**April 3, 2006 --4:00 p.m.**

*Latent variables, uncertainty and evidence.*

In many areas of science, our models involve latent variables
which cannot be observed. Often these variables are such that,
were we able to observe, them the testing of scientific hypotheses
would be straightforward. A classical example is that of Bernoulli
trials (tosses of a fair coin) observed with error. While every
student knows how to construct a test that the coin is fair,
how should uncertainty in observation be taken into account?

Recently, the notions of fuzzy p-values and confidence levels
have been introduced into the statistics literature as a way
to describe the uncertainty inherent in a randomized test. In
latent variable problems, the natural definition of a fuzzy
p-value is the distribution, given observed data, of that function
of latent variables that would be the p-value were the latent
variables observed. This notion puts our uncertainty directly
onto the p-value scale, and permits simultaneous expression
of the strength of the evidence and our uncertainty.

Several simple examples that show both the flexibility and
the usefulness of the approach will be discussed. These examples
require no more than knowledge of the binomial distribution
and the classical p-value, yet are sufficient to show how the
approach provides a new approach to uncertainty in broad areas
of scientific inference.

**April 4, 2006 --4:00 p.m.**

*Uncertainty in inheritance and the detection of genetic linkage*

It has long been recognized that genetic analysis would
be simple if we could observe directly the inheritance of genome
from parents to offspring. However, in human genetic analyses,
this inheritance is often uncertain, even at highly polymorphic
and well sampled genome positions. While Monte Carlo methods
in general, and Markov chain Monte Carlo in particular, permit
imputation of latent variables of scientific interest, simple
integration over imputations loses information regarding our
uncertainty.

As discussed in the first lecture, fuzzy p-values describe
the uncertainty inherent in a randomized test. Using this idea,
and taking as our latent variables the unobservable patterns
of inheritance in pedigrees, we apply this idea to show how
fuzzy p-values can summarize both the strength of evidence for
linkage and the uncertainty about that evidence. The approach
also provides a solution to the long-standing problem of providing
a global significance level for the multiple dependent tests
performed in testing for linkage.

We show how realizations from the fuzzy p-value distribution
may be obtained efficiently with only two sets of Monte Carlo
realizations, one from the unconditional distribution of latent
inheritance patterns, and the other conditional on observed
marker data. No resimulation of marker data is required, and
the procedure, being conditional of the observed marker data,
shares with permutation-based tests a partial robustness to
the genetic map and assumed allele frequencies of the markers.

Elizabeth Thompson received a B.A. in Mathematics (1970), a
Diploma in Mathematical Statistics (1971), and Ph.D. in Statistics
(1974), from Cambridge University, UK. In 1974-5 she was a NATO/SRC
postdoc in the Department of Genetics, Stanford University.
From 1975-81 she was a Fellow of King's College, Cambridge,
and from 1981-5 was Fellow and Director of Studies in Mathematics
at Newnham College. From 1976-1985 she was a University Lecturer
in the Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge. She joined the faculty of the University
of Washington in December 1985, as a Professor of Statistics.

From 1988 to 2004, Dr. Thompson was also Professor of Biostatistics.
Since Spring 2000, she has been an Adjunct Professor in Genetics
(now Genome Sciences) at the University of Washington, and an
Adjunct Professor of Statistics at North Carolina State University.

At the University of Washington, Dr. Thompson was Chair of
the Department of Statistics from 1989-94, and was Graduate
Program Coordinator in Statistics, 1995-8, and 1999-2000. From
1990 to 2002 she was a member of the QERM Interdisciplinary
Graduate Program faculty, and served as the alternate QERM Graduate
Program Coordinator for 1998-9. From 1999-2002 she was also
a member of the interdisciplinary faculty group in Computational
Molecular Biology, but since 1999 has focussed primarily on
the development of research and education in Statistical Genetics
at the University of Washington .