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on a specific parametric model. Second, we will later build on the probability model
to define a formal decision problem for the selection of a final list of tripeptide-tissue
pairs. For this, and for a simulation study to validate the model, we have to rely on
efficient and fast computations.
These two competing desiderata lead us to consider a semi-parametric mixture
model. We will use a mixture of parametric models, with a nonparametric prior
on the mixing measure. Under the Bayesian paradigm, nonparametric priors refer
to probability models on probability distributions. The non-parametric prior on the
mixing measure greatly generalizes the underlying parametric model, in much the
same way as a mixture of independent normal kernels can approximate arbitrary
multivariate distributions in kernel density estimates. Similar semi-parametric mix-
ture models have successfully been applied for Bayesian inference in a variety of other
applications, including, for example, Müller and Rosner (1997), Mukhopadhyay and
Gelfand (1997), and Kleinman and Ibrahim (1998). The special case of binary out-
comes has been discussed, among many others, by Basu and Mukhopadhyay (2000).
We start with a sampling model for Ni conditional on assumed mean counts
across stages for the peptide-tissue combination i. Conditional on the mean counts
we assume independent Poisson sampling. In anticipation of the final inference goal
we parameterize the mean counts as (μi, μiβi, μiδβ, allowing us to describe increasing
mean counts by the simple event 1 < βi < δi. We write Poi(x ∣ m) to indicate a
Poisson distributed random variable x with mean m.
p(Nil, Ni2, Ni3 I μi, βi, δi) = Poi(l¼ι ∣ μi) Poi(2Vi2 ∣ μiβi) Poι(Ni3 ∣ μiδi) (3.26)
for i = 1,.. .n. The sampling model includes different poison-slopes for each stage,
in contrast to the model proposed by Ji et al. (2007). The parameter μi can be
thought as the expected count mean of the pair i across the three stages if we were
not enriching the tripeptide library at every stage. We extend (3.26) to the desired
semi-parametric mixture model by assuming a non-parametric prior for a random
effects distribution for θl ≡ (∕¾, δl). Let G(b, d) denote a bivariate random probability