  We study the group test for DNA library screening based on probabilistic
approach. Group test is a method of detecting a few positive items from among a
large number of items, and has wide range of applications. In DNA library
screening, positive item corresponds to the clone having a specified DNA
segment, and it is necessary to identify and isolate the positive clones for
compiling the libraries. In the group test, a group of items, called pool, is
assayed in a lump in order to save the cost of testing, and positive items are
detected based on the observation from each pool. It is known that the design
of grouping, that is, pooling design is important to %reduce the estimation
bias and achieve accurate detection. In the probabilistic approach, positive
clones are picked up based on the posterior probability. Naive methods of
computing the posterior, however, involves exponentially many sums, and thus we
need a device. Loopy belief propagation (loopy BP) algorithm is one of popular
methods to obtain approximate posterior probability efficiently. There are some
works investigating the relation between the accuracy of the loopy BP and the
pooling design. Based on these works, we develop pooling design with small
estimation bias of posterior probability, and we show that the balanced
incomplete block design (BIBD) has nice property for our purpose. Some
numerical experiments show that the bias correction under the BIBD is useful to
improve the estimation accuracy.
