Lab |

[ 2a.2 ] Visualization

[ 2a.3 ] Transformations

[ 2a.3.1 ] Marginalization

[ 2a.3.2 ] Pushforward

[ 2a.3.3 ] Law of the unconscious statistician

[ 2a.3.4 ] Change of measure

[ 2a.4 ] Functionals

[ 2a.4.1 ] Key definitions

[ 2a.4.2 ] Mode

[ 2a.4.3 ] Quantile family

[ 2b.1 ] Mean-covariance classes

[ 2b.1.1 ] Mean vector

[ 2b.1.2 ] Covariance matrix

[ 2b.1.3 ] Equivalence class

[ 2b.1.4 ] Z-score

[ 2b.2 ] Visualization

[ 2b.2.1 ] Spectral decomposition

[ 2b.2.2 ] Mean-covariance ellipsoid

[ 2b.2.3 ] Principal component analysis

[ 2b.3 ] Affine transformations

[ 2b.3.1 ] Mean vector

[ 2b.3.2 ] Covariance matrix

[ 2b.3.3 ] Equivalence class

[ 2b.3.4 ] Marginalization

[ 2b.3.5 ] Z-score invariance

[ 2b.3.6 ] Higher order moments

[ 2b.4 ] Elicitability

[ 2b.4.1 ] Individual versus joint

[ 2b.4.2 ] Minimum volume ellipsoid

[ 2b.4.3 ] Minimum z-score parameters

[ 3.1 ] Distributions geometry

[ 3.1.1 ] Fisher metric: length and volume

[ 3.1.2 ] Flatness and geodesics

[ 3.1.3 ] Duality: potentials and Legendre transformations

[ 3.1.4 ] Distance and divergence

[ 3.2 ] Exponential distributions geometry

[ 3.3 ] Scenario-probability distribution geometry

[ 4a.1 ] Independence

[ 4a.2 ] Conditioning

[ 4a.2.1 ] Conditional distribution

[ 4a.2.2 ] Bayes theorem

[ 4a.2.3 ] Compound distributions

[ 4a.3 ] Conditional independence

[ 4a.4 ] Innovation

[ 4a.4.1 ] Multivariate quantile

[ 4a.4.2 ] Independent component analysis

[ 4a.4.3 ] Decomposition

[ 4a.5 ] Conditional functionals

[ 4a.5.1 ] Mode

[ 4a.5.2 ] Mean family

[ 4a.5.3 ] Quantile family

[ 4a.5.4 ] Elicitable functionals

[ 4b.1 ] Orthogonality

[ 4b.2 ] Linear projection

[ 4b.2.1 ] Geometrical interpretation

[ 4b.2.2 ] Iterated projections

[ 4b.2.3 ] From ∥ projection to conditioning

[ 4b.3 ] Partial orthogonality

[ 4b.4 ] Linear innovation

[ 4b.4.1 ] Orthogonalization

[ 4b.4.2 ] Gram-Schmidt

[ 4b.4.3 ] Orthonormal components

[ 4b.4.4 ] Decomposition

[ 5a.1 ] Univariate results

[ 5a.2 ] Definition and properties

[ 5a.2.1 ] Grades

[ 5a.2.2 ] Copula

[ 5a.2.3 ] Sklar’s theorem

[ 5a.2.4 ] Copula invariance

[ 5a.3 ] Special classes of copulas

[ 5a.3.1 ] Elliptical copulas

[ 5a.3.2 ] Archimedean copulas

[ 5a.4 ] Implementation

[ 5a.4.1 ] Copula-marginal separation

[ 5a.4.2 ] Copula-marginal combination

[ 5b.1 ] Measures of dependence

[ 5b.1.1 ] Schweizer-Wolff measure

[ 5b.1.2 ] Mutual information

[ 5b.2 ] Measures of concordance

[ 5b.2.1 ] Kendall’s tau

[ 5b.2.2 ] Spearman’s rho

[ 5b.3 ] Correlation

[ 5b.3.1 ] Related definitions

[ 5b.4 ] Points of interest, pitfalls, practical tips

[ 5b.4.1 ] Schweizer and Wolff measure via simulations

[ 6.1 ] Statistical decision problems

[ 6.1.1 ] State

[ 6.1.2 ] Action

[ 6.1.3 ] Loss function

[ 6.1.4 ] Reward

[ 6.2 ] Stochastic dominance

[ 6.2.1 ] Strong dominance

[ 6.2.2 ] Weak dominance

[ 6.2.3 ] Higher order dominance

[ 6.2.4 ] Stochastic dominance failure

[ 6.3 ] Bayes risk

[ 6.3.1 ] Problem without inputs

[ 6.3.2 ] Problem with inputs

[ 6.3.3 ] Frequentist approach

[ 6.3.4 ] Bayesian approach

[ 6.3.5 ] A unified view

[ 6.3.6 ] Admissibility

[ 6.3.7 ] Model and estimation risk

[ 6.4 ] Minimax and other approaches

[ 6.4.1 ] Minimax

[ 6.4.2 ] Minimax versus Bayes

[ 6.4.3 ] Generalizations

[ 6.5 ] Points of interest

[ 6.5.1 ] Randomized decisions

[ 6.5.2 ] Improper priors

[ 6.5.3 ] Generalized entropy and excess risk

[ 6.5.4 ] Propriety

[ 6.5.5 ] Stochastic optimization

[ 7a.1 ] Affine equivariance

[ 7a.1.1 ] Key definitions

[ 7a.1.2 ] Mode/modal dispersion

[ 7a.1.3 ] Median/interquantile range

[ 7a.1.4 ] Multivariate extensions

[ 7a.2 ] Variational principles

[ 7a.2.1 ] Key definitions

[ 7a.2.2 ] Elicitable and quadrangle features

[ 7a.2.3 ] Quantile family

[ 7a.2.4 ] Mean absolute deviation family

[ 7a.2.5 ] Median absolute deviation

[ 7a.2.6 ] Other locations/dispersions

[ 7a.2.7 ] Multivariate extensions

[ 7a.3 ] Points of interest, pitfalls, practical tips

[ 7a.3.1 ] Visualization via uncertainty bands

[ 7b.1 ] Key concepts

[ 7b.1.1 ] Probability space

[ 7b.1.2 ] Random variable

[ 7b.1.3 ] Expectation

[ 7b.1.4 ] Radon-Nikodym derivative

[ 7b.1.5 ] Abstract distributions

[ 7b.1.6 ] Conditional probability

[ 7b.2 ] L2 spaces of random variables

[ 7b.2.1 ] Inner product

[ 7b.2.2 ] Length, distance and angle

[ 7b.2.3 ] Visualization

[ 7b.2.4 ] Geometry of random vectors

[ 7b.2.5 ] Projection

[ 7b.2.6 ] Covariance (improper) inner product

[ 7b.3 ] Abstract conditional expectation

[ 7b.3.1 ] Partitions of the sample space

[ 7b.3.2 ] Probability conditional on a partition

[ 7b.3.3 ] Discretization of random variables

[ 7b.3.4 ] Conditional discretization of random variables

[ 7b.3.5 ] Abstract Bayes theorem

[ 7c.1 ] Normal distribution

[ 7c.1.1 ] Pdf, cdf and characteristic function

[ 7c.1.2 ] Moments

[ 7c.1.3 ] Conditional distribution

[ 7c.1.4 ] Stochastic representations

[ 7c.1.5 ] Affine equivariance

[ 7c.1.6 ] Matrix-normal distribution

[ 7c.2 ] Elliptical distributions

[ 7c.2.1 ] Fundamental concepts

[ 7c.2.2 ] Student t

[ 7c.2.3 ] Cauchy

[ 7c.2.4 ] Uniform inside the ellipsoid

[ 7c.2.5 ] Uniform on the ellipsoid

[ 7c.2.6 ] Affine equivariance

[ 7c.2.7 ] Stochastic representations

[ 7c.2.8 ] Generation of elliptical scenarios

[ 7c.2.9 ] Scenario generation with dimension reduction

[ 7c.3 ] Chi-squared, gamma and Wishart distributions

[ 7c.3.1 ] Chi-squared distribution

[ 7c.3.2 ] Gamma distribution

[ 7c.3.3 ] Wishart distribution

[ 7c.3.4 ] Inverse-Wishart distribution

[ 7c.4 ] Scenario-probability distributions

[ 7c.4.1 ] Types of scenario-probability distributions

[ 7c.4.2 ] Probability mass and density function

[ 7c.4.3 ] Transformations and generalized expectations

[ 7c.4.4 ] Cumulative distribution function

[ 7c.4.5 ] Quantile

[ 7c.4.6 ] Moments and other statistical features

[ 7c.5 ] Categorical distributions

[ 7c.5.1 ] Discriminant variables

[ 7c.5.2 ] Probabilities parametrization

[ 7c.6 ] Exponential family distributions

[ 7c.6.1 ] Normal distribution

[ 7c.6.2 ] Categorical distribution

[ 7c.7 ] Mixture distributions

[ 7c.7.1 ] Binary case

[ 7c.7.2 ] Multinomial case

[ 7c.8 ] Stable, additive and infinitely divisible distributions

[ 7c.8.1 ] Stable distributions

[ 7c.8.2 ] Additive distributions

[ 7c.8.3 ] Infinitely divisible distributions

[ 7c.9 ] Other notable distributions

[ 7c.9.1 ] Lognormal distribution

[ 7c.9.2 ] Quadratic-normal distribution

[ 7c.10 ] Moment-matching scenarios

[ 7c.10.1 ] Twisting scenarios

[ 7c.10.2 ] Twisting probabilities

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