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conjugate families. A simple inspection of the data suggests that there are outliers.
The semi-parametric nature of the model allows for possible heterogeneity of the data
and robustifies inference. A limitation of the model is that it assumes all pairs to be a
priori exchangeable. Instead, a more realistic model should assume that the peptide
A in tissue T is more likely to behave similar to the same peptide A in other tissue
or similar to any other peptide in the same tissue T.

In the same chapter, we propose an inference from a decision theoretic point of
view. At the moment of selecting the list of pairs to report to the biologist collaborator
for further research and interpretation, we face a massive multiplicity problem. For
each peptide∕tissue pair we are testing the alternative hypothesis of increasing means.
The most commonly used multiplicity adjustment is Bonferroni’s Criterion. This
procedure works well for testing few hypotheses, but it is too conservative when
the number of hypotheses is very large. A more recent method to address massive
multiplicity problems is by controlling the (frequentist) False Discovery Rate (FDR).
FDR is the expected proportion of false positives in the list of reported comparisons
(“discoveries”). We control the related posterior FDR, i.e., the posterior expected
proportion of false discoveries. By controlling I mean establishing an upper bound
on the posterior expected FDR while maximizing the number of reported pairs in
the list. This Bayesian procedure can be characterized as a Bayes rule under a
utility function that considers statistical significance. But the utility function ignores
biological significance. That is, it only takes into account whether or not the pair has
(significantly) increasing counts, but ignores the size of this increase. This observation
leads us to consider an alternative FDR that does account for biologic significance,
simply by modifying the underlying utility function.

In chapter 4 I address inference for non-exchangeable experimental units. The
motivating application is to a clinical trial for rare sarcomas, including patients from
n = 12 different disease subtypes with very slow accrual for some of the subtypes.
A practical clinical trial design requires borrowing of strength across the disease

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