  The diversity of the immune repertoire is initially generated by random
rearrangements of the receptor gene during early T and B cell development.
Rearrangement scenarios are composed of random events -- choices of gene
templates, base pair deletions and insertions -- described by probability
distributions. Not all scenarios are equally likely, and the same receptor
sequence may be obtained in several different ways. Quantifying the
distribution of these rearrangements is an essential baseline for studying the
immune system diversity. Inferring the properties of the distributions from
receptor sequences is a computationally hard problem, requiring enumerating
every possible scenario for every sampled receptor sequence. We present a
Hidden Markov model, which accounts for all plausible scenarios that can
generate the receptor sequences. We developed and implemented a method based on
the Baum-Welch algorithm that can efficiently infer the parameters for the
different events of the rearrangement process. We tested our software tool on
sequence data for both the alpha and beta chains of the T cell receptor. To
test the validity of our algorithm, we also generated synthetic sequences
produced by a known model, and confirmed that its parameters could be
accurately inferred back from the sequences. The inferred model can be used to
generate synthetic sequences, to calculate the probability of generation of any
receptor sequence, as well as the theoretical diversity of the repertoire. We
estimate this diversity to be $\approx 10^{23}$ for human T cells. The model
gives a baseline to investigate the selection and dynamics of immune
repertoires.
