lesson
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lesson topics
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Part 1: Probability theory and probability models
1
29/10/2019
Introduction to the course.
Review of the basic concepts of probability. Basic
combinatorics. Stirling's formula.
2
2
31/10/2019
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. Independent events. Statistical independence
and dimensional reduction.
4
3
7/10/2019
Elementary examples of Bayesian
inference. Conditional probabilities in stochastic modeling:
gambler's ruin and probabilistic diffusion models. Discrete
and continuous forms of gambler's ruin. Overview of random
walk models in physics.
6
4
12/11/2019
Proof 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. 8
5
14/11/2019
Introduction to statistical visualization and
to exploratory statistics.
10
6
19/11/2019
Uniform distribution. Buffon's
needle.
Review of dispersion (variance) and its properties.
Chebyshev's inequality. 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. 12
7
21/11/2019
Brief review of common
probability models. 1. The uniform distribution; 2. the
Bernoulli distribution; 3 the binomial distribution. 4. the
geometric distribution; 5. the negative binomial distribution.
5. the hypergeometric distribution; 6. the multinomial
distribution .
14
8
26/11/2019
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. Properties of the
normal distribution.
16
9
28/11/2019
Distributions (ctd):
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.
18
10
3/12/2019
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. Bertrand's
paradox. Overview of Jaynes' solution of Bertrand's paradox.
20
11
5/12/2019
Introduction to probability
generating and characteristic functions. Generating and
characteristic functions of some common distributions. PGF of
the Galton-Watson branching process.
22
12
10/12/2019
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).
24
Part
2: Statistical inference
13
12/12/2019
Descriptive and exploratory statistics. Sample mean, sample
variance, estimate of covariance and correlation coefficient.
PDF of sample mean for exponentially distributed data.
26
14
17/12/2019
Order statistics. Box plots. Outliers. Violin plots. Rug
plots. Kernel density plots.
Introduction to the Monte Carlo method. Early history of the
Monte Carlo method. Pseudorandom numbers.Uniformly distributed
pseudorandom numbers. Transformation method. Transformation
method and the trasformation of differential cross sections.
Acceptance-rejection method. 28
15
19/12/2019
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.
30
16
7/01/2020
Statistical bootstrap.
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)32
17
7/01/2020
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.
34
18
9/01/2020
Maximum likelihood method 2.
Cramer-Rao-Fisher bound. Variance of ML estimators.
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.
36
19
10/01/2020Maximum
likelihood method 3. Non-uniqueness of the likelihood
function. 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. Extended
maximum likelihood. Examples. Introduction to ML with binned
data.
38
20
13/01/2020
Introduction to confidence
intervals. Confidence intervals and confidence level. Detailed
analysis of the Neyman construction of confidence intervals (link
to the Neyman paper). Maximum likelihood method 4.
Example of ML with binned data: decay rate in radioactive
decay. Example of ML with binned data: power laws.
40
21
14/01/2020
Maximum likelihood method 5.
Example of confidence level estimation with the graphical
method.
Chi-square and its relation to ML. Very brief overview of
chi-square and least squares fits. Chi-square distribution. 42
22
14/01/2020
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.
44
23
16/01/2020
p-values and multiple testing (Bonferroni inequality, Sidak
correction and Bonferroni correction. The False Detection Rate
(FDR) and the Benjamini-Hochberg procedure. The principal
component method (PCA): singular value decomposition;
statistical independence of rotated variates; ranking of
variates and dimensional reduction. Example of pattern
recognition performed with the help of PCA (link
to the paper by Sirovich and Kirby).
46