  Tumorigenesis is an evolutionary process which involves a significant number
of genomic rearrangements typically coupled with changes in the gene copy
number profiles of numerous cells. Fluorescence in situ hybridization (FISH) is
a cytogenetic technique which allows counting copy numbers of genes in single
cells. The study of cancer progression using FISH data has received
considerably less attention compared to other types of cancer datasets.
  In this work we focus on inferring likely tumor progression pathways using
publicly available FISH data. We model the evolutionary process as a Markov
chain in the positive integer cone Z_+^g where g is the number of genes
examined with FISH. Compared to existing work which oversimplifies reality by
assuming independence of copy number changes, our model is able to capture
dependencies. We model the probability distribution of a dataset with
hierarchical log-linear models, a popular probabilistic model of count data.
Our choice provides an attractive trade-off between parsimony and good data
fit. We prove a theorem of independent interest which provides necessary and
sufficient conditions for reconstructing oncogenetic trees. Using this theorem
we are able to capitalize on the wealth of inter-tumor phylogenetic methods. We
show how to produce tumor phylogenetic trees which capture the dynamics of
cancer progression. We validate our proposed method on a breast tumor dataset.
