Mogens Bladt og Mathias Drton afholder tiltrædelsesforelæsninger med efterfølgende reception. Mogens Bladt lægger ud kl. 14:00 og 14:45 følger Mathias Drton. Kl. 15:30 er der reception for de to; instituttet inviterer alle på en forfriskning foran AUD 1.

Inaugural lectures from Mogens Bladt and Mathias Drton followed by a reception

Mogens Bladt starts at 14:00, followed at 14:45 by Mathias Drton. At 15:30 there will be a reception for the two; The Department invites everyone to a refreshment in front of AUD 1.

Mogens Bladt’s lecture:

Multivariate phase-type distributions

Abstract: Matrices have proven their power and convenience in many fields of mathematics being a convenient way of keeping track of multiple parameters. In probability theory, likewise, a class of so-called matrix-exponential distributions can be defined by essentially replacing the scalar parameter of the exponential distribution by a matrix. This class of distributions coincide with the distributions on the positive reals having rational Laplace transforms. An important sub-class of these distributions are generated by Markov processes on a finite state-space which eventually get absorbed into a single state. The time until the absorption occurs is said to have a phase-type distribution.

We review the construction of phase-type distributions and a multivariate extension and exemplify their use by a recent application in population genetics. It turns out that a number of coalescent models may conveniently be described in terms of univariate and multivariate phase-type distributions, and thereby be considered as special cases of a generic phase-type coalescent model. Properties and formulae of interest in the generic model are expressed compactly in terms of functions of matrices, by which we obtain a greater level of transparency and potential for performing further calculations by drawing on known results from phase-type theory. In particular, we provide methods for calculating the distributions of tree heights, total branch lengths and the number of segregating sites, both with or without recombinations present, as well as closed-form formulae for (joint) moments of any order.

Multivariate phase-type distributions may find applications in a number of other fields like for example life-insurance models with multiple categories.

Mathias Drton’s lecture:

Maximum likelihood thresholds of multivariate statistical models

Abstract: Statistical analysis of multivariate data relies heavily on models based on the multivariate normal distributions. An interesting invariant of such Gaussian models is the maximum threshold, by which we mean the minimal sample size needed for almost sure existence of the maximum likelihood estimator. I will review some examples in the area of graphical models and then discuss ongoing work that studies the thresholds of models whose covariance matrices are Kronecker products.