The BIOSI project, based in Limerick and Galway, invites you to the following seminars to take place as follows:
Friday, May 7th, 2pm
ROOM CG054, MAIN BUILDING
UNIVERSITY OF LIMERICK
2pm Chris Cannings (University of Sheffield) The Coalescent
3pm Florian Frommlet (University of Vienna) Asymptotic optimality properties of multiple testing and model selection procedures under sparsity
The abstracts are given below
The Coalescent
Chris Cannings,
School of Mathematics and Statistics,
University of Sheffield
The WrightFisher model of "genetic drift" considers a population of
fixed size N, with nonoverlapping generations. Each individual in the
population at time t+1 is the offspring of any particular individual in the population
at time t with probability 1/N, (mutually) independently of the parentage of other
individuals. It is possible in this and in a wider class of
models (Cannings,1974) to specify the eigenvalues and
eigenvectors in a fairly full fashion and thus to study various
aspects of the process.
Kingman(1982) introduced a major insight
for the study of such processes, the Coalescent. Instead of
looking at whole generations with time running forward the
coalescent runs time backward. Since the number of individuals in
generation t who are actually, rather than potentially, parents of some set of k
individuals in generation t+1, is ˜ k (with nonzero probability for
<k), the ancestry of any set of individuals is a tree running
backwards in time to a MRCA (most recent common ancestor of the
set). This is the coalescent, and the study of the genetic drift process
reduces to the study of this tree.
A brief survey of some of the features of this process will include
the probabilities of various tree topologies, and the time to the
MRCA.
The coalescent approach will then be used to tackle two problems.
(1) Suppose that individuals have a type specified by an integer,
and that a parent of type x produces an offspring of type x1,
x or x+1 with probabilities u/2, 1u and u/2. We
wish to specify aspects of the distribution of the types in the
population at some time n. This model correspond to genetic
situations in which an individual has some number of copies of a
genetic unit, and that due to errors in the copying process required
for producing an offspring, that offspring may have a slightly
different number of copies. We derive certain results for the moments of
this and a related normalised process.
(2) The branch from an individual backwards in time to the
coalescent tree is called an external edge. Some results regarding
the lengths of such edges will be derived.
Malgorzata Bogdan, Arijit Chakrabarti, Florian Frommlet, Jayanta K. Ghosh
Asymptotic optimality properties of multiple testing and model selection procedures under sparsity
Asymptotic optimality of a large class of multiple testing rules is investigated using
the framework of Bayesian Decision Theory. A normal scale mixture model is considered, leading to an asymptotic
framework which can be naturally motivated under the assumption of sparsity,
where the proportion of ``true'' alternatives converges to zero. Within this setup optimality of a rule is proved by
showing that the ratio of its Bayes risk and that of the
Bayes oracle (a rule which minimizes the Bayes risk) converges to one.
The class of fixed threshold multiple testing rules which are asymptotically optimal is fully characterized as well
as the class of optimal rules controlling the Bayesian False Discovery Rate (BFDR).
Furthermore, conditions are provided under which the popular BenjaminiHochberg procedure is asymptotically optimal.
It is shown that for a wide class of sparsity levels, the threshold of the former can
be approximated very well by a nonrandom threshold.
Apart from multiple testing the problem of model selection for multiple regression under sparsity is considered.
Under the assumption of orthogonality and for known variances results from multiple testing immediately translate into the regression setting,
where the scale mixture model is extended to a more general class of priors.
We illustrate asymptotic optimality properties of modified versions of the Bayesian Information Criterion (mBIC),
where we specifically discuss modifications allowing to control FDR. Finally optimality of mBIC in the case of unknown variances is proven.
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