lesson
date
lesson topics
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Part 1: Probability theory and probability models
1
3/10/2017
Introduction to the course.
Review of the basic concepts of probability. Basic
combinatorics.
2
2
5/10/2017
Set theory representation of
probability space. Addition law for probabilities of
incompatible events. Addition law for probabilities of
non-mutually exclusive events. Conditional probabilities.
Bayes' theorem.
4
3
17/10/2017
Independent events. Statistical
independence and dimensional reduction. Overview of random
walk models in physics. Conditional probabilities in
stochastic modeling: gambler's ruin and probabilistic
diffusion models. Discrete and continuous forms of gambler's
ruin.
6
4
19/10/2017
Statement of the first and
second Borel-Cantelli lemmas.
The transition from sample space to random variables. Review
of basic concepts and definitions on discrete and continuous
random variables. Uniform distribution. Buffon's needle.
Review of dispersion (variance) and its properties.
Chebyshev's inequality. 8
5
24/10/2017
From Chebyshev's inequality to the weak law of
large numbers. Other inequalities in probability theory:
Markov's inequality, generalized Chebyshev's inequality.
Strong law of large numbers.
Moment generating function. Brief review of common
probability models. 1. The uniform distribution; 2. the
Bernoulli distribution; 3 the binomial distribution.10
6
26/10/2017
Distributions (ctd): 4. the
geometric distribution; 5. the negative binomial distribution.
5. the hypergeometric distribution; 6. the multinomial
distribution
12
7
31/10/2017
Distributions (ctd): 7. Poisson
distribution; memoryless random processes; Examples: lottery
tickets and the Poisson statistics; cell plating
statistics in biology; 8. exponential distribution. Example:
paralyzable and non-paralyzable detectors. 9. The De Moivre -
Laplace theorem and the normal distribution.
14
9
7/11/2017
Distributions (ctd): Properties
of the normal distribution. Transformations of random
variables. Sum of random variables (convolution). Functions of
random variables. Approximate transformation of random
variables: error propagation. Linear (orthogonal)
transformation of random variables. The multivariate normal
distribution.
16
10
9/11/2017
Distributions (ctd): Other
important distributions (lognormal, gamma, beta, Rayleigh,
logistic, Laplace, Cauchy). Example of a complex model used to
setup a null hypothesis: the distribution of nearest-neighbor
distance.
18
11
13/11/2017
Distribution of
nearest-neighbor distance (ctd.). Bertrand's paradox. Overview
of Jaynes' solution of Bertrand's paradox. Generating
functions.
20
12
14/11/2017
Introduction to probability
generating and characteristic functions. Generating and
characteristic functions of some common distributions. PGF of
the Galton-Watson branching process.
22
13
16/11/2017
Photomultiplier noise. Poisson distribution as limiting case
of a binomial distribution. Moments of a distribution.
Skewness and kurtosis. Mode and median. Properties of
characteristic functions. The Central Limit Theorem
(CLT). The Berry-Esseen theorem. Additive and
multiplicative processes.
24
Part
2: Statistical inference
14
21/11/2017
Descriptive and exploratory statistics. Sample mean, sample
variance, estimate of covariance and correlation coefficient.
PDF of sample mean for exponentially distributed data.
26
15
23/11/2017
Statistical estimators in the German Tank Problem.
Limitations of the standard deviation as a descriptor of the
width of a distribution. Shannon entropy and information
concentration in pdf's. Box plots. Outliers. Violin plots. Rug
plots. Kernel density plots.
Principal Component Analysis (PCA).
28
15
28/11/2017
Principal Component Analysis with the use of Singular Value
Decomposition (SVD). Face recognition as an example of PCA and
SVD.
Introduction to the Monte Carlo method. Early history of the
Monte Carlo method. Pseudorandom numbers. 30
16
5/12/2017
Uniformly distributed
pseudorandom numbers. Transformation method. Transformation
method and the trasformation of differential cross sections.
Acceptance-rejection method. Example: one-dimensional
diffusion process with absorption.
Monte Carlo method examples: Examples: generation of angles in
the e+e- -> mu+mu- scattering; generation of angles in the
Bhabha scattering. The structure of a complete MC program to
simulate low-energy electron transport. 32
17
7/12/2017
The structure of a complete MC
program to simulate low-energy electron transport
(ctd.).
Statistical bootstrap.
34
18
12/12/2017
(11-13) Introduction to Bayesian
methods. Example of Bayesian parametric estimate, estimate of
probability in a binomial model (duality with Beta
distribution, considerations on different priors, importance
of wise choice of prior, information embedded in prior
distribution and effect of data, Bernstein-Von Mises theorem)
(link to presentation)
36
19
12/12/2017
(14-16) Maximum likelihood method 1.
Connection with Bayes' theorem. Point estimators. Properties
of estimators. Consistency of the maximum likelihood
estimators. Asymptotic optimality of ML estimators. Bartlett's
Identities. Cramer-Rao-Fisher bound. Variance of ML
estimators.
38
20
14/12/2017
Maximum likelihood method 2.
Efficiency and Gaussianity of ML estimators. Introduction to
Shannon's entropy. Information measures based on the Shannon's
entropy: Kullback-Leibler divergence, Jeffreys distance,
Fisher information.
40
21
19/12/2017
(9-11)
Maximum likelihood method 3.
Non-uniqueness of the likelihood function. Introduction to
confidence intervals. Confidence intervals and confidence
level. Detailed analysis of the Neyman construction of
confidence intervals (link
to the Neyman paper). Confidence intervals for the
sample mean of exponentially distributed samples. Confidence
intervals for the correlation coefficient of a bivariate
Gaussian distribution from MC simulation. More properties of
the likelihood function. Graphical method for the variance
of ML estimators. Example with two counting channels.
42
22
20/12/2017
(9-11)
Maximum likelihood method 4.
Extended maximum likelihood. Examples. Introduction to ML with
binned data. Example of ML with binned data: decay rate
in radioactive decay. Chi-square and its relation to ML. Very
brief overview of chi-square and least squares fits.
Chi-square distribution.
44
23
20/12/2017
(14-16)
Hypothesis tests, significance level. Examples. Critical
region and acceptance region. Errors of the first and of the
second kind. p-value and rejection of the null hypothesis.
Chi-square as a test statistic. Neyman-Pearson lemma.
Confidence intervals and the Feldman-Cousins construction (link
to the FC paper). CLb, CLs+b and CLs.
46
24
11/01/2017
Final seminar by Diego Tonelli:
Statistical Methods for the Large Hadron Collider (link
to slides).
48