  Model selection in latent block models has been a challenging but important
task in the field of statistics. Specifically, a major challenge is encountered
when constructing a test on a block structure obtained by applying a specific
clustering algorithm to a finite size matrix. In this case, it becomes crucial
to consider the selective bias in the block structure, that is, the block
structure is selected from all the possible cluster memberships based on some
criterion by the clustering algorithm. To cope with this problem, this study
provides a selective inference method for latent block models. Specifically, we
construct a statistical test on a set of row and column cluster memberships of
a latent block model, which is given by a squared residue minimization
algorithm. The proposed test, by its nature, includes and thus can also be used
as the test on the set of row and column cluster numbers. We also propose an
approximated version of the test based on simulated annealing to avoid
combinatorial explosion in searching the optimal block structure. The results
show that the proposed exact and approximated tests work effectively, compared
to the naive test that did not take the selective bias into account.
