalities

2) Frequentist inference: point estimators, confidence intervals, hypothesis testing.

3) Bayesian inference: conjugate inference, numerical models, data assimilation.

Prerequisites:

_basic set theory (inclusion, union, intersection, difference of sets)

_basic analysis (infinite series, calculus)  

_matrix algebra

_some knowledge of probabilistic foundations (discrete probability, Gaussian distributions) would be helpful.

References:

_ "All of Statistics, a concise course in statistical inference", Larry Wasserman

_ "Probability and statistics, 4th edition",  DeGroot and Schervish

_ "Bayesian Theory", José M. Bernardo, Adrian F.M. Smith

_ « Applied Statistical inference, likelihood and Bayes », Leonhard Held and Daniel Sabanés Bové

_ « Probabilistic forecasting and Bayesian Data Assimilation », Sebastian Reich and Colin Cotter