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Title 1: Sampling bias in logistic models

This talk is concerned with regression models for the effect of
covariates on correlated binary and correlated polytomous responses.
In the conventional construction of generalized linear mixed models,
correlations are induced by a random effect, additive on the logistic scale.
The joint distribution $p_{\bfx}(\bfy)$ obtained by integration
depends on the covariate values $\bfx$ on the sampled units.
The thrust of this talk is that the conventional formulation
may not be appropriate for natural sampling schemes in which the
sampled units arise from a random process such as sequential recruitment.
The conventional analysis incorrectly predicts parameter attenuation
due to the random effect, thereby giving a misleading impression of the
magnitude of treatment effects.
The error in the GLMM analysis is a subtle consequence
of selection bias that arises from random sampling of units.
This talk will describe a non-standard but mathematically natural formulation
in which auto-generated units are subsequently
selected by an explicit sampling plan.
For a quota sample in which the $\bfx$-configuration is pre-specified,
the model distribution coincides with $p_{\bfx}(\bfy)$ in the GLMM.
However, if the covariate configuration is random, for example the values
obtained by simple random sampling from the available population,
the conditional distribution $p(\bfy \given \bfx)$ is different from $p_\bfx(\bfy)$.
By contrast with conventional models,
conditioning on~$\bfx$ is not equivalent to stratification by~$\bfx$.
The implications for likelihood computations and estimating equations
will be discussed.

{\tt www.stat.uchicago.edu/\tie pmcc/reports/bias.pdf}
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Title 2: Partition models and cluster processes

This talk will begin with a gentle introduction to the notion of an
exchangeable random partition, continuing with a more detailed discussion
of the Ewens process and some of its antecedents.
The concept of an exchangeable cluster process will be described,
the main example being the Gauss-Ewens process.
Some applications of cluster processes will be discussed,
including problems of classification or supervised learning, and cluster analysis.
The talk will conclude with some remarks about exchangeable fragmentation trees
and tree-based cluster processes.
(The talk is based partly on joint work with Jie Yang,
Jim Pitman and Matthias Winkel)

{\tt www.stat.uchicago.edu/\tie pmcc/reports/class.pdf}
{\tt www.stat.uchicago.edu/\tie pmcc/reports/clusters.pdf}
{\tt www.stat.uchicago.edu/\tie pmcc/reports/Gibbs.pdf}
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Title 3:  Some remarks on fiducial inference

The fiducial argument is Fisher's attempt to make inferential statements about
the likely value of parameters without recourse to a specific prior distribution
on the parameter space.
In its original form using pivotal quantities, the fiducial argument is 
vitrually indistinguishable from Neyman's theory of confidence intervals.
Fisher strove to distance his theory from that of confidence intervals
by emphasizing correctly the importance for inference
of ancillary statistics and recognizable subsets.
Despite many efforts by Fisher, Fraser and Barnard and others over the years,
the passage from a pivotal statistic to a probability distribution on
the parameter space remains a conceptual stumbling block.
In this talk, I will address the question of whether, in any circumstances,
parametric inference is possible without a prior distribution.
The view initially taken is similar to that of Hora, Buehler, Geisser and Dawid,
that inference and prediction are indistinguishable activities.
From this point of view, the parameter space is relatively unimportant,
and the observation space becomes the focus of inferential activity.
The goal, therefore, is how best to replace the {\it set\} of processes $\{P_\theta\}$
with a single process $Q$ to be used for inferential purposes.
Each mixture $P_\nu = \int P_\theta\, \nu(d\theta)$ is a Bayesian option,
which Fisher sought to avoid.
In a group-generated model, another option is to use the invariant process,
which is unique, or its fiducial extension, which is effectively unique.
Connections with marginal likelihood, REML and Kriging will be discussed.
(This talk is based partly on joint work with V.~Vovk.)

\bye