Browsing by Snapshot Mathematical Subject "Probability Theory and Statistics"
Now showing items 120 of 22

The Algebraic Statistics of an Oberwolfach Workshop
[SNAP2018001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180227)Algebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by ... 
Biological shape analysis with geometric statistics and learning
[SNAP2022008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221025)The advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold ... 
Counting selfavoiding walks on the hexagonal lattice
[SNAP2019006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190604)In how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these socalled selfavoiding ... 
Cutoff Phenomenon: Surprising Behaviour in Card Shuffling and other Markov Chains
[SNAP2023005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20231221)This snapshot compares two techniques of shuffling a deck of cards, asking how long it will take to shuffle the cards until a “wellmixed deck” is obtained. Surprisingly, the number of shuffles can be very different for ... 
Determinacy versus indeterminacy
[SNAP2020004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200422)Can a continuous function on an interval be uniquely determined if we know all the integrals of the function against the natural powers of the variable? Following Weierstrass and Stieltjes, we show that the answer is yes ... 
Domino tilings of the Aztec diamond
[SNAP2015016EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)Imagine you have a cutout from a piece of squared paper and a pile of dominoes, each of which can cover exactly two squares of the squared paper. How many different ways are there to cover the entire paper cutout with ... 
The Enigma behind the Good–Turing formula
[SNAP2021008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210716)Finding the total number of species in a population based on a finite sample is a difficult but practically important problem. In this snapshot, we will attempt to shed light on how during World War II, two cryptanalysts, ... 
Estimating the volume of a convex body
[SNAP2018015EN] (Mathematisches Forschungsinstitut Oberwolfach, 20181230)Sometimes the volume of a convex body needs to be estimated, if we cannot calculate it analytically. We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape. 
FokusErkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse
[SNAP2016008DE] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Viele epileptische Anfälle entstehen in einer begrenzten Region im Gehirn, dem sogenannten Anfallsursprung. Eine chirurgische Entfernung dieser Region kann in vielen Fällen zu Anfallsfreiheit führen. Aus diesem Grund ist ... 
The KadisonSinger problem
[SNAP2014008EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)In quantum mechanics, unlike in classical mechanics, one cannot make precise predictions about how a system will behave. Instead, one is concerned with mere probabilities. Consequently, it is a very important task to ... 
Limits of graph sequences
[SNAP2019010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190904)Graphs are simple mathematical structures used to model a wide variety of reallife objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent ... 
Quantum diffusion
[SNAP2015014EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)If you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see ... 
Random matrix theory: Dyson Brownian motion
[SNAP2020002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200415)The theory of random matrices was introduced by John Wishart (1898–1956) in 1928. The theory was then developed within the field of nuclear physics from 1955 by Eugene Paul Wigner (1902–1995) and later by Freeman John ... 
Random permutations
[SNAP2019007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190712)100 people leave their hats at the door at a party and pick up a completely random hat when they leave. How likely is it that at least one of them will get back their own hat? If the hats carry name tags, how difficult ... 
Randomness is Natural  an Introduction to Regularisation by Noise
[SNAP2024002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20240522)Differential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic ... 
Searching for structure in complex data: a modern statistical quest
[SNAP2021003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210329)Current research in statistics has taken interesting new directions, as data collected from scientific studies has become increasingly complex. At first glance, the number of experiments conducted by a scientist must ... 
Solving inverse problems with Bayes' theorem
[SNAP2022006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220905)The goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as ... 
Statistics and dynamical phenomena
[SNAP2014006EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)A friend of mine, an expert in statistical genomics, told me the following story: At a dinner party, an attractive lady asked him, "What do you do for a living?" He replied, "I model." As my friend is a handsome man, the ... 
Topological recursion
[SNAP2018002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180305)In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last decade. After introducing ... 
Visual analysis of Spanish male mortality
[SNAP2015012EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)Statistical visualization uses graphical methods to gain insights from data. Here we show how a technique called principal component analysis is used to analyze mortality in Spain over about the last hundred years. This ...