  Current statistical methods in differential proteomics analysis generally
leave aside several challenges, such as missing values, correlations between
peptide intensities and uncertainty quantification. Moreover, they provide
point estimates, such as the mean intensity for a given peptide or protein in a
given condition. The decision of whether an analyte should be considered as
differential is then based on comparing the p-value to a significance
threshold, usually 5%. In the state-of-the-art limma approach, a hierarchical
model is used to deduce the posterior distribution of the variance estimator
for each analyte. The expectation of this distribution is then used as a
moderated estimation of variance and is injected directly into the expression
of the t-statistic. However, instead of merely relying on the moderated
estimates, we could provide more powerful and intuitive results by leveraging a
fully Bayesian approach and hence allow the quantification of uncertainty. The
present work introduces this idea by taking advantage of standard results from
Bayesian inference with conjugate priors in hierarchical models to derive a
methodology tailored to handle multiple imputation contexts. Furthermore, we
aim to tackle a more general problem of multivariate differential analysis, to
account for possible inter-peptide correlations. By defining a hierarchical
model with prior distributions on both mean and variance parameters, we achieve
a global quantification of uncertainty for differential analysis. The inference
is thus performed by computing the posterior distribution for the difference in
mean peptide intensities between two experimental conditions. In contrast to
more flexible models that can be achieved with hierarchical structures, our
choice of conjugate priors maintains analytical expressions for direct sampling
from posterior distributions without requiring expensive MCMC methods.
